Math 121 Assignment 2
Due Friday January 22
1. Find the function f that satisfies the equation
Z 1
f (t) dt.
2f (x) + 1 = 3
x
2. Find the area of
(a) the plane region bounded between the two curves y = x42 and
y = 5 − x2 .
(b) Find the area of the closed loop of the curve y 2 = x4 (2 + x) that
lies to the left of the origin.
3. Evaluate the integrals
Z 4√
(a)
9t2 + t4 dt.
Z0
t
t
2
2
(b)
cos
sin
dt.
5
2
Z
(c)
cos4 x dx.
Z
dx
(d)
.
x
e +1
Z 2
x dx
.
(e)
4
0 x + 16
Z πp
2
(f )
1 − sin(θ) dθ.
0
4. Use mathematical induction to show that for every positive integer
k,
n
X
nk+1
nk
k
j =
+
+ Pk−1 (n),
k+1
2
j=1
where Pk−1 is a polynomial of degree at most k − 1. Deduce from
this that
Z a
ak+1
xk dx =
.
k+1
0
5. Does the function
Z 2x−x2
1
F (x) =
dt
cos
1 + t2
0
have a maximum or minimum value? Justify your answer.
1
2
6. (a) If m, n are integers, compute the integrals
Z π
Z π
Z π
sin mx cos nx dx.
sin mx sin nx dx,
cos mx cos nx dx,
−π
−π
−π
(b) Suppose that for some positive integer k,
k
a0 X
f (x) =
+
(an cos nx + bn sin nx)
2
n=1
holds for all x ∈ [−π, π]. Find integral formulas for the coefficients a0 , an , bn with the integrand involving f of course.
(Remark: The coefficients a0 , an , bn are called the Fourier coefficients of f . Fourier coefficients arise in a variety of contexts,
such as communications and signal processing. If f is a musical note, then the integers n for which an or bn are nonzero are
precisely the frequencies comprising the note.)