Vv156 Honors Calculus II

Assignment 4

This assignment has a total of (34 points).

Note: Unless specified otherwise, you must show the details of your work via logical reasoning for each exercise.

Simply writing a final result (whether correct or not) will receive 0 point.

Exercise 4.1 [Ste10, p. 385]

(a) (1 pt) If f is continuous on [a, b], use −|f(x)| ≤ f(x) ≤ |f(x)| to show that

Z b

a

f(x) dx

≤

Z b

a

|f(x)| dx

(b) (1 pt) Use the result of prevous part to show that

Z 2π

0

f(x) sin(2x) dx

≤

Z 2π

0

|f(x)| dx

(2 pts)

Exercise 4.2 [Ste10, p. 395] The error function

erf(x) = 2

√

π

Z x

0

e

−t

2

dt

is used in probability, statistics, and engineering.

(a) (1 pt) Show that

Z b

a

e

−t

2

dt =

1

2

√

π[erf(b) − erf(a)]

(b) (1 pt) Show that the function y = e

x

2

erf(x) satisfies the differential equation y

′ = 2xy + 2/

√

π.

(2 pts)

Exercise 4.3 [Ste10, p. 396] The sine integral function

Si(x) = Z x

0

sin t

t

dt

is important in electrical engineering. [The integrand f(t) = (sin t)/t is not defined when t = 0, but we know that its

limit is 1 when t → 0. So we define f(0) = 1 and this makes f a continuous function everywhere.]

(a) (1 pt) Sketch the graph of Si.

(b) (1 pt) At what values of x does this function have local maximum values?

(c) (1 pt) Find the coordinates of the first inflection point to the right of the origin.

(d) (1 pt) Does this function have horizontal asymptotes?

(e) (1 pt) Solve the following equation (for x) correct to one decimal place:

Z x

0

sin t

t

dt = 1

(5 pts)

Exercise 4.4 [Ste10, p. 396] Evaluate the limit by first recognizing the sum as a Riemann sum for a function defined

on [0, 1].

(a) (1 pt) limn→∞

Xn

i=1

i

4

n5

.

(b) (1 pt) limn→∞

1

n

Xn

i=1

r

i

n

Page 1 of 2

(2 pts)

Exercise 4.5 [Ste10, p. 396] If f is continuous and g and h are differentiable functions, find a formula for

d

dx

Z h(x)

g(x)

f(t) dt

(2 pts)

Exercise 4.6 [Ste10, p. 396] Find a function f and a number a such that

6 + Z x

a

f(t)

t

2

dt = 2√

x for all x > 0.

(2 pts)

Exercise 4.7 [Ste10, p. 414] Evaluate the indefinite and definite integral.

Z

e

tan x

sec2

(i) x dx

Z

sin(ln x)

x

(ii) dx

Z √

cot x csc2

(iii) x dx

Z

dx

√

1 − x

2 arcsin x

(iv)

Z 2

1

e

1/x

x

2

(v) dx

Z π/3

−π/3

x

4

(vi) sin x dx

Z e

4

e

dx

x

√

ln x

(vii) Z 1

0

dx

(1 + √

x)

4

(viii)

(8 pts)

Exercise 4.8 [Ste10, p. 412] If f ∈ C

0

(R), show that

(a) (2 pts) Z b

a

f(−x) dx =

Z −a

−b

f(x) dx.

(b) (2 pts) Z b

a

f(x + c) dx =

Z b+c

a+c

f(x) dx.

(c) (2 pts) Z π

0

xf(sin x) dx =

π

2

Z π

0

f(sin x) dx.

(d) (2 pts) Z π/2

0

f(cos x) dx =

Z π/2

0

f(sin x) dx.

(8 pts)

Exercise 4.9 [Ste10, p. 412] Evaluate the definite integral.

(a) (1 pt) Z π

0

x sin x

1 + cos2 x

dx.

(b) (1 pt) Z π/2

0

cos2 x dx.

(c) (1 pt) Z π/2

0

sin2 x dx.

(3 pts)

References

[Ste10] J. Stewart. Calculus: Early Transcendentals. 7th ed. Cengage Learning, 2010 (Cited on pages 1, 2).

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