CALC TEST (PLEASE SHOW WORK EXTENSIVELY)

Final Exam

Math 261: Applied Calculus I

May 7, 2020

Concepts. (2 points each)

Answers here should be as simple and non-technical as you can make them.

1. Suppose ∫ 5 0 f(x) dx = 10. What does this mean? That is, what information is

the answer, 10, giving us?

2. We started this course by studying limits. Why was this necessary? How does

the idea of a limit relate to both derivatives and integrals? (Hint: Recall some of

our key equations from this course.)

3. Suppose f(t) describes the true number of individuals in the U.S infected with

COVID-19 at any given moment t. What would the derivative of f(t) describe?

In other words, how would you interpret the meaning of f′(t) in this context?

[Hint: Consider an actual graph of coronavirus infections. What does the slope

at any point on the curve tell us?]

4. (True/False) There’re an infinite number of functions whose derivative is f(x) =

x2.

5. (True/False) If f(x) is continuous at x0, then limx→x0 f(x) must exist.

6. (True/False) If ∫ 8 1 f(x) > 0, then f(x) > 0 for all x in the interval [1, 8]. That is,

the curve f(x) lies entirely above the x axis on the interval [1, 8].

7. (True/False) It is possible to have f′′(x0) = 0 but for there not to be an inflection

point at x0.

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Computation

1. For the following functions, list whether or not ∫ b a f(x) dx is positive, zero, or

negative (4 points):

First function:

Second function:

2. Answer the following based on the graph below (16 points)

(a) limx→−2 F (x) =

(b) limx→−2+ F (x) =

(c) limx→−∞ F (x) =

(d) At which points x is F (x) not continuous?

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3. Evaluate the following limits (4 points each):

(a) limx→5 x 2 − 3x + 8 =

(b) limx→1 5 =

(c) limx→3 x2−9 x−3

4. Find the derivatives of the following functions (4 points each):

(a) f(x) = 20x 1 2

(b) f(x) = (10x4 + 20x− 5)(3x + 2). (No need to simplify your answer)

(c) f(x) = 2x−3 x

5. Find or evaluate the following (6 points each):

(a) Find ∫

4 dx.

(b) Find ∫

(3x− 2)2 dx

(c) Evaluate ∫ 3 1

(3t2 + 7) dt

(d) Evaluate ∫ 2 1

(4t3 − 1) dt

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6. Suppose f(x) = 2x3 − 3x2 − 12x + 12. Sketch the function using the below pro- cedure.

Find f′(x) and f′′(x) (2 points):

Find the critical points and classify each as producing a relative max, relative

min, or neither (provide any relative max/mins as well) (4 points):

Sketch the graph (no need to show concavity) (4 points):

On which intervals is the function increasing, and which is it decreasing? (4

points):

7. Sketch a graph that matches the following description (4 points):

f(x) is increasing and concave down on (−∞, 1), f(x) is increasing and concave up on (1,∞).

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Extra credit (5 points.) Suppose f(t) describes the true number of individuals in the

U.S infected with COVID-19 at any given time t. What would the integral of f(t)

from January 1st till May 1st describe? In other words, how would you interpret the

meaning of ∫ May−01−2020 Jan−01−2020

f(t) dt

in this context? [Hint: Consider an actual graph of coronavirus infections. What does

the area under the curve from January till May tell you?]