MATH 151 (17-19), Dr. Z. , Answers for the Practice For Midterm II,

These posted answers refer to the Nov. 20, 7:30pm version. If you have an earlier version, please download the latest version of the Practice for Midterm II.

Disclaimer: Even I make mistakes. If you find any errors, please E-mail me: zeilberg at math dot rutgers dot edu . (Subject: MathIsFun). I will pay 1 dollar to the first person to detect any error (untlil Sunday Nov. 21, 8:00pm). Of course, if your answer diagrees, it is more likely that you made a mistake, so please check it three times before you report an error.

Nov. 21, 2004, 8:11am: Prashan found an error in 5(moderate-iii), it should be 2 + .03/e^{2}. Here is the corrected version.

1. (warm-up) (a) f(x) = 12x^{2} + 6 , (b) f(x) = \frac{20}{x^{6}} , (c) f(x) = \frac{−4}{x^{2}} .

1. (moderate) (a) f′′(t) = −2 sin t − t cos t , (b) f′′(t) = e^{t} − 4t cos t + (t^{2} − 2) sin t , (c) f′′(s) = (s^{2} + 4s + 2)e^{s} , (d) f′′(x) = \frac{−4}{(x+3)^{3}} + 2e^{x}.

1. (hard) (a)

f′′(x) = \frac{6x(2 + x^{6} + (1 − x^{6})^{3/2})}{(\sqrt{1 − x^{6}})^{3}}

(b)

f′′(x) = −\frac{1 + 2x tan^{−1} x}{((1 + x^{2}) tan^{−1} x)^{2}}

2. (warm-up) : Horiz. Asymp.: none. Vertical Asympt.: none. decreasing: −∞ < x < 2 ; increasing: 2 < x < ∞; local max: none; local min: (2, −3); concave up: −∞ < x < ∞; concave down: never; inflection point: none. (Sorry, I am too lazy to prepare a diagram on the computer or to scan).

2. (medium) Horiz. Asymp.: none. Vertical Asymp.: none. decreasing: 1 < x < 2; increasing: −∞ < x < 1 AND 2 < x < ∞; local max: (1, 6); local min: (2, 5); concave up: 3/2 < x < ∞; concave down: −∞ < x < 3/2; inflection point: (3/2, 11/2). (Sorry, I am too lazy to prepare a diagram on the computer or to scan).

2. (hard) Horiz. Asymp.: none. Vertical Asympt.: none. decreasing: −∞ < x < 1; increasing: 1 < x < ∞ ; local max: none; local min: (1, 4); concave up: −∞ < x < 0 AND 2/3 < x < ∞; concave down: 0 < x < 2/3; inflection points: (0, 5) (horizontal inflection point!) and (2/3, 119/27). (Sorry, I am too lazy to prepare a diagram on the computer or to scan).

3. (warm-up-i) Horiz. Asymp.: y = 0 (on both ends!). Vertical Asymp. x = 1. decreasing: −∞ < x < 1 AND 1 < x < ∞ ; increasing: never. local max: none; local min: none; concave up: 1 < x < ∞ ; concave down: −∞ < x < 1; inflection points: none. (Sorry, I am too lazy to prepare a diagram on the computer or to scan).

3. (warm-up-ii) Horiz. Asymp.: y = 0 (on both ends!). Vertical Asymp. x = 2. decreasing: 2 < x < ∞ ; increasing: −∞ < x < 2 ; local max: none; local min: none; concave up: −∞ < x < 2 AND 2 < x < ∞ ; concave down: never ; inflection points: none. (Sorry, I am too lazy to prepare a diagram on the computer or to scan).

3. (warm-up-iii) Horiz. Asymp.: y = 0 (on both ends!). Vertical Asymp. x = −1. decreasing: never; increasing: −∞ < x < −1 AND −1 < x < ∞; local max: none; local min: none; concave up: −∞ < x < −1 ; concave down: −1 < x < ∞ ; inflection points: none. (Sorry, I am too lazy to prepare a diagram on the computer or to scan).

3. (moderate-i) Horiz. Asymp.: y = 1 (on both ends!). Vertical Asymp. x = 1. decreasing: −∞ < x < 1 AND 1 < x < ∞ ; increasing: never. local max: none; local min: none; concave up: 1 < x < ∞ concave down: −∞ < x < 1 inflection points: none. (Sorry, I am too lazy to prepare a diagram on the computer or to scan).

3. (moderate-ii) Horiz. Asymp.: y = 2 (on both ends!). Vertical Asymp. x = −1. increading: −∞ < x < −1 AND −1 < x < ∞ ; decreasing: never. local max: none; local min: none; concave up: −∞ < x < −1 concave down: −1 < x < ∞ inflection points: none. (Sorry, I am too lazy to prepare a diagram on the computer or to scan).

3. (hard-i) Horiz. Asymp.: y = 0 (on both ends!); Vertical Asymp. x = −2 AND x = 2; increasing: never; decreasing: −∞ < x < −2 AND −2 < x < 2 AND 2 < x < ∞ ; local max: none; local min: none; concave up: −2 < x < 0 AND 2 < x < ∞ concave down: −∞ < x < −2 AND 0 < x < 2. inflection points: (0, 0). (Sorry, I am too lazy to prepare a diagram on the computer or to scan).

3. (hard-ii) Horiz. Asymp.: y = 0 (only from the right!); Vertical Asymp.: none; increasing:−∞ < x < 1/2 ; decreasing: 1/2 < x < ∞; local max: (1/2, 1/(2e)); local min: none; concave up: 1 < x < ∞ concave down: −∞ < x < 1 inflection points: (1, 1/e^{2}). (Sorry, I am too lazy to prepare a diagram on the computer or to scan).

4. (war-up) R is decreasing at a rate of 5 Ohms per second.

4. (moderate-i) R is decreasing at a rate of 59/36 Ohms per second.

4. (moderate-ii) The radius is increasing at a rate of 1 centimeter per second.

4. (moderate-iii) The bottom of the ladder is moving away from the wall at a speed of 4/3 feet per second.

4. (hard) The water level is rising at a speed of \frac{5}{24} meters per second.

5. (warm-up) 1.02.

5. (moderate-i) (\sqrt{2}/2) · (1.01)

5. (moderate-ii) 3\frac{1}{108} or \frac{325}{108}.

5. (moderate-iii) 2 + .03/e^{2}.

6. (warm-up) x_{2} = 2.

6. (moderate-i) x_{2} = 4/3, x_{3} = 20/63.

6. (moderate-ii) x_{2} = 95/48, x_{3} = 1152220351/532329100.

7. (warm-up) It is continuous in [0, 1] and differentiable in (0, 1) (every polynomial is!, everywehere!). c = 0.5.

7. (moderate) It is continuous in [0, 1] and differentiable in (0, 1); c = (3.2)^{1/4}.

7. (hard) ln x is continuous and differentiable whenever x > 0, in particular in the interval [e^{2}, e^{3}]; c = e^{2}(e − 1).

8. (warm-up) min value is 1; max value is 2.

8. (moderate) min value is 0; max value is 49.

9. (warm-up-i) 10 and 10.

9. (warm-up-ii) The maximal possible area is 1250 square feet, the dimensions are 25 by 50 feet.

9. (medium-i) (500)^{1/3} × (500)^{1/3} × 2 · (500)^{1/3}.

9. (medium-ii) (2, \sqrt{2}).

9. (hard) radius is (150)^{1/3} inches and height is 2 · (150)^{1/3} inches.

9. (hard) (550)^{1/3} × (550)^{1/3} × (1000)/(550)^{2/3} cm.

10. (warm-up) x^{2} + 3x + C

10. (moderate-i)

f(x) = \frac{x^{2}}{2} − \frac{1}{x} + C

10. (moderate-ii)

f(x) = −3 cos x + 4 sin x + e^{x} + C

10. (hard-i)

f(x) = \frac{3x^{2}}{2} + 5 sin^{−1} x + C

10. (hard-ii)

f(x) = 3\frac{x^{2}}{2} + 5 tan^{−1} x + C

11. (warm-up) f(x) = x^{2}.

11. (moderate-i) f(x) = 2x^{2} + 2/x^{3} + 2.

11. (moderate-ii) f(x) = − cos x + 3.

11. (hard-i) f(x) = x^{4}/12 + x^{2}/2 + 2x/3 − 1/4.

11. (hard-ii) f(x) = x^{4}/24 − x^{3}/6 + 5x^{2}/4 − x/6 + 1/24.