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 a counter-example, ect. as appropriate. a) If F (t) and G(t) are antiderivatives of the function f (t) with F (0) = 1 and G(0) = 3 then F (2) - G(2) = 1. b) If h(t) > 0 for 0 ≤ t ≤ 1, then the function H (x)= Z x 0 h(t)dt is concave up for 0 ≤ x ≤ 1. c) If Z 2 0 g(t)dt = 6 then Z 3 2 3g(2t - 4)dt =9. d) d dx Z sin x -x 2 e t 3 dt = cos xe x 3 +2xe x 3 2. Problem 2 a) Compute Z 1 0 f 0 (x) sin(2πx)dx where f (x) is given by the following table Figure 1 b) Find Z x 3 cos x 2 dx Date : Today. 1
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HOMEWORK 1 I hate victims who respect their executioners. -Jean
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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.
1
2 DONDI ELLIS
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.
4 DONDI ELLIS
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.