Math 1320 Midterm Review
1. Find the volume of the solids bounded by the given curves.
3
(a) x = , y = 1, y = 5, x = 0, about the y -axis.
y
√
(b) y = 2x, y = 0, x = 2, about the x-axis.
√
(c) y = 2x 3 , y = 2 x, about y = 2.
(d) y = 3x, y = 3x 2 , about y = 6.
(e) y = 3x, y = 3x 2 , about the x-axis.
(f) y = 4x − x 2 , y = 3, about x = 1.
(g) x = 1 + (y − 2)2 , x = 2, about the x-axis.
2. Find the exact length of the curve.
8
(a) x = 5 + 4t 2 , y = 7 + t 3 , 0 ≤ t ≤ 2.
3
√ t/2
t
(b) x = e − 2t, y = 4 2e , −2 ≤ t ≤ 1.
(c) x = y 3/2 , 0 ≤ y ≤ 1.
3. Find the average value of the function f (x) = 4 − x 2 on the interval [−1, 2].
4. The work (in Joules) accumulated by pushing against a linear spring to move from position
0 to x, is given by the function W (x) = 10x 2 . Find the force exerted by the spring when
compressed to x = 3. [Hint: the spring’s neutral position is at x = 0. Write down the
integral for W (x).]
5. If 6J of work is needed to stretch a linear spring from 10cm to 12cm and another 10J is
needed to stretch it from 12cm to 14cm, what is the natural length of the spring?
6. Coulomb’s Law states that the electrical force between two charged objects is inversely
proportional to the square of the distance between them:
F (r ) =
kq1 q2
,
r
where k is a constant, q1 and q2 are the charges of the two objects in Coulombs, and r
is the distance between the objects in meters.
Calculate the work required to move the charged objects from a distance of 20m apart
to a distance of 5m apart. (Your answer will be in terms of k, q1 , and q2 .)
7. Solve the following differential equations.
dy
= 2x sec y , y (1) = π/2.
dx
(b) y 0 = 1 + x + y + xy , y (0) = 10.
dy
x
(c)
= , y (1) = 3.
dx
y
dy
(d) 2y
= 2x + 2, y (0) = −2.
dx
(a)
8. A cake is removed from an oven at 210◦ F and left to cool at room temperature, which is
70◦ F. After 30min the temperature of the cake is 140◦ F. When will it be 100◦ F assuming
the process follows Newton’s Law of Cooling?
9. Verify that y (t) = cos(t)e t is a solution of
y 00 − 2y 0 + 2y = 0.
10. A tank contains 1000L of a solution consisting of 100kg of salt dissolved in water. Pure
water is pumped into the tank at the rate of 5L/s and the mixture - kept uniform by
stirring - is pumped out at the same rate. How long will it be until only 10kg of salt
remains in the tank?
11. In an elementary chemical reaction, single molecules of two reactants A and B form a
molecule of product C. A + B −→ C. The Law of Mass Action states that the rate of
reaction is proportional to the product of the concentrations of A and B,
d[C]
= k[A][B].
dt
Thus, if the initial concentrations are [A] = a moles/L and [B] = b moles/L and we
write x = [C] then we have
dx
= k(a − x)(b − x).
dt
Assuming that a 6= b, find x as a function of t. Use the fact that the initial concentration
of C is 0.
12. Bouncing ball A ball is let go at a height of 1m from the ground and bounces up and
down forever. Each bounce is 2/3 the previous bounce’s height, going on toward infinity.
Compute the complete distance traveled, summing both up and down directions of the
ball’s path. (Use series!)
13. An animal population is growing in such a way that each new generation is 1.5 times as
large as the previous generation. Suppose there are 100 animals in the first generation.
How many will there be in the fifth? (Use sequences!)
14. Find the limit of the sequences, if it exists, or justify its divergence.
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cos(3n)
√ .
(a) an =
π+ 3n
(b) bn =
(2n − 2)!
.
(2n + 1)!
(c) an =
(2n + 1)!
.
(2n − 1)!
(e) an =
n+1
n
n
.
(f) bn = ln(π 2 n3 + 1) − ln(4n3 − 7).
1
(g) a1 = 1, an+1 = (an + 3).
3




1
1
1
(h) 1,
,... .
,
1 ,
 2 + 1 2 + 2+1

2 + 2+1 1
(d) an = ln(7n2 + 2) − ln(en + 2).
2+1
15. Find the limit of the series.
∞ X
∞
X
5n
.
(a)
n+1
7
n=3
(b)
∞
X
2n+3 − 5n+1
n=1
7n−1 2n
2
1
(c)
+ n .
n(n
+
1)
3
n=3
∞
X
1
1 + 3n
(d)
+
.
en
2n
n=3
.
16. Determine whether the series is convergent or divergent. Show your work, justify, use
theorems, . . .
∞
X
2 + (−1)n
√
(h)
.
3
n
n
n=1
∞
X
3n2 − 1
.
2+3
7n
n=1
∞ n+1
X
3
3
(b)
+
.
n
5
n
n=2
(a)
(c)
(d)
(e)
(f)
(g)
(i)
∞
X
n7/2
.
4+3
n
n=1
∞
X
n=1
∞
X
n=1
∞
X
n=1
∞
X
n=1
(j)
5/2
5n + 3n
.
n7 − 2
(k)
2 + cos n
.
3n
(l)
(−1)n n2
.
n4 + 1
(m)
1
.
n ln n
(n)
∞
X
n3 + 2
.
3+1
3n
n=1
∞
X
7n + 5n+1
n=1
∞
X
n=1
∞
X
n=1
∞
X
n=1
∞
X
n=1
6n+2
(−2)n
.
n
(−2)n
√
.
2n + 3
(−1)n n5
.
7n
2n
.
(2n + 1)!
17. Determine whether the series is absolutely convergent.
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.
(a)
∞
X
(−2)n
n!
n=1
.
(b)
∞
X
(−3)n
n=2
n
.
18. Find the radius and interval of convergence.
(a)
(b)
(c)
∞
X
2(x − 5)n
.
n
n=1
∞
X
n=2
∞
X
n=1
(d)
n 2n
(−1) x
.
(2n)!
(e)
n
x
.
n!
(f)
∞
X
n=1
∞
X
n=1
∞
X
n=1
n!
.
(x − 2)n
3n (x + 4)n
√
.
n
(4x + 1)n
.
n2
19. Use differentiation or integration to find the power series, and interval of convergence of
the following.
1+x
(a) ln(x 2 ).
.
(c) ln
1−x
−2x
1
(d)
.
.
(b)
2
(1 − x)
(1 + x 2 )2
20. Find the Taylor Series and the radius of convergence about x = a for the following
functions.
1
, a = −3.
x
(d) x − x 3 , a = −2.
(a) cos x, a = π.
(c)
(b) e x , a = 2.
21. Find the first three non-zero terms in the McLaurin series of the following.
(b) x − x 3 .
(a) cot x.
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