math test from calculus 1 (3)

MAT 136: Calculus I - Spring 2020

MAT 136: Calculus I - Spring 2020 Exam 2

NAME: Class grade: Test grade /90

Instructions: Answer each of the following questions completely. To receive full credit, you must show sufficient work for each of your answers (unless stated otherwise). How you reach your answer is more important than the answer itself.

1.a) Find the local linearization of f(x) = x3 − 2x2 + 5 at x = 5 (5 points)

b) Estimate f(6) and f(0) using part a). (2 points)

c) Are any of the estimates from part b) useful? Ex- plain why or why not. (3 points)

2 Using calculus techniques find the absolute maximum(s) and absolute minimum(s) of f(x) = x3 − 2x2 + 15x on the interval [−1, 2] (8 points)

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MAT 136: Calculus I - Spring 2020

3. Find the point c in the interval [−1, 3] that satisfies the mean value theorem for f(x) = x3 − 2x2. (8 points)

4. Find the inflection points of f(x) = x4 − 4x3 + 18x2 − 6x + 1

5. Find the following limits. State any theorems used while taking the limits.

a) lim x→0+

ln(x) sin(x) b) lim

x→2

x2 − 5x + 6x x2 + 2x− 8

c) lim x→0

xx d) lim x →∞

−x3 + 5x2 − 2 x2 + 3x

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MAT 136: Calculus I - Spring 2020

6. Draw a graph with the following properties. (10 points)

Property Interval

f ′(x) > 0

f ′(x) < 0

f ′′(x) > 0

f ′′(x) < 0

(−∞,−3), (−1, 0), (5,∞) (−3,−1), (0, 5) (−∞,−3), (−3, 0) (0, 7), (7,∞)

Asymptote at x = −3

lim x→−∞

f(x) = 0 lim x→∞

f(x) = 1

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MAT 136: Calculus I - Spring 2020

7. A company is making a square based box as shown below. The cost for the material varies depending on the side. The cost for material on the top is $0.20 per sq ft for the top, $0.10 per sq ft for the bottom and $0.30 per sq ft for the sides. The company wants the box to have an volume of 81in3. What should the dimensions be to minimize the cost of the box? (10 points)

8. After the following statements write T for True or F for false. No rational is needed for your answers. (2 points each)

a) If f is a continuous function with f(0) = −1 and f(2) = 1 there must be a value c in the interval (0, 2) such that f(c) = 0.

b) If lim x→∞

f(x) = ∞ and lim x→∞

g(x) = 0 then lim x→∞

f(x)g(x) = 0.

c) All absolute maximums happen at a local maximum.

d) A point of inflection can also be a local maximum.

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MAT 136: Calculus I - Spring 2020

9. Let the graph on the left depict f ′. On the right give a rough sketch of f. (5 points)

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