HOMEWORK 1

DONDI ELLIS

I hate victims who respect their executioners.

-Jean-Paul Sartre (French Existentialism Writer/Philosopher)

1. Problem 1

These are True and False problems. Please do not just say True or False,but also say why or why not as well. You may state a theorem or use acounter-example, ect. as appropriate.

a) If F (t) and G(t) are antiderivatives of the function f(t) with F (0) = 1and G(0) = 3 then F (2)−G(2) = 1.

b) If h(t) > 0 for 0 ≤ t ≤ 1, then the function H(x) =

∫ x

0h(t)dt is

concave up for 0 ≤ x ≤ 1.

c) If

∫ 2

0g(t)dt = 6 then

∫ 3

23g(2t− 4)dt = 9.

d) ddx

(∫ sinx

−x2

et3dt)

= cosxex3

+ 2xex3

2. Problem 2

a) Compute

∫ 1

0f ′(x) sin(2πx)dx where f(x) is given by the following

table

Figure 1

b) Find

∫x3 cosx2dx

Date: Today.

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3. Problem 3

Given the graph of f ′(x). Sketch a graph of f(x) on the provided axesgiven that f(1) = 0. On your graph, label any local maxima, minima, andpoints of inflection. Make sure that the concavity of the graph of f(x) isvisible in your graph.

Figure 2

4. Problem 4

These are True and False problems. Please do not just say True or False,but also say why or why not as well. You may state a theorem or use acounter-example, ect. as appropriate.

a) If F (x) is an antiderivative of an even function f(x), then F (x) mustalso be an even function.

b) If G(x) is an antiderivative of g(x) and (G(x)−F (x))′ = 0, then F (x)is an antiderivative of g(x).

c) Let f(t) = bt+ ct2 with b > 0 and c > 0, then Left(n) ≤∫ 10

0f(t)dt for

all n.

HOMEWORK 1 3

Note that Left(n) denotes a left hand Reimann sum using n subdivisions.

d) The average of an even function f(x) over the interval [−a, a] is equalto twice its average over the interval [0, a].

5. Problem 5

The graph of an odd function f is shown below.

Figure 3

a) Let F (x) be the antiderivative of f(x) with the property that F (3) =−2. Use the graph of f(x) to compute the following values of F (x).

Figure 4

b) Sketch the graph of F (x) from x = −7 to x = 7. Label all points ofinflection.

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Figure 5

6. Problem 6

Use the graphs of f(x) and g(x) to find the EXACT values of A,B, andC. Show all your work.

Figure 6

a) A =

∫ 6

−3|f(x)|dx

b) B =

∫ 2

0xg′(x2)dx

c) C =

∫ 3

02xg′(x)dx

Freedom is what you do with what’s been done to you.

-Jean-Paul Sartre (French Existentialism Writer/Philosopher)