| Appropriate Rogawski
section(s) Problem #
Problem description with a link to PDF of the problem. With Javascript on, a problem synopsis appears when the mouse covers this entry. When off, the synopsis appears with the section information. |
Plain TEX file | Links to needed pictures, diagrams, or graphs |
|---|---|---|
| An
introductory workshop: translate words &
pictures #3U A trapezoid with prescribed perimeter and base angles is given: graph and analyze the area as a function of the base length. CLICK FOR PDF. |
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| An introductory workshop: translate words &
pictures. #1A
A cylinder inscribed in a sphere: graph and analyze the volume of the cylinder. CLICK FOR PDF. |
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| An introductory workshop: translate words &
pictures. #nan5
A rectangular picture and its frame are analyzed. CLICK FOR PDF. |
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§1.1, Real
Numbers, Functions, and Graphs #3C A function is given graphically. Several algebraic transformations are made to it. Students are asked to graph the results and give domains and ranges. CLICK FOR PDF. |
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| §1.1, Real
Numbers, Functions, and Graphs #2Y
A graph of |x–|x–3|| is requested, and then students are asked for a piecewise description of the function. CLICK FOR PDF. |
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| §1.1, Real Numbers, Functions, and Graphs
#nan1
What's the minimum distance between a pair of 5 points in the unit square? CLICK FOR PDF. |
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| §1.1, Real Numbers, Functions, and Graphs
#nan2
Find the radius of a circle tangent to another circle and two lines. CLICK FOR PDF. |
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| §1.1, Real Numbers, Functions, and Graphs
#nan3
A circle is characterized using distances to two points. CLICK FOR PDF. |
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| §1.1, Real Numbers, Functions, and Graphs
#nan4
The vertices of a triangle constructed from a line segment and a perpendicular bisector are requested. CLICK FOR PDF. |
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| §1.1, Real Numbers, Functions, and Graphs
#nan6
Sketches of two functions defined using absolute value are requested. Discussion of the values for x large (+ and –) are requested. CLICK FOR PDF. |
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| §1.3, The Basic Classes of Functions
#2X
A formula (square root of a linear fractional function) is given. A graph and the domain and range of the function are requested. CLICK FOR PDF. |
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| §1.3, The Basic Classes of Functions
#nan7
A parabola and a line interswect in a single point. Are there other such lines through the given point? CLICK FOR PDF. |
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| §1.4, Trigonometric Functions
#2J
The height of a building is deduced from a distance measurement and two angle measurements. CLICK FOR PDF. |
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| §1.4, Trigonometric Functions
#sg11
Some exploration of sin(1/x) and its graph. CLICK FOR PDF. |
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| §1.5, Inverse Functions #1L
Two graphs are requested. Trig/inverse trig identities are needed to show that the functions are indeed identical. CLICK FOR PDF. |
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| §1.5, Inverse Functions #nan10
Find the functions inverse to four functions given by algebraic formulas. CLICK FOR PDF. |
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| §1.6, Exponential and Logarithmic Functions
#1T
Find the domain and range of (arctan(ln(sqrt(x)–1)))3, and find the formula for its inverse. CLICK FOR PDF. |
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| §1.6, Exponential and Logarithmic
Functions #2P
Some "loglog graph paper" is given, and students must discover that a power relationship (y=AxB) appears as a straight line of slope B on this graph paper. Further copies of the graph paper may be printed to help with writeups if this problem is assigned. CLICK FOR PDF. |
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| §1.6, Exponential and Logarithmic
Functions #2L
Approximate values of ln(2) and ln(10) are given. How many decimal digits should be expected in 240? CLICK FOR PDF. |
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| §1.6, Exponential and Logarithmic Functions and
§1.7, Technology: Calculators and
Computers #2Q
Students are asked to graph x.1, e.01x, and ln(x) in varying intervals of positive numbers. They must find intervals where each of the graphs is "on top" and discuss what happens for x large and for x near 0+. Some answers to the first three parts are given as comments in the TEX file. CLICK FOR PDF. |
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| §1.6, Exponential and Logarithmic Functions
#nan9
Students are asked to solve some equations, most involving logs and exponentials. CLICK FOR PDF. |
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§1.7, Technology: Calculators and Computers #3D
Four computer-drawn graphs of x2 centered at (2,4) are displayed. Students are asked to explain the results, and to give an example of a "straight line" which could not result from this process. This problem was suggested by G. Cherlin. Some answers are given as comments in the TEX file. Possibly relevant to a later graphing section. CLICK FOR PDF. |
w3D3.eps w3D4.eps |
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| §1.7, Technology: Calculators and Computers #1S
Experimenting with the graphs of xne–x and xe–nx for n=1,2,3. CLICK FOR PDF. |
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§2.4, Limits and
Continuity #3F
Tax rates of a country are given. Describe the continuous tax function and graph it. CLICK FOR PDF. |
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§2.4, Limits and
Continuity #3V The squaring function has values changed at several points, and students are asked to compute limits of the resulting function. CLICK FOR PDF. |
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| §2.5, Limits and
Continuity #5D
Four limit situations are described which all play on falsifying the conditions of continuity. Graphs are requested of functions satisfying these situations. CLICK FOR PDF. |
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| §2.5, Limits and
Continuity #5E
A piecewise defined function with two parameters is given. Graphs are required for two pairs of values of the parameters, one given, and the other requested to make the given function continuous. CLICK FOR PDF. |
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| §2.5, Limits and
Continuity #nan16
Consider the behavior of sqrt(2–sqrt(4–x2))/x near x=0. A graph and algebraic analysis are requested. CLICK FOR PDF. |
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§2.6, Trigonometric
Limits #5G
Consider the function (cot(x)(1–cos(2x))/x graphically near 0, and then use algebra and sine's limit to find the limit as x→0. CLICK FOR PDF. |
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§2.7, Intermediate Value Theorem #3W
The roots of 2x=x2 are investigated. The graphs of two increasing, continuous functions which intersect {2|3|4} times are requested. CLICK FOR PDF. |
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| §2.7, Intermediate Value
Theorem #5C
Graph x3 and a specific bounded continuous function in an appropriate interval, appeal to IVT for a root, and approximate the root. Understand what happens in general. CLICK FOR PDF. |
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| §2.7, Intermediate Value
Theorem #5F
Use the Intermediate Value Theorem to analyze the pseudo-Fermat equation 4x+5x=6x. Generalize. This problem would also work with Newton's Method. CLICK FOR PDF. |
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| §2.7, Intermediate Value
Theorem #nan8
Tabular information is given about a function and students are asked to investigate the number of roots. Two specific examples are given to explore the situation. CLICK FOR PDF. |
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| §2.7, Intermediate Value
Theorem #nan12
Students are asked why a specific cubic polynomial has a root in [0,1]. This polynomial is perturbed and students are asked to reconsider the number of roots. CLICK FOR PDF. |
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| §3.1, Definition of the Derivative
#1R
A ball is thrown in the air. Compute the velocity at various times, and decide when the ball is highest. CLICK FOR PDF. |
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| §3.1, Definition of the Derivative
#1Q
Students must use a graphing calculator here. One function is 3x at (0,1). The other function is 6x arctan((ln x)/(x3+2)) at (1,0). Graphs of the functions and two secant lines are requested, as well as tables of slopes of secant lines. Then equations for the tangent lines are requested. CLICK FOR PDF. |
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§3.2, The Derivative as a Function #3E
A rational function with two parameters is given. Find parameter values so that a given line is tangent at a given point. Graph the resulting curve with the line. CLICK FOR PDF. |
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§3.2, The Derivative as a
Function #3O
Find lines tangent to a parabola passing through a given point. Then graph the lines and the parabola. CLICK FOR PDF. |
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§3.2, The Derivative as a Function #3G
Find C so that Cx2 and 1/x2 are orthogonal. Graph the resulting curves. CLICK FOR PDF. |
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| §3.2, The Derivative as a
Function #1C
A graph of a (bizarre!) function is given, and students are asked about continuity, differentiability, and the graph of the derivative. See also problems 49 and 74 in §3.2. CLICK FOR PDF. |
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| §3.2, The Derivative as a
Function
#2Z
Find c so that the graph of |x|+c "supports" the graph of x2. CLICK FOR PDF. |
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| §3.2, The Derivative as a
Function
#nan11
Find the center and radius of a circle tangent to y=x2 at both (1,1) and (–1,1). CLICK FOR PDF. |
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| §3.2, The Derivative as a
Function
#nan13
Lines tangent to a cubic at two points are given. The specific cubic must be found, and graphs of the cubic and lines are requested. CLICK FOR PDF. |
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| §3.2, The Derivative as a
Function
#nan14
The parabola y=x2 is flipped and pushed up: what "push" is needed to have orthogonal intersections? Then the parabola is translated left/right: what translates give orthogonal intersections? CLICK FOR PDF. |
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§3.3, Product and Quotient Rules #1B
Tabular information is given about two functions and their derivatives. Values of algebraic combinations of the functions and derivatives of these combinations are requested. See also problems 49–52 of §3.3. CLICK FOR PDF. |
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| §3.3, Product and Quotient Rules #3Z
A formula (quotient of quadratic and exp) is given. Then a graph is requested, and a tangent line and estimates of {largest|smallest} values. The derivative verifies the estimates. CLICK FOR PDF. |
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§3.4, Rates of Change #3J
Students are asked to model a simple physical "scenario" described in words (position, velocity) with a graph. CLICK FOR PDF. |
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§3.4, Rates of Change #3X
Four containers with identical volume but different profiles are given. They are filled at a constant rate. Students graph and analyze (continuous? differentiable?) the liquid's height in each container. Volume change is constant here but see also problems 52–54 of §4.4. CLICK FOR PDF. |
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| §3.7, The Chain Rule
#2H
Tabular information is given about a function and its first two derivatives. Find values of the function and the first two derivatives for f(x2), (f(x))2, and f(f(x)). Also there's a linear approximation part which can be omitted if the material has not yet been covered. CLICK FOR PDF. |
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§3.7, The Chain Rule #3S
Velocity is given by a formula in terms of distance. Acceleration and graphs of velocity as functions of both time and distance are requested. CLICK FOR PDF. |
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§3.8, Implicit
Differentiation #3K
Find the "bounding box" of a tilted ellipse. CLICK FOR PDF. |
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| §3.8, Implicit
Differentiation #4E
Find the largest circle touching (0,0) and above y=x2. CLICK FOR PDF. |
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| §3.8, Implicit
Differentiation #4X
A point moves on a parabola. The x and y velocities are studied. What happens if the horizontal velocity is constant? What happens if the vertical velocity is constant? CLICK FOR PDF. |
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| §3.8, Implicit
Differentiation #nan15
Two families of curves are defined. Students are asked to sketch some members of each family and then are requested to verify that the members of the families intersect orthogonally. CLICK FOR PDF. |
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| §3.9, Derivatives of Inverse
Functions #1J
Tangent lines of combinations of arcsin and arctan are made parallel by numerical solution of an equation. CLICK FOR PDF. |
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| §3.9, Derivatives of Inverse
Functions #1K
Verify that a given integer linear fractional transformation is invertible. Compute values of the functions (original and inverse) and derivatives at corresponding points. CLICK FOR PDF. |
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§3.9, Derivatives of Inverse
Functions #1I
The value and first and second derivatives of the function inverse to x5+x3+x at 3 are requested. CLICK FOR PDF. |
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| §3.11, Related Rates
#1N
The radii of an annular region are changing. Is the area of the annulus increasing or decreasing? CLICK FOR PDF. |
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| §3.11, Related Rates #3T A ladder slides down a wall. The end of the ladder goes faster than the speed of sound, and then, faster than light: a simple mathematical model extrapolated to silliness. See also problems 19–23 of §3.11. CLICK FOR PDF. |
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| §3.11, Related Rates
#2K
A car is driving away from a flagpole at known speed, and is pulling up a flag. How fast is the flag going up? CLICK FOR PDF. |
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§4.1, Linear Approximation and Applications #3P
Linear approximation and powers of ten create an apparent coincidence which students must explain. CLICK FOR PDF. |
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§4.1, Linear Approximation and Applications #3L
Total resistance of a parallel circuit with three resistances, and the approximate change in the total resistance as several resistances are separately changed. CLICK FOR PDF. |
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| §4.1, Linear Approximation and Applications #1U
A curve is implicitly defined by a polynomial equation. Linear approximation is used to locate points on the curve near a given point on the curve. CLICK FOR PDF. |
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| §4.1, Linear Approximation and Applications #sg48
A classical formula for body surface area due to Dubois & Dubois is analyzed. What happens if body weight changes? What happens if body height changes? Understand how linear approximation can give some information. CLICK FOR PDF. |
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| §4.2, Extreme Values
#1V
A parameterized collection of cubics is given. They are investigated as the max & min merge into an inflection point and then no horizontal tangent. CLICK FOR PDF. |
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§4.2,
Extreme Values #3A
This problem is a more subtle version of the model workshop problem included in the Rutgers version of the textbook. A piecewise-defined function is analyzed and minimized. CLICK FOR PDF. |
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| §4.3, The Mean Value Theorem and Monotonicity
and §4.1, Linear Approximation and Applications
#1O
A graph of f´ is given. Unrelated questions about the maximum of f on an interval and the linearization of f at a point are asked. CLICK FOR PDF. |
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§4.3, The Mean Value Theorem and Monotonicity #3Q
Formulas for the derivatives of two functions are given. Where are the critical points? Find and identify the extrema. CLICK FOR PDF. |
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| §4.3, The Mean Value Theorem and Monotonicity
#1W
Definition of a fixed point. Computation of fixed points for three functions, and graphs. Show that if f´(x)<1 for all x, then (MVT) there can be only one fixed point. CLICK FOR PDF. |
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§4.3, The Mean Value Theorem and Monotonicity #3I
The formula for the derivative of a function is given. Students are asked if f(0)<f(1). (The formula for the function itself is rather complicated.) CLICK FOR PDF. |
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§4.3, The Mean Value Theorem and Monotonicity #1G
The Chain Rule, the Intermediate Value Theorem, and the Mean Value Theorem are invoked. CLICK FOR PDF. |
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§4.3, The Mean Value Theorem and Monotonicity #3R
The graph of the derivative of a function is given, and its value at a point. Careful use of the Mean Value Theorem, combined with the Intermediate Value Theorem, shows the existence of a root of the function. An elaborate problem. CLICK FOR PDF. |
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§4.3, The Mean Value Theorem and Monotonicity #nan17
arctan(x+1)–arctan(x–1)+arctan(x2/2) is Π/2 always. CLICK FOR PDF. |
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§4.4, The Shape of a Graph #1F
The graph of solubility of sodium sulfate (real data!) is given. The graph is peculiar, and various questions are asked about it. CLICK FOR PDF. |
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§4.4, The Shape of a Graph #1H
Real data: the monthly rate of inflation for the U.S. in the last half of 1920 is given. What do these numbers mean about the first and second derivatives of consumer prices? What is a real-world translation of the numbers in terms of prices? CLICK FOR PDF. |
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§4.4, The Shape of a Graph #4W
Students are asked to find the lowest degree that a polynomial could have with the given wiggly graph. CLICK FOR PDF. |
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| §4.4, The Shape of a Graph
#1M
A point on an implicitly defined curve is given. What is the concavity of the curve at that point? (Second derivative via implicit differentiation; a good example where an explicit formula for y in terms of x exists and is very unwieldy.) CLICK FOR PDF. |
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§4.4, The Shape of a Graph #3N
A first derivative graph is given, and students must find where the function is {in|de}creasing, has local extrema, is concave {up|down}, and has points of inflection. The graph of the original function is requested. CLICK FOR PDF. |
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| §4.4, The Shape of a
Graph #2R
A parameterized family of functions is given: (x2+c)ex. Graphs drawn with the help of a machine motivate investigation of the number of inflection points of these graphs. CLICK FOR PDF. |
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| §4.4, The Shape of a
Graph #nan18
Students must identify and explain their answer to this question: which of three given graphs has third derivative always positive? CLICK FOR PDF. |
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| §4.5, Graph Sketching and
Asymptotes #1E
Two functions are given by "simple" formulas, and graphs are requested. These are algebraic models for functions which resemble step functions. CLICK FOR PDF. |
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| §4.5, Graph Sketching and
Asymptotes #2V
A graph of e–1/x is requested. Students must find a tangent line, asymptotes, and the inflection point. CLICK FOR PDF. |
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| §4.6, Applied Optimization
#2M
Find the largest area of a trapezoid with three sidelengths given. Many problems in section 4.6 are similar and could make good workshop problems. CLICK FOR PDF. |
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| §4.6, Applied Optimization
#2I
Find the shortest line segment tangent to a parabola and connecting to the x-axis. An algebraically intricate problem. CLICK FOR PDF. |
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| §;4.7, L'Hôpital's
Rule #2W
Three examples of ∞–∞ indeterminate forms. CLICK FOR PDF. |
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§4.7, L'Hôpital's Rule #3H
Three sorting algorithms are given with their average running times. Which is better for various values of inputs, and which is best asymptotically as list size goes to ∞? CLICK FOR PDF. |
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| §4.8, Newton's Method
#1P
Looks at the behavior of Newton's method for two initial guesses. A calculator is needed for both computational and graphical work. CLICK FOR PDF. |
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§4.8, Newton's
Method #3M
Asks for a Newton's method approximation to an equation, then discussion about the positive roots of 1/(1+x2)=tan(x). Also problem 28 in this section would probably be a good workshop problem. CLICK FOR PDF. |
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§4.9, Antiderivatives #1D
A velocity function is defined by piecewise data. It must be graphed. Then the resulting position function must be computed and graphed. CLICK FOR PDF. |
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| §4.9, Antiderivatives
#2S
Two scenarios for a car stopping: in the first, deceleration and speed are given, and in the second, deceleration and stopping distance are given. CLICK FOR PDF. |
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| §5.1, Approximating and Computing Area
#1Y
The sum of the sixth powers is given, tested, and used to compute an area. CLICK FOR PDF. |
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| §5.1, Approximating and Computing Area
#7G
Find a finite sum which approximates the area enclosed by the x-axis, y=sqrt(3x+6x4), and x=1 with an error less than 10–10. CLICK FOR PDF. |
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| §5.1, Approximating and Computing Area
#2F
The weight of a metal bar is approximated using samples of its varying density. CLICK FOR PDF. |
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| §5.1, Approximating and Computing Area
#2G
Samples of varying water flow taken at various times are given, and an approximation of the total water flow is requested. Two data sets are shown. CLICK FOR PDF. |
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| §5.2, The Definite Integral
#1Z
Two functions of the decimal representation of a number are defined, and their definite integrals are requested. CLICK FOR PDF. |
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| §5.2, The Definite Integral
#2N
Graphs and integrals of 2–|x|, |2–|x||, and |2–x| on [–1,3]. CLICK FOR PDF. |
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| §5.3, The Fundamental Theorem of Calculus, Part I
#2O
Graphs and integrals of x3–x, |x3–x|, and x3–|x| on [–1,2]. CLICK FOR PDF. |
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| §5.3, The Fundamental Theorem of Calculus, Part I
#2D
Solve for constants to get an antiderivative of xex and therefore evaluate a definite integral. CLICK FOR PDF. |
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| §5.3, The Fundamental Theorem of Calculus, Part I
#2E
Differentiate a sum of lns to verify the value of a definite integral. CLICK FOR PDF. |
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| §5.4, The Fundamental Theorem of Calculus, Part II
#2A
A profile function is given geometrically, and a function is defined using it as integrand with variable upper bound. Some values and the maximum value are requested. CLICK FOR PDF. |
§5.4, The Fundamental Theorem of Calculus, Part II
#2B
A profile function with a discontinuity is given graphically. Requested is information, including a graph, continuity, and differentiability, about a definite integral with a variable upper bound. CLICK FOR PDF. |
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| §5.4, The Fundamental Theorem of Calculus, Part II
#2T
A profile curve is given with some additional information. A value of the definite integral with variable upper bound is requested, as is the equation of a tangent line to the graph of the integral. CLICK FOR PDF. |
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| §5.4, The Fundamental Theorem of Calculus, Part II
#2U
F(x) is the integral from 0 to x of e(t2). Two limits are requested (L'Hôpital's Rule is used). Numerical answers are given as comments in the TEX file. CLICK FOR PDF. |
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| §5.5, Net or Total Change as the Integral of a
Rate #2C
A holding tank is filled with a periodic function, and emptied by two pipes, one open only half the time. The tank has an initial content. What happens after a day? Or later? CLICK FOR PDF. |
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| §5.6, Substitution Method #5S
The values of several definite integrals of a function are given, and students are asked to deduce values of other definite integrals using substitutions. CLICK FOR PDF. |
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| §5.6, Substitution Method #5R
Students compute antiderivatives of sqrt(1+sqrt(x)) and sqrt(1+sqrt(1+sqrt(x))). The latter can't be computed by current hand-held devices. They are asked to discuss the antidifferentiation of sqrt(1+sqrt(1+sqrt(1+sqrt(x)))) which currently (December 2007) can't be done with the default settings of Maple and Mathematica. CLICK FOR PDF. |
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| §5.8, Exponential Growth and Decay
#4A
A certain radioactive substance when affected by (hypothetical) gamma ray flux has faster decay. Determine when to turn on the flux to obtain a specific amount of the substance after a specific time. CLICK FOR PDF. |
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| §5.8, Exponential Growth and Decay
#5I
"Lab data" for growth of bacteria has some missing entries. The missing information must be reconstructed. Some of the text's problems in this section would be good workshop problems. CLICK FOR PDF. |
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| §6.1, Area Between Two Curves
#1X
Area between y=ex, y=12, and x=0 is computed and compared to the definite integral of ln x between 1 and 12. CLICK FOR PDF. |
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| §6.1, Area Between Two Curves
#5T
The definite integral of a difference of a cubic curve and a sine curve is computed and interpreted as a comparison of the area of two regions. CLICK FOR PDF. |
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| §6.1, Area Between Two Curves
#4B
The area below y=1/x3 with x between 1 and 2 is divided in half with a vertical line and then with a horizontal line. CLICK FOR PDF. |
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| §6.1, Area Between Two Curves
#4T
Compare these areas in the first quadrant: between x+y=1 and x2+y2=1, and between x+y=1 and sqrt(x)+sqrt(y)=1. CLICK FOR PDF. |
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| §6.2, Setting Up Integrals: Volume, Density,
Average Value #4C
A parameterized profile curve which, when revolved, models a raindrop, is given. What value of the parameter results in a raindrop with volume=1? CLICK FOR PDF. |
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| §6.2, Setting Up Integrals: Volume, Density,
Average Value #6Q
A region in the plane whose boundary is part of the x-axis and a parabolic arc is revolved to create three solids, whose pictures and volumes are requested. CLICK FOR PDF. |
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| §6.2, Setting Up Integrals: Volume, Density,
Average Value #7H
A region in the plane whose boundary is a line segment and a parabolic arc is the base for several solids with known cross-sections. The area of the region and the volumes of the solids are requested. CLICK FOR PDF. |
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| §6.2, Setting Up Integrals: Volume, Density,
Average Value #4D
The time average and distance average of the velocity of a freely falling body are computed. They are different. CLICK FOR PDF. |
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| §6.2, Setting Up Integrals: Volume, Density,
Average Value #6Y
The average temperature according to the weather bureau is compared with the average of the temperature function (for a sine function and for a linear function). CLICK FOR PDF. |
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| §6.2, Setting Up Integrals: Volume, Density,
Average Value #7U
The contrast between the maximum value and the average value of nx(n2) on [0,1] is investigated as n→∞. Then explain the result. CLICK FOR PDF. |
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| §6.2, Setting Up Integrals: Volume, Density,
Average Value #sg2
A region with a parameter in the first quadrant is described. Its area is requested, as is a volume of revolution. A specified value of the parameter is requested which makes the volume equal to 1. CLICK FOR PDF. |
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§6.5, Work and Energy #3Y
The work to empty three different containers is compared. CLICK FOR PDF. |
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| §6.5, Work and
Energy #4V
The work needed to move an electron is analyzed. A parameter goes to ∞ in the last part of the problem (like escape velocity or an improper integral). CLICK FOR PDF. |
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| §6.5, Work and
Energy #sg1
A textbook problem on filling a tank is expanded and analyzed. CLICK FOR PDF. |
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| §6.5, Work and
Energy #sg4
The work done emptying a cone filled with fluid having varying density is requested. CLICK FOR PDF. |
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| §7.1, Numerical
Integration #6N
Find an approximation of the definite integral of cos(x2) from 0 to 1 using the Midpoint Rule, with error <10–6. CLICK FOR PDF. |
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| §7.1, Numerical
Integration #6L
The graph of a function and its first four derivatives are given. An estimate of the definite integral from the graph is requested, following by use of the graphical information to get error bounds for the Trapezoid Rule and for Simpson's Rule. Note that the default location for four of the graphs is on the next page. CLICK FOR PDF. |
w6L1.eps w6L2.eps w6L3.eps w6L4.eps |
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§7.2, Integration by Parts #5U
A region is bounded by y=ln(x), y=0, x=1, and x=e. Find its area, and the volume of the solids obtained by revolution about the x- and y-axes. CLICK FOR PDF. |
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| §7.2, Integration by
Parts #6I
Two definite integrals which can be computed using integration by parts. CLICK FOR PDF. |
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§7.2, Integration by Parts #3B
Integration by parts using function information given by a table. CLICK FOR PDF. |
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§7.2, Integration by Parts #4R
The function e–Ax is considered on [1,2]. As A→∞, the integral of f(x), xf(x), and (1/1+5x48)f(x) all→0. The second result needs integration by parts, and the third is done by estimation. CLICK FOR PDF. |
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| §7.2, Integration by
Parts #4S
The function exsin(Nx) is considered on the interval [0,1]. Graphs for N=5, 10, and 100 are requested, as is the definite integral of the function. What happens as N→∞? CLICK FOR PDF. |
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| §7.2, Integration by
Parts #5O
Integrate cos(mx)cos(nx) from 0 to 2Pi. Use this to find a sum of cosines when given the values of the integrals of the function multiplied by some cosines (an elementary aspect of Fourier series). CLICK FOR PDF. |
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| §7.2, Integration by
Parts #5Q
Find the line through the origin which best approximates (least mean square error) sin(x) on [0,1]. Sketch the line and sin(x) and the tangent line at x=0 to sin(x) on [0,1]. CLICK FOR PDF. |
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| §7.3, Trigonometric Integrals #5W
Find some antiderivatives of xi(cos(xj))k where i, j, and k are 1 or 2. CLICK FOR PDF. |
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| §7.3, Trigonometric Integrals #7V
Compute the average value of (sin(Ax))3 on [0,2] and investigate what happens as A→∞. Then explain the result. CLICK FOR PDF. |
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| §7.4, Trigonometric Substitution
#4L
Find the area inside two ellipses. CLICK FOR PDF. |
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| §7.4, Trigonometric Substitution
#5V
Find the volume inside part of a cylinder three ways (geometry, calculator computation, and an integral via trig substitution). CLICK FOR PDF. |
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| §7.4, Trigonometric Substitution
#6X
Two similar-looking antiderivatives are requested. One can be done with a simple substitution and the other needs a more elaborate trig subsitution. CLICK FOR PDF. |
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| §7.4, Trigonometric Substitution
#7S
Three standard antiderivatives which are done using trig substitutions. CLICK FOR PDF. |
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| §7.6, The Method of Partial Fractions
#4M
The area of a region under (5+x)/(x2+4x+3) is computed exactly and by calculator. The volume obtained when the region is revolved about the x-axis is computed exactly and by calculator. CLICK FOR PDF. |
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| §7.6, The Method of Partial Fractions
#5P
The integrals from 1 to 2 of 1/x2 and 1/(x(x–m)) and 1/(x2+m) for n and m small and positive are computed. What happens as m→0+ and n→0+? Illustrate with a sketch. CLICK FOR PDF. |
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| §7.6, The Method of Partial Fractions
#6H
Rationalizing substitutions are used to compute three antiderivatives including the antiderivative of secant. CLICK FOR PDF. |
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| §7.6, The Method of Partial Fractions
#6V
The value of a definite integral done with partial fractions is given and students must verify this value. CLICK FOR PDF. |
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| §7.7, Improper Integrals
#5X
Consider the region bounded by y=0, y=x–a, and x=1. For which a's does this region have finite area? Revolve the region about the x-axis (respectively, the y-axis). For which a's are each of these volumes finite? CLICK FOR PDF. |
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| §7.7, Improper Integrals
#4I
Sketch and find the area in the first quadrant bounded by y=(1+e–x)2 and y=(1+e–2x)2. CLICK FOR PDF. |
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| §7.7, Improper Integrals
#4J
Sketch and find the area of the three-sided region in the first quadrant bounded by the y-axis, y=sec(x), and y=tan(x). CLICK FOR PDF. |
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| §7.7, Improper Integrals
#5H
Explain why two rational integrals, one improper, are equal. Either compute both integrals or show they are equal using a change of variables (substitution). CLICK FOR PDF. |
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| §7.7, Improper Integrals
#5M
Integrals from 0 to ∞ of xj/(1+x4), j=1..4, are given. Which converge? Compute one convergent integral. CLICK FOR PDF. |
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| §7.7, Improper Integrals
#5N
Integrals of xne–x2 from 0 to ∞ are investigated. Students must find a reduction formula using integration by parts. CLICK FOR PDF. |
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| §7.7, Improper Integrals
#6M
The integral of x–ln(x) from 0 to ∞ is investigated. The limits of the integrand as x→0+ and x→∞ are requested. The integral is evaluated depending on the value of the integral of e–x2 from –∞ to ∞: an integral which "can't" be computed using the default configuration of Maple. CLICK FOR PDF. |
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| §7.7, Improper Integrals
#6O
A method to numerically approximate the integral of e–x2 from 0 to ∞ is discussed. Then students are asked to compute the integral from 0 to ∞ of x2e–x2. CLICK FOR PDF. |
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| §7.7, Improper Integrals
#7K
The integrals from 1 to ∞ of ln(x)/xj (here j=1, 1/2, and 3) are analyzed. CLICK FOR PDF. |
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| §7.7, Improper Integrals
#sg50
Several examples of computing improper integrals using the technique insert a parameter, differentiate, and then fix the parameter are given. CLICK FOR PDF. |
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| §7.7, Improper Integrals
#sg18
Here the integral ∫0∞(sin(x)/x)dx is evaluated by a complicated use of differentiation under the integral. This problem may be too difficult to assign, but the technique here and in sg50, the previous problem, is discussed nicely by K. Conrad in this paper: www.math.uconn.edu/~kconrad/blurbs/analysis/diffunderint.pdf (the URL is also available as a comment in the TEX file). CLICK FOR PDF. |
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| §8.1, Arc Length and Surface
Area #5Y
The ratio area/perimeter2 is computed for the region bounded by y=x2 and y=4. This ratio is compared to similar numbers for some other regions. CLICK FOR PDF. |
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| §8.1, Arc Length and Surface
Area #5Z
Is the area of the surface obtained by revolving y=e–x, x>0, about the x-axis finite? Only yes/no is asked, but an exact answer is given as a TeX comment. CLICK FOR PDF. |
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| §8.4, Taylor
Polynomials #7B
A quadratic approximation for x5 near x=2 is found. An error bound in the interval [2,2.1] is requested, accompanied by a graph. CLICK FOR PDF. |
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| §8.4, Taylor
Polynomials #7C
The fourth degree Taylor polynomial for 1/sqrt(1–x) is requested. An error bound in the interval [–.5,.5] is requested, accompanied by several graphs. CLICK FOR PDF. |
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| §8.4, Taylor
Polynomials #sg5
Students are requested to exhibit various close rational approximations to sin(.4), sin(1.4), and e2.8. Only the error estimates must be computed! CLICK FOR PDF. |
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| §8.4, Taylor
Polynomials #sg6
A polynomial with rational coefficients approximating sqrt(x) in [3,5] with error<.01 is requested. CLICK FOR PDF. |
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| §8.4, Taylor
Polynomials #sg7
Suppose f(x)=ex2+sin(x). Information about values of f(j)(0) is given, along with some graphs. Then Taylor polynomials of f(x) centered at 0 are requested along with an error estimate. CLICK FOR PDF. |
sg7b.eps sg7c.eps |
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| §9.1, Solving Differential Equations
#4U
The equation A´=sqrt(A)(9–A) is analyzed as a model of tissue culture growth. CLICK FOR PDF. |
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| §9.1, Solving Differential Equations
#6Z
A separable ODE is given, and the solution of a specific initial value problem is requested. The student must graph the solution curve and state its domain. CLICK FOR PDF. |
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| §9.1, Solving Differential Equations
#6S
Decide which two of four ODE's are separable and then solve an initial value problem for each of them. Also explain why one of the other ODE's is not separable. CLICK FOR PDF. |
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| §9.1, Solving Differential Equations
#6T
A function explodes at b if lim→b–f(x) is +∞ or –∞. After some examples, solutions of y´=y2 have their "explosions" analyzed (that is, how domains of solution curves depend on initial conditions is investigated). CLICK FOR PDF. |
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| §9.2, Models involving y´=k(y–b)
#6T
Find the initial temperature of an object subject to Newton's Law of Cooling if two data points are known. Some of the text's problems in this section would be good workshop problems. CLICK FOR PDF. |
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| §9.3, Graphical and Numerical Methods
#6K
A separable ODE is analyzed. Which values of the integration constant correspond to solutions? What is the domain of the solution curves? How does the direction field interact with solution curves? CLICK FOR PDF. |
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| §9.3, Graphical and Numerical Methods
#6W
The direction field for a first order linear ODE is given. The equation can't be solved explicitly using standard functions. Students are asked to sketch a solution curve and to analyze any possible critical points of the solution curve. CLICK FOR PDF. |
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| §10.1, Sequences
#4Y
Suppose A and B are positive. Analyze the sequences (An+Bn)1/n and (An+B)1/n. CLICK FOR PDF. |
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| §10.1, Sequences
#4Z
Repeated square roots of 5 followed by subtracting 1 and multiplying by powers of 2 approximate ln(5). Explain this, and show how ln's can be "computed" generally with this method. CLICK FOR PDF. |
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| §10.1, Sequences
#5J
Several variations on the sequential definition of e are given. Numerical values for the first terms are requested, and then the limiting values are found using L'Hôpital's Rule. CLICK FOR PDF. |
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| §10.1, Sequences
#5K
A recursively defined sequence is analyzed graphically, and students are asked to cite a specific result which will guarantee convergence ("Monotone bounded sequences ...") and must find the limit. CLICK FOR PDF. |
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| §10.1, Sequences
#5L
A recursively defined sequence is analyzed algebraically. It is increasing and bounded. Students are asked to cite a specific result which will guarantee convergence ("Monotone bounded sequences ...") and must find the limit. CLICK FOR PDF. |
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| §10.1, Sequences
#6P
Several "simple" sequences converging to 0 are analyzed numerically to detect how rapidly they converge to 0. CLICK FOR PDF. |
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| §10.2, Summing an Infinite Series
#5A
Students eat portions of a loaf of bread (three distinct "scenarios"). What each student eats is the sum of a geometric series. CLICK FOR PDF. |
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| §10.2, Summing an Infinite Series
#5B
A "fractal" figure is created in a unit square. The resulting object has finite area and infinite perimeter. CLICK FOR PDF. |
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| §10.2, Summing an Infinite Series
#7J
The centers of the sides of a square are joined to form a smaller square. In turn, the centers of the sides of that square are joined ... What is the total length of the figure? (Our only pictex figure.) CLICK FOR PDF. |
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| §10.2, Summing an Infinite Series
#sg52
Is every sequence the sequence of partial sums of a series? An example is requested. CLICK FOR PDF. |
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| §10.3, Convergence of Series with Positive Terms
#7A
Inequalities relating to the Integral Test are used to investigate partial sums of divergent series whose jth terms are 1/j, 1/sqrt(j), and 1/(jln(j)). CLICK FOR PDF. |
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| §10.3, Convergence of Series with Positive Terms
#7D
Inequalities relating to the Integral Test are used to investigate infinite tails of convergent series whose jth terms are 1/j5 and je–2. Finally, an approximation of the whole sum of one of the series is requested. CLICK FOR PDF. |
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| §10.3, Convergence of Series with Positive Terms
#4G
Several computed partial sums of a series are reported, and students are asked to verify that the series converges to a specific number, with an error tolerance. The Integral Test should be used.The next problem, 6R, is a companion to this problem. CLICK FOR PDF. |
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| §10.3, Convergence of Series with Positive Terms
#6R
Several computed partial sums of a series are reported, and students are asked to verify that the series converges to a specific number, with an error tolerance. A comparison to a geometric series is suggested. This is a companion to the previous problem, 4G. CLICK FOR PDF. |
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| §10.3, Convergence of Series with Positive Terms
#6J
A series from a text by Bromwich (about a century old!) is investigated. p-series and comparison are needed. CLICK FOR PDF. |
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| §10.3, Convergence of Series with Positive Terms
#sg51
Another series from a text by Bromwich (about a century old!) is investigated. It is a hypergeometric-like series. CLICK FOR PDF. |
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| §10.3, Convergence of Series with Positive Terms
#7R
Two series are given. Verification that they converge using comparison is requested as are error estimates for the difference between the fifth partial sum and the sum. CLICK FOR PDF. |
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| §10.3, Convergence of Series with Positive Terms
#sg49
∫01x–xdx=∑n=1∞1/nn. CLICK FOR PDF. |
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| §10.4, Absolute and Conditional Convergence
#4H
Two series involving ln's are given, one convergent and one divergent. Specific partial sums for each series are requested. CLICK FOR PDF. |
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| §10.4, Absolute and Conditional Convergence
#4O
Consider a series whose absolute value is a geometric series with 3 as first term and 1/3 as ratio. Students are asked if sign choices (+ or –) can make the series diverge, sum to 3.5, or sum to 2.25. CLICK FOR PDF. |
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| §10.4, Absolute and Conditional Convergence
#4N
A Fourier sine series is analyzed. The graph of a partial sum is given. Questions about convergence, periodicity, and the accuracy of the graph are asked. A comment in the TEX file gives the sum of the series (verification is possible with Euler's identity). CLICK FOR PDF. |
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| §10.4, Absolute and Conditional Convergence
#4Q
Two series of constants which converge to ln 2 are given. Students are asked which converges "faster". CLICK FOR PDF. |
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| §10.4, Absolute and Conditional Convergence
#7Q
Three series are offered as candidates for the alternating series test. For those series which qualify, partial sums within 10–8 are requested, and the value of one such partial sum is desired. CLICK FOR PDF. |
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| §10.5, The Ratio and Root Tests
#4K
Several computed partial sums of a series are reported, and students are asked to verify that the series converges to a specific number, with an error tolerance. A quantitative version of the Ratio Test can be used. CLICK FOR PDF. |
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| §10.5, The Ratio and Root Tests
#7P
Students are asked to compare the rates of growth of several functions. Ideas needed combine the ratio test and perhaps L'Hôpital's Rule. See also the discussion of "<<" in section 4.7. CLICK FOR PDF. |
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| §10.6, Power Series
#4P
Is there a power series with interval of convergence (0,1]? Is there a power series with interval of convergence (0,∞)? CLICK FOR PDF. |
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| §10.6, Power Series
#4F
A Bessel function series is investigated. After verifying convergence, students are asked to show that one value of the series is positive and another value is negative. CLICK FOR PDF. |
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| §10.6, Power Series
#7E
The geometric series formula is used to find power series for several functions. One of these series is then used to write a definite integral as the sum of an infinite series. CLICK FOR PDF. |
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| §10.6, Power Series
#6U
A power series is analyzed: its interval of convergence, and an ODE it satisfies. Solving the ODE gives a formula for the sum of the series. A possible continuation asks for an error bound (using alternating series) between the function and a specific partial sum. CLICK FOR PDF. |
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| §10.6, Power Series
#7F
Two graphs are given, one with a jump discontinuity and one with a cusp. In each case, what is the largest possible radius of convergence of a power series whose sum is the function with displayed graph? CLICK FOR PDF. |
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| §10.6, Power Series
#7T
Find the radius and intervals of convergence for three power series. The boundary behavior is different for the three. CLICK FOR PDF. |
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| §10.6, Power Series
#sg31
A problem from Knuth's The Art of Computer Programming. It relates power series "convolved" with the harmonic series and an integral: an abstract problem difficult for students. CLICK FOR PDF. |
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| §10.7, Taylor Series
#7L
The first 10 terms of the Maclaurin series of (1+x+x2)sin(x3) are requested, as well as the 10th derivative of this function at 0. Some of the text's problems in this section, especially the integrals, would be good workshop problems. CLICK FOR PDF. |
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| §10.7, Taylor Series
#7M
Use Maclaurin series to compute a limit which would take 9 applications of L'Hôpital's Rule. Some of the text's problems in this section, especially the integrals, would be good workshop problems. CLICK FOR PDF. |
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| §11.1, Parametric
Equations #6A
Equations defining a collection of parametric curves with a loop and self-intersection are given, and students must identify the unique curve in this collection where the self-intersection is orthogonal. Some of the text's problems in this section would be good workshop problems. CLICK FOR PDF. |
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| §11.1, Parametric
Equations #6B
Five different parametric descriptions (using "simple" geometric parameters) are requested for y=x2. Five is too much, and perhaps select three of the five for a workshop problem. Some of the text's problems in this section would be good workshop problems. CLICK FOR PDF. |
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| §11.1, Parametric
Equations #6F
A parametric curve must be sketched, and the tangent lines which are parallel to the coordinate axes and to y=x are requested. Some of the text's problems in this section would be good workshop problems. CLICK FOR PDF. |
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| §11.1, Parametric
Equations #7W
Find the bounding box of a ``tilted ellipse''. This is a parametric version of problem 3K for section 3.8. CLICK FOR PDF. |
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| §11.1, Parametric Equations #sg8
Find the point closest to the origin on a given parametric curve which has a portion of its graph displayed. CLICK FOR PDF. |
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| §11.1, Parametric Equations #sg10
Two particles are moving. Their paths cross twice. Do the particles ever collide? CLICK FOR PDF. |
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| §11.2, Arc Length and
Speed #6G
A parameterized collection of parametric curves is given. Students must sketch some representative curves, find some tangent lines, and find the length of a loop in one of them. CLICK FOR PDF. |
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| §11.3, Polar
Coordinates #6C
Find the description of the boundary of a "Norman window" (a rectangle with a semicircle on top) in polar coordinates. CLICK FOR PDF. |
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| §11.3, Polar
Coordinates #6E
Find the points of intersection of a lemniscate and a cardioid. Sketch the curves together. CLICK FOR PDF. |
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| §11.3, Polar
Coordinates #sg9
Use integration in polar coordinates to compute the area of a particular triangle. CLICK FOR PDF. |
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| §11.4, Area and Arc Length in Polar
Coordinates #6D
Unwind a spool of thread to create a "heart" (a cardioid) and find a value of the parameter describing the heart to fit the length, and then compute the area. CLICK FOR PDF. |
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| §11.4, Area and Arc Length in Polar
Coordinates #7O
What percent of the area of a cardioid is above the x-axis? What percent of the area of a cardioid is inside the unit circle? CLICK FOR PDF. |
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| §11.4, Area and Arc Length in Polar
Coordinates #7N
Deduce speed formula in polar coordinates given the formula in rectangular coordinates. Then deduce the connection between speed and the magnitude of acceleration for uniform circular motion. CLICK FOR PDF. |
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§12.3, Dot Product and the Angle Between
Two Vectors #ers1
Estimating ||a+b|| given information about ||a|| and ||b||; getting information about the angle between a and b given information about ||a||, ||b||, and ||a+b||. CLICK FOR PDF. |
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§12.4, The Cross
Product #sg22
Can dot and cross products be "canceled"? (Separately and together.) CLICK FOR PDF. |
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§12.4, The Cross
Product #ers2
An identity involving dot and cross products is deduced from sin2+cos2=1, and then verified with a component calculation. CLICK FOR PDF. |
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§12.5, Planes in
Three-Space #sg16
Find the equations of two planes given some strange information about them. CLICK FOR PDF. |
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§12.5, Planes in
Three-Space #ers4
Find a sphere of given radius tangent to two planes intersecting obliquely. CLICK FOR PDF. |
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§13.2, Calculus of Vector-Valued
Functions #sg23
Can (1,2,3) on a tangent line to the twisted cubic <t,t2,t3>. An optional version (see the TeX file) includes a (possibly) helpful picture of the situation. CLICK FOR PDF. |
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§13.2, Calculus of Vector-Valued
Functions #ers5
A spaceship travels on a path. Show that it can "hit" the origin if the engine is turned off at some time. (A complement to the twisted cubic problem!) CLICK FOR PDF. |
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§13.4,
Curvature #sg17
Graph the curvature of a race track which is displayed. CLICK FOR PDF. |
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§13.4,
Curvature #sg20
A curve which spirals at each "end" towards two circles of different radius is given, and a graph of the curvature is requested. CLICK FOR PDF. |
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§13.4,
Curvature #sg34
The curvature of <cos(t),sin(t),t2> is requested. Although it "is" a circle in its first two coordinates, the curvature →0 as t→∞. An explanation is requested. CLICK FOR PDF. |
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§13.4,
Curvature #sg36
A circular arc and a line segment are given in "disguised" algebraic form, and the curvature is requested, along with some geometric explanation. CLICK FOR PDF. |
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§13.4,
Curvature #sg37
No graph y=f(x) can have curvature≥1 always. A difficult problem. CLICK FOR PDF. |
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§13.4,
Curvature #sg46
A curve on the unit sphere must have curvature at least 1. A somewhat difficult problem. CLICK FOR PDF. |
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§14.1, Functions of Two of More
Variables #sg27
An absurd problem modeled after post office specifications for allowable package sizes. The answer (the domain of allowable package sizes) is a bit surprising. (One of my favorite problems.) CLICK FOR PDF. |
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§14.3, Partial
Derivatives #ers5
Use Clairaut's Theorem to get information about functions given first partial derivatives. CLICK FOR PDF. |
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§14.4, Differentiability, Linear
Approximation, and Tangent
Planes #sg38
The functions xy and xyz are analyzed for their domains and some linear approximations. CLICK FOR PDF. |
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§14.4, Differentiability, Linear
Approximation, and Tangent
Planes #sg45
Coefficients of a polynomial are symmetric functions of it roots. In this problem, students are asked to write these functions for a cubic polynomial, and then find the first derivatives and accompanying linear approximations. They are then asked to consider the inverse problem, and find linear approximations to the roots if the coefficients are perturbed. CLICK FOR PDF. |
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| §14.4, Differentiability, Linear Approximation, and Tangent Planes #ers7 A tangent plane to a surface intersects the xy-plane in a line which is parallel to the line tangent to a certain level curve. Explain. CLICK FOR PDF. | ||
| §14.4, Differentiability, Linear Approximation, and Tangent Planes #ers8 Use linear approximation to analyze the error in a product of four numbers between 0 and 50 produced by rounding. CLICK FOR PDF. | ||
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§14.5, The Gradient and Directional
Derivatives #sg21
Find a vector tangent to the curve of intersection of two surfaces. CLICK FOR PDF. |
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§14.5, The Gradient and Directional
Derivatives #ers12
The gradient of the 2x2 determinant is requested, followed by a inquiry of how best to increase the value of the determinant at specific entries. CLICK FOR PDF. |
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§14.5, The Gradient and Directional
Derivatives #ers14
A lovely pictex image of the level curves of a function is given, and information about ∇f is requested. CLICK FOR PDF. |
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| §14.6, The Chain Rule #sg12
Two computations of mixed partial second derivatives (Clairaut's Theorem) done using implicit differentiation. CLICK FOR PDF. |
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| §14.6, The Chain Rule #ers11
Clairaut's Theorem is checked for a function defined by a complicated abstract formula. CLICK FOR PDF. |
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| §14.6, The Chain Rule #sg32
Compute derivatives of F if F(x,y,z)=f(xz2+y3). CLICK FOR PDF. |
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| §14.6, The Chain Rule #sg39
Compute derivatives of R if R(x,y)=v(x+y2). CLICK FOR PDF. |
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| §14.6, The Chain Rule #sg40
Compute derivatives of g if g(t)=Q(t3,t5). CLICK FOR PDF. |
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| §14.6, The Chain Rule #sg43
This problem is too difficult and too long and suitable almost surely only for ambitious honors students. It first discusses the linearity of the one-dimensional wave equation, and then introduces the Korteweg-de Vries equation, together with some historical setting. KdV is non-linear. CLICK FOR PDF. |
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§14.6, The Chain
Rule #ers10
If t=f(u,v,w), with f differentiable, and that u=x–y, v=y–z, and w=z–x, then t solves the PDE tx+ty+tz=0. CLICK FOR PDF. |
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§14.6, The Chain
Rule #ers9
A bug crawls on a metal plate with specific temperature. What are the first and second derivatives of the temperature with respect to time at a specific time and location. CLICK FOR PDF. |
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§14.6, The Chain
Rule #ers13
Investigation of the Fundamental Solution of the two-dimensional Heat Equation. CLICK FOR PDF. |
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§14.6, The Chain
Rule #erc1
Differentiation and homogeneous functions of two variables. Similar problems occur in many textbooks. CLICK FOR PDF. |
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| §14.7, Optimization in Several Variables #sg13
Investigation of the critical points of solutions to the 2-dimensional heat equation. CLICK FOR PDF. |
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| §14.7, Optimization in Several Variables #sg24
Two parameterized lines are given, and the distance between the points specified by parameter values is investigated. Are there critical points, and, if so, what kind? What is the geometric feature of the critical point found? CLICK FOR PDF. |
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| §14.7, Optimization in Several Variables #sg25
A rectangle is dissected into four rectangles by two lines parallel to its sides. Find the configurations so that the sums of the areas of the subrectangles are largest and smallest. CLICK FOR PDF. |
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| §14.7, Optimization in Several Variables #sg26
Discovering the least squares line for a (finite) collection of data points. This is section 14.7's problem 44. CLICK FOR PDF. |
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| §14.7, Optimization in Several Variables #sg28
Is there a function of three variables which has the twisted cubic <t,t2,t3> as its set of critical points? CLICK FOR PDF. |
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| §14.7, Optimization in Several Variables #ers16
A function with a simple local and absolute maximum is perturbed and students are asked to locate one critical point of the perturbed function and explain the situation. CLICK FOR PDF. |
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| §14.7, Optimization in Several Variables #ers17
Gene Speer's effort (the creator is identified by the presence of a pictex image!) to introduce 251 students to the special flavor of linear programming problems. situation. CLICK FOR PDF. |
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§14.8, Lagrange Multipliers: Optimizing
with a Constraint
#sg19
Find the hottest and coldest points on the closed unit disc with a temperature distribution given. CLICK FOR PDF. |
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§14.8, Lagrange Multipliers: Optimizing
with a Constraint
#sg41
Find the max of 3x+5y on x2+y2=1 with an appropriate picture. Now do the same analysis with constraint xn+yn=1 (n>0). What happens as n→∞? What happens as n→0+? Draw the appropriate pictures. This problem is delicate but can be rewarding. CLICK FOR PDF. |
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§14.8, Lagrange Multipliers: Optimizing
with a Constraint
#ers15
The nearly canonical "minimize the price of a rectangular box with given cost for the sides" Lagrange multiplier problem is given a bit abstractly here. CLICK FOR PDF. |
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§15.2, Double Integrals over More General
Regions #sg14
Which doubly iterated integrals correspond to the computation of a geometric volume? CLICK FOR PDF. |
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§15.2, Double Integrals over More General
Regions #sg30
Three doubly iterated integrals which can be computed by interchanging the order of integration. CLICK FOR PDF. |
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§15.2, Double Integrals over More General
Regions #sg42
Consideration of the improper integrals over R2 of e–|x+y| and e–|x|–|y|. CLICK FOR PDF. |
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§15.2, Double Integrals over More General
Regions #ers20
Four double integrals of (y-x2)ex2+y2 are considered, and the student should order them in terms of their values, smallest to largest. CLICK FOR PDF. |
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§15.2, Double Integrals over More General
Regions #ers21
A provisional definition of convergent improper integral for nonnegative functions over bounded two-dimensional regions is given. Students then must consider whether three specific improper integrals of functions over the unit square converge. CLICK FOR PDF. |
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§15.2, Double Integrals over More General
Regions #ers19
Set up the iterated integrals needed to compute a double integral over a region between two non-concentric circles. CLICK FOR PDF. |
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§15.3, Triple
Integrals #sg15
Find the correct ordering of an iterated triple integral whose limits are given. Compute it. CLICK FOR PDF. |
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§15.3, Triple
Integrals #sg29
A "simple" iterated triple integral over a tetrahedron must be rewritten in the opposite order. CLICK FOR PDF. |
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§15.3, Triple
Integrals #sg47
A stick is broken into three parts and painted. Students are asked to find the probability (the "chance") that certain inequalities occur among the pieces. CLICK FOR PDF. |
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§15.4, Integration in Polar, Cylindrical,
and Spherical
Coordinates #ers24
The center of gravity of a plate with uniform density in the first quadrant between two circles centered at the origin is requested. When is this point actually inside the region? CLICK FOR PDF. |
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§15.4, Integration in Polar, Cylindrical,
and Spherical
Coordinates #ers25
Similar to ers24 but in three dimensions: find the center of mass of a region in the first octant between two spherical shells centered at the origin. If the radii get close to one another, what happens to this center of mass? CLICK FOR PDF. |
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§15.4, Integration in Polar, Cylindrical,
and Spherical
Coordinates #ers22
Compute a specific iterated double integral three ways: directly, by interchanging the order, and by conversion to polar coordinates. CLICK FOR PDF. |
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§15.5, Change of
Variables #sg33
Verify the value of a double integral using Change of Variables. CLICK FOR PDF. |
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§16.2, Line
Integrals #ers26
Compare the values of the line integrals over two given curves using three given integrands over two given curves. Do this both by "reasoning" (without evaluation) and by direct computation. CLICK FOR PDF. |
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§16.3, Conservative Vector Fields #sg53
The student is asked to construct a 2-dimensional vector field from geometric information, and then to compute a specific line integral involving this vector field. CLICK FOR PDF. |
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§16.4, Parameterized Surfaces and Surface
Integrals #sg44
The average height of a bat flying around a hemispherical region is computed, and that's contrasted with the average height of a slug crawling over the curved surface of the hemisphere. CLICK FOR PDF. |
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§16.5, Surface Integrals of Vector
Fields #ers28
Is a certain flux integral positive, negative, or zero? CLICK FOR PDF. |
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§17.1, Green's Theorem #sg35
Use Green's Theorem to compute a line integral. CLICK FOR PDF. |
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§17.1, Green's Theorem #sg3
Use Green's Theorem to compute a line integral (variation: only the area and first moments of the region are given). CLICK FOR PDF. |
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§17.1, Green's Theorem #ers27
Explain why Green's Theorem cannot be used to compute two line integrals but can be used to compute the difference between them. CLICK FOR PDF. |
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