Math 126-104
Study guide for Final
Fall 2014
Carter
1. Definitions and Theorems.
(a) Be able to define the definite integral:
Z
N
X
b
f (x) dx = lim
N →∞
a
where ∆x = (b − a)/N and
x∗j
f (x∗j )(∆x)
j=1
∈ [xj−1 , xj ].
(b) Be able to state the Fundamental Theorem of Calculus:
I. Let y = f (t) denote a continuous function that is defined on a closed interval [a, b].
Define
Z x
f (t) dt.
A(x) =
a
Then
A0 (x) = f (x).
Or
II. Let y = f (t) denote a continuous function that is defined on a closed interval [a, b].
Suppose that G0 (x) = f (x). Then
Z
b
f (x) dx = G(b) − G(a).
a
(c) Be able to define the limit of a sequence:
lim an = L
n→∞
if and only if for every > 0 there is an integer N such that if N ≤ n we have
|an − L| < .
2. Integrals
(a) Compute the area that lies between the curves:
i. y = x3 − x2 + 1 and y = 1.
√
ii. y = x2 and y = x for x ∈ [0, 2].
iii. y = sin (x) and y = cos (x) for x ∈ [0, π2 ]
R1
(b) i. 0 x3 + x1/3 dx
R2
ii. 1 x12 dx
R4
iii. 1 (3x4 + 5x3 − 4x2 + 7) dx
R
iv. dx
ex
Rπ
(c) i. 0 (1 + cos (x)) dx
R
ii. ex cos (ex ) dx
1
(d)
(e)
(f)
(g)
(h)
3
iii.
R
x2 ex dx
iv.
R
sin (x) cos5 (x) dx
i.
R
xex dx
ii.
R
x2 sin (2x) dx
iii.
R
ex cos x dx
iv.
R
x ln (x) dx
v.
R
arcsin (x) dx
vi.
R
arctan (x) dx
i.
R
sec4 (x) dx
ii.
R
sin5 (x) dx
iii.
R
sin2 (x) cos2 (x) dx
iv.
R
2 sin (x) cos (x) dx
i.
R
ii.
R
iii.
R
iv.
R
i.
R
ii.
R
iii.
R
iv.
R
v.
R
√ dx
36−x2
√ dx
9+x2
√ dx
x x2 −25
x dx
x2 +49
1
dx
(x+1)(x−1)2
1
(x+1)(x−1) dx
1
dx
(x2 +1)(x−1)
dx
x2 +5x−6
(3x−5) dx
(x−2)(x+3)
i.
R1
ii.
iii.
iv.
v.
dx
√
0
x
R∞
−x2 dx
0 2xe
R∞ 1
1 (x5/4 ) dx
R∞ 1
0 (x2 +1) dx
R ∞ 1 −x/2
dx
0 2e
3. (a) Compute the volume obtained by rotating the region bounded by the curves y = x2 ,
y = 0, x = 1 about the x-axis.
(b) Compute the volume obtained by rotating the region bounded by the curves y = 2x,
y = 0, and x = 2 about the y-axis.
(c) Compute the volume obtained by rotating the region bounded by the curves y = e−x ,
y = 0, x = 0, x = 1 about the x-axis.
√
(d) Compute the volume obtained by rotating the region bounded by the curves y = 2,
x = y 2 , and x = 0 about the x-axis.
2
(e) Find the volume of the tetrahedron, by considering the triangular cross sections perpen-
3
4
y
z
5
x
dicular to the x-axis.
3
4. (a) Find the length of the
curve y = x 2 from x = 0 to x = 4. Recall that the formula for
Rbq
arc-length is L = a 1 + (f 0 (x))2 dx.
(b) Compute the length of the curve of the from 0 ≤ x ≤ 3 of the function
3
1
y = (x2 + 2) 2
3
y
−y
(c) Find the area of the surface generated by revolving the curve x = e +e
, for 0 ≤ y ≤
2
ln(2), about the y-axis.
√
(d) Find the surface area that is obtained by rotating y = 64 − x2 about the x-axis for
x ∈ [0, 8].
2
meters. Assuming that Hooke’s
5. (a) A force of 2-Newtons will stretch a rubber band 100
Law applies, how far with a force of 4-Newtons stretch the rubber band? How much
work does it take to stretch the rubber band this far? Recall: Hooke’s law states that
the force required to stretch a spring is proportional to the length that it is stretched.
(b) A force of 25-Newtons will stretch a spring 5 meters beyond its natural length. Assuming
that Hooke’s Law applies, how much work does it take to stretch the spring 10 meters
beyond its natural length? Recall: Hooke’s law states that the force required to stretch
a spring is proportional to the length that it is stretched.
3
(c) The half-life of the plutonium isotope is 24,360 years. If 10 grams of plutonium is released
into the atmosphere by a nuclear accident, how many years will it take for 80% of the
isotope to decay?
(d) A leaky bucket is being lifted 20 meters. It originally holds 10 kilograms of water, and
at the end of its journey it holds 5 kilos. How much work does it take to lift the bucket?
6. Determine which of the following sequences converge. If the sequence does converge, find the
limit.
n
(a) an = 1 + 13
(b) an =
(c) an =
(d) an =
n2 +n+3
3n2 +2n+1
n3
n5 +3n3 −2
cos(πn)
n
(e) an = ln(n) − ln(n + 1)
(f) an = (−1)n + n1
n
(g) an = 1 + −2
n
n
n
(h) an = n+3
(i) an =
(j) an =
(−8)n
n!
√
n
55n
7. For each of the series determine a formula for the nth partial sum:
(a)
1
1
1
+
+ ··· +
+ ···
2·3 3·4
(n + 1) · (n + 2)
(b)
1+
1
1
1
+
+ ··· + n + ···
5 25
5
(c)
9
9
9
+
+ ··· + n + ···
10 100
10
(d)
2 3 4
2
2
2
−
+ −
+ −
+ ···
3
3
3
8. Use any test that you like to determine if the given series converges.
(a)
∞
X
ln n
√
n
n=2
(b)
∞
X
n=1
1
n100,001/100,000
4
(c)
∞
X
n=1
n2
n
+1
3n
1
−1
(d)
∞
X
n=1
(e)
√
∞
X
n
2
n +3
n=1
(f)
∞
X
1
n!
n=1
(g)
∞ X
3 n
1+
n
n=1
(h)
X n!
(2n)!
n=1
(i)
∞
X
1
2/3
n
n=1
(j)
∞
X
1
n n2 − 1
n=2
√
(k)
∞ n
X
2
3
n=0
(l)
∞
X
1
1
√ −√
n
n+1
n=1
(m)
∞
X
n2 (n + 2)!
n=1
n!32n
(n)
∞
X
1
n1.1
n=1
5
(o)
∞ X
n=1
−2
1+
n
n
(p)
∞
X
n=0
n2
n
+1
(q)
∞
X
(−3)n
n!
n=0
9. Compute the interval of convergence for the series:
(a)
∞
X
(x − 2)n
3n
n=0
(b)
∞
X
(−1)n (x + 6)n
n · 3n
n=0
(c)
∞
X
nxn
n=1
(d)
∞
X
xn
(2n)!
n=1
10. Find the 4th Taylor polynomial, P4 , generated by f (x) =
1
x
at center a = 2
11. Use substitution and power series operations to find the Taylor series about x=0 of the
following functions:
x
(a) e− 2
(b) sin
πx
2
(c) x2 cos (3x)
(d)
1
3−x
(e) ln(1 + 34 x)
12. Compute the parametrization of the line segment from (1, 2) to (3, −8) for t ∈ [0, 1].
13. Give a parametrization of the ellipse
interval t ∈ [0, 2π].
x2
25
+
y2
9
= 1 that travels once counter-clockwise in an
14. Give the equation of the tangent line to the curve at the given point
6
(a)
x(t) = 2 cos (t) y(t) = 2 sin (t)
at t = π/6.
(b)
x(t) = 2t2 + 3
y(t) = t4
at t = −1.
(c)
x(t) = t + et
y(t) = 1 − et
at t = 0.
15. areas of polar graphs (and the graphs of the functions) such as
(a) r = 1 + cos (θ)
(b) one petal of r = cos (3θ)
(c) r = sin (θ) for θ ∈ [0, π].
7