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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 R 5
0
f(x) dx = 10. What does this mean? That is, what information is
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
0
(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) =
x
2
.
5. (True/False) If f(x) is continuous at x0, then limx→x0
f(x) must exist.
6. (True/False) If R 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
00(x0) = 0 but for there not to be an inflection
point at x0.
1
2
Computation
1. For the following functions, list whether or not R 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?
3
3. Evaluate the following limits (4 points each):
(a) limx→5 x
2 − 3x + 8 =
(b) limx→1 5 =
(c) limx→3
x
2−9
x−3
4. Find the derivatives of the following functions (4 points each):
(a) f(x) = 20x
1
2
(b) f(x) = (10x
4 + 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 R
4 dx.
(b) Find R
(3x − 2)2 dx
(c) Evaluate R 3
1
(3t
2 + 7) dt
(d) Evaluate R 2
1
(4t
3 − 1) dt
4
6. Suppose f(x) = 2x
3 − 3x
2 − 12x + 12. Sketch the function using the below procedure.
Find f
0
(x) and f
00(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, ∞).
5
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
Z 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?]

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