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MATH 151 (17-19), Dr. Z. ,  {\bf Practice For Midterm II}, 
First Posted: Nov. 14, 2004.
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This version: Nov. 20, 7:30pm. Some typos have been corrected.
The posted answers refer to {\it this version}. If you have an earlier
version, please download the latest version.
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For each kind of question, there is a {\it warm-up version}, that
you should do first.  The warm-up is too easy to be a typical
exam question. This is followed by {\it moderate versions} 
and {\it hard versions}, both of which (kinds!,
not the very same question) are possible exam questions.

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{\bf 1}. (warm-up)  Find the second derivative of 
(a) $f(x)=x^4+3x^2+4 \quad$, (b) $f(x)=1/x^4 \quad$,
(c) $f(x)=4 \ln x \quad $.
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{\bf 1}. (moderate)
Find the second derivative of 
(a) $f(t)=t \cos t \quad$,
(b) $f(t)=e^t-t^2 \sin t \quad $,
(c) $f(s)=s^2e^s \quad$,
(d) $(x+1)/(x+3)+ 2e^x$.
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{\bf 1}. (hard) Find the second derivative of 
(a) $f(x)=2\sin^{-1} (x^3) +x^3 \quad $;
(b) $\ln( \tan^{-1} x ) \quad $ .

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{\bf 2}. (warm-up) Sketch the function
$$
f(x)= x^2-4x+1\quad,
$$
indicating all asymptotes (if applicable), local extrema, inflection points,
(if applicable),
intervals of increase and decrease, and concave up/down.
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{\bf 2}. (medium) Sketch the function
$$
f(x)= 2x^3-9x^2+12x+1 \quad,
$$
indicating all asymptotes (if applicable), local extrema, inflection points,
intervals of increase and decrease, and concave up/down.
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{\bf 2}. (hard) 
Sketch the function
$$
f(x)= 3x^4-4x^3+5 \quad,
$$
indicating all asymptotes (if applicable), local extrema, inflection points,
intervals of increase and decrease, and concave up/down.
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{\bf 3}. (warm-up-i) 
Sketch the curve
$$
y= {{1} \over {x-1}} \quad,
$$
indicating all asymptotes (if applicable), local extrema, inflection points,
intervals of increase and decrease, and concave up/down.
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{\bf 3}. (warm-up-ii) 
Sketch the curve
$$
y= {{1} \over {(x-2})^2} \quad,
$$
indicating all asymptotes (if applicable), local extrema, inflection points,
intervals of increase and decrease, and concave up/down.
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{\bf 3}. (warm-up-iii) 
Sketch the curve
$$
y= {{-1} \over {x+1}} \quad,
$$
indicating all asymptotes (if applicable), local extrema, inflection points,
intervals of increase and decrease, and concave up/down.
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{\bf 3}. (moderate-i) 
Sketch the curve
$$
y= {{x} \over {x-1}} \quad,
$$
indicating all asymptotes (if applicable), local extrema, inflection points,
intervals of increase and decrease, and concave up/down.
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{\bf 3}. (moderate-ii) 
Sketch the curve
$$
y= {{2x} \over {x+1}} \quad,
$$
indicating all asymptotes (if applicable), local extrema, inflection points,
intervals of increase and decrease, and concave up/down.
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{\bf 3}. (hard-i) 
Sketch the curve
$$
y= {{x} \over {x^2-4}} \quad,
$$
indicating all asymptotes (if applicable), local extrema, inflection points,
intervals of increase and decrease, and concave up/down.
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{\bf 3}. (hard-ii) 
Sketch the curve
$$
y= xe^{-2x} \quad,
$$
indicating all asymptotes (if applicable), local extrema, inflection points,
intervals of increase and decrease, and concave up/down.
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{\bf 4}. (war-up) If two resistors with resistances $R_1$, $R_2$
are connected in series then the total resistance $R$, measured
in ohms ($\Omega$), is given by
$$
R=R_1+R_2
$$
If $R_1,R_2$ are decreasing at rates of $2 \,\, \Omega/s$,
$3 \,\, \Omega/s$, 
respectively, how fast
is $R$ changing when $R_1=2 \,\,\Omega$ and $R_2=3 \,\,\Omega$.
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{\bf 4}. (moderate-i) If three resistors with resistances $R_1$, $R_2$
$R_3$
are connected in parallel then the total resistance $R$, measured
in ohms ($\Omega$), is given by
$$
{{1} \over {R}}=
{{1} \over {R_1}}+{{1} \over {R_2}}
+{{1} \over {R_3}}
$$
If $R_1,R_2,R_3$ are decreasing at rates of $2 \,\, \Omega/s$,
$3 \,\, \Omega/s$, $5 \,\, \Omega/s$
respectively, how fast
is $R$ changing when $R_1=1 \,\,\Omega$ and $R_2=1/2 \,\,\Omega$.
$R_3=1/3 \,\,\Omega$.
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{\bf 4}. (moderate-ii) The volume of a spherical ball is expanding at a rate of
$400 \pi \, cm^2/s$. How fast is the radius changing when the radius is
$10 \, cm$?
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{\bf 4}. (moderate-iii) The top of  15-ft ladder is sliding down a vertical wall
at a rate of $1 \,ft/sec$. How fast is the bottom of the
ladder moving away from the wall
when the top is $12$ ft above
the floor.
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{\bf 4}. (hard)
A water trough is of length $30$ meters
and a cross-section has the shape of an isosceles trapezoid
that has width $10$ meters at the bottom, $20$  meters at the top,
and  height $5$ meters. If the trough is being filled with
water at a rate of $10\, m^3/min.$ , how fast is the water
level rising when the water had depth $3$ ?
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{\bf 5}. (warm-up) Use differentials to estimate $1.01^2$.
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{\bf 5}. (moderate-i) Use differentials to estimate 
$\sin( {{\pi} \over {4}}+ .01)$.
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{\bf 5}. (moderate-ii) Use differentials to estimate $82^{1/4}$.
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{\bf 5}. (moderate-iii) Use differentials to estimate $\ln(e^2+.03)$.
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{\bf 6}. (warm-up)
Use one step of Newton's method to find an
approximation for the root of the equation $x^2-3=0$, taking
$x_1=1$.
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{\bf 6}. (moderate-i)
Use two steps of Newton's method to find an
approximation for the root of the equation $x^3=3x-4$, taking
$x_1=0$.
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{\bf 6}. (moderate-ii)
Use two steps of Newton's method to find an
approximation for the root of the equation $x^5=x^4+15$, taking
$x_1=2$.
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{\bf 7}. (warm-up) Verify that the function
$$
f(x)=x^2
$$ 
satisfies the hypothesis of the Mean Value Theorem on the closed
interval $[0,1]$. Then find all numbers $c$ that satisfy the
conclusion of the Mean Value Theorem.
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{\bf 7}. (moderate) Verify that the function
$$
f(x)=x^5+1
$$ 
satisfies the hypothesis of the Mean Value Theorem on the closed
interval $[0,2]$. Then find all numbers $c$ that satisfy the
conclusion of the Mean Value Theorem.
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{\bf 7}. (hard) Verify that the function
$$
f(x)=\ln x
$$ 
satisfies the hypothesis of the Mean Value Theorem on the closed
interval $[e^2,e^3]$. Then find all numbers $c$ that satisfy the
conclusion of the Mean Value Theorem.
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{\bf 8}. (warm-up)
Find the absolute maximum and minimum values of
$f(x)=x^2+1$ on the interval $[-1,1]$.
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{\bf 8}. (moderate)
Find the absolute maximum and minimum values of
$f(x)=4x^5-5x^4+1$ on the interval $[0,2]$.
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{\bf 9}. (warm-up-i) Find two numbers whose sum is $20$ and
whose product is as big as possible.
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{\bf 9}. (warm-up-ii) You have $100$ feet of fence and
you have to enclose a rectangular plot bordering
a straight river (no fence along the river).
What is the largest possible area that you can achieve, and
what are the dimensions?
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{\bf 9}. (medium-i) A closed box, with square base, is made to have a volume 
of $1000 \, cm^3$. The material for the bottom and top is
twice as expensive as the material for the sides.
What are the dimensions that will minimize the total cost of the material?
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{\bf 9}. (medium-ii) Find the point on the curve $y=\sqrt{x}$ that
is closest to the point $({{3} \over {2}},0)$.
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{\bf 9}. (hard) What are the radius and height of a cylinder
whose volume is $300\pi$ cubic inches, that
minimizes the surface area.
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{\bf 9}. (hard) An open box, with square base, is made to have a volume 
of $1000 \, cm^3$. 
The material for the bottom  costs $\$ 1$ per square meter, 
the material for the front and back costs $\$ 2$ per square meter
the material for the left side costs $\$ 3$ per square meter
the material for the right side costs $\$ 4$ per square meter
What are the dimensions that will minimize the total cost of the material?
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{\bf 10}. (warm-up) Find the most general antiderivative
$$
f(x)= 2x+3
$$ 
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{\bf 10}. (moderate-i) Find the most general antiderivative
$$
f(x)= {{x^3+1} \over {x^2}}
$$ 
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{\bf 10}. (moderate-ii) Find the most general antiderivative
$$
f(x)=3\sin x + 4 \cos x + e^x
$$ 
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{\bf 10}. (hard-i) Find the most general antiderivative
$$
f(x)=3x+5(1-x^2)^{-1/2}
$$ 
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{\bf 10}. (hard-ii) Find the most general antiderivative
$$
f(x)=3x+ {{5} \over {1+x^2}}
$$ 
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{\bf 11}. (warm-up) Find $f$ if $f'(x)=2x$, $ f(1)=1$.
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{\bf 11}. (moderate-i) Find $f$ if $f'(x)=4x-6/x^4, x>0, f(1)=6$.
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{\bf 11}. (moderate-ii) Find $f$ if $f'(x)=\sin x, x>0, f(\pi/2)=3$.
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{\bf 11}. (hard-i) Find $f$ if $f''(x)=x^2+1$, $f(1)=1,f'(1)=2$.
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{\bf 11}. (hard-ii) Find $f$ if $f'''(x)=x-1$, $f(1)=1,f'(1)=2,f''(1)=2$.





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