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) (1pt)Iff iscontinuouson[a,b],use−|f(x)|≤f(x)≤|f(x)|toshowthat
b b
f ( x ) d x ≤ | f ( x ) | d x
aa (b) (1 pt) Use the result of prevous part to show that
2π 2π
f (x) sin(2x) dx ≤ |f (x)| dx
00 Exercise 4.2 [Ste10, p. 395] The error function
erf(x) = √ is used in probability, statistics, and engineering.
(a) (1 pt) Show that
b2 1√
e−t dt = 2
(b) (1 pt) Show that the function y = ex2 erf(x) satisfies the differential equation y′ = 2xy + 2/√π.
(2 pts)
Exercise 4.3 [Ste10, p. 396] The sine integral function
x sint
(2 pts)
(5 pts)
n→∞ n
i=1
n
n i4
i=1
n
5.
a
2x2
π0
e−t
dt
Si(x)=
is important in electrical engineering. [The integrand f(t) = (sint)/t is not defined when t = 0, but we know that its
t dt
limit is 1 when t → 0. So we define f(0) = 1 and this makes f a continuous function everywhere.]
0
(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: x sin t dt = 1
0t
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) (1pt) lim
n→∞
(b)(1pt)lim1n i
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π[erf(b) − erf(a)]
(2 pts)
Exercise 4.5 [Ste10, p. 396] If f is continuous and g and h are differentiable functions, find a formula for
d h(x)
dx f(t)dt
g(x)
(2 pts)
Exercise 4.6 [Ste10, p. 396] Find a function f and a number a such that
sec xdx dx
dx x4 sin x dx
(iii) (vii)
cotxcsc xdx dx
(ii) (vi)
(iv) √ (viii)
6+
π/3 −π/3
x f(t) √
t2 dt=2 x forallx>0. Exercise 4.7 [Ste10, p. 414] Evaluate the indefinite and definite integral.
(2 pts)
tanx (i) e
2 e1/x
2
sin(lnx) x
√ e4
e xlnx
2
dx √1−x2arcsinx
a
(v)
1
x2
1 0
dx
(1 + √x)4
(8 pts)
Exercise 4.8 [Ste10, p. 412] If f ∈ C0(R), show that b −a
(a) (2pts)
(b) (2pts)
(c) (2pts)
(d) (2pts) (8 pts)
f(−x)dx =
a −b
b a
f(x)dx. b+c
f(x+c)dx=
π ππ
xf(sinx)dx = 2 00
π/2 π/2 f(cosx)dx=
00
Exercise 4.9 [Ste10, p. 412] Evaluate the definite integral.
1+cos2 x dx. π/2
cos2xdx. sin2xdx.
(a) (1pt)
(b) (1pt)
(c) (1pt) (3 pts)
References
π xsinx
a+c
f(x)dx.
f(sinx)dx.
f(sinx)dx.
0
0
π/2
0
[Ste10] J. Stewart. Calculus: Early Transcendentals. 7th ed. Cengage Learning, 2010 (Cited on pages 1, 2).
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