Math 112 FINAL EXAM, Fall 2024
Read each problem carefully. Show all your work for each problem! No Calculators!
1. (12) Evaluate the following integrals:
Z π/x
(a)
2
dx.
x2 − 1
Z
sec2 x tan x dx,
(b)
0
2. (a) (6) Use the shell method to find the volume of the solid generated by revolving the region
bounded by the curves y = 2/x, y = 2 and x = 2 about the y-axis.
(b) (6) Find the length of the curve y = 13 (x2 + 2)3/2 from x = 0 to x = 3.
3. (12) Evaluate the following integrals:
Z
(a)
xe2x dx, (b)
Z p
4 − x2 dx.
4. (a) (8) Use known series to write down the first 3 non-zero terms in the Maclaurin series (Taylor
2
series about a = 0) for the function f (x) = x3 e−x /(1 + x2 ).
(b) (4) Use the series found in part (a) to determine f (5) (0), i.e. the fifth derivative of f (x) at
x = 0. Do NOT attempt to differentiate f (x) five times!
5. (a) (5) Find the constant a such that the following equation is true
∞
X
31−n an+1 = 18.
n=0
(b) (7) Evaluate the following improper integral:
Z ∞
0
e−x
dx.
1 + e−x
6. (12) Determine if the series converge or diverge. State which test you are using.
(a)
∞ X
n + en n=1
n3 + en
∞
X
(b)
√
n+2
n+1
n=1
!4
.
7. (a) (10) Find the radius of convergence and interval of convergence for
∞
X
2n (x − 1)n+2
n=1
(b) (4) Given f (x) = x +
n
.
x2 x3 x4
+
+
+ · · · with 0 < x < 1, then f ′ (x) = ? (circle one)
2
3
4
(a) ex ,
(b)
1
,
1−x
(c) e−x ,
(d)
1
1+x
8. (a) (8) Find the area of the polar region that lies inside the cardioid r = 2 + 2 sin θ and outside
the circle r = 2.
(b) (6) Find an equation for the line tangent to the curve x =
√
t
+ 2, y = t at the point t = 4.
4