Math 166
December 10, 2015
Final exam study guide
The final exam will take place Thursday, December 17 from 4:30 to 6:30pm
in MacKay 0117. The exam will consist of 10 problems, 6 from Chapter 10, 4
from Chapter 11. There will be no problems directly from previous chapters,
but knowledge of past material will probably be necessary to do the exam. The
only topic from section 10.10 that could be covered is binomial series.
Old finals
Final exams from some recent past semesters are posted here:
http://orion.math.iastate.edu/dept/CoursePages/165-6/
10.1 - 10.6
Here is a summary of the types of series and convergence tests we have encountered:
1. nth-term test
2. Geometric series
3. p-series
4. Tests for series with positive terms: integral test, direct comparison test,
limit comparison test, ratio test, root test
5. Absolute convergence test
1
6. Alternating series test
For sample problems see the previous study guide
http://tchumley.public.iastate.edu/m166f15/studyguide3.pdf
10.7 Power series
Topics
• Find the interval of convergence of a power series
• Find the center and radius of the interval of convergence
Key technique
• Use the ratio test to find the interior of the interval of convergence
• Use a separate ad hoc convergence test to determine convergence at the
endpoints
Exercises
Find the interval of convergence and the center and radius of the interval for each
of the following power series. Also state where the series converges absolutely
and where it converges conditionally
1.
2.
∞
X
2n xn
n!
n=0
∞
X
n!(x − 3)n
n=0
3.
∞
X
(2x + 1)n+1
2n + 3
n=0
2
10.8 Taylor series
Topics
• Find the Taylor series of f centered at x = a using the formula
∞
X
f (n) (a)
(x − a)n .
n!
n=0
• Find the Taylor series of f by manipulating a known Taylor series.
Common Maclaurin series
∞
X
1
=
xn ,
1 − x n=0
for |x| < 1.
∞
X
1
(−1)n xn ,
=
1 + x n=0
ln(1 + x) =
∞
X
(−1)n+1 n
x ,
n
n=1
∞
X
1
=
(−1)n x2n ,
1 + x2
n=0
−1
tan
for |x| < 1.
for |x| < 1.
for |x| < 1.
∞
X
(−1)n 2n+1
x=
x
,
2n + 1
n=0
ex =
∞
X
xn
,
n!
n=0
for |x| < 1.
for all x.
sin x =
∞
X
(−1)n 2n+1
x
,
(2n + 1)!
n=0
cos x =
∞
X
(−1)n 2n
x ,
(2n)!
n=0
for all x.
for all x.
Exercises
1. Find the first four terms of the Taylor series of f centered at a where
(a) f (x) = 1/x,
a=3
3
(b) f (x) = sin x,
a = π/6
2. Find the Maclaurin series for
(a) f (x) = x2 cos x2
2
(b) f (x) =
(1 − x)3
10.9 Convergence of Taylor series
Topics
• Taylor’s theorem: if f is (n + 1)-times differentiable in an interval I
containing the point a, then
f (x) = Pn (x) + Rn (x)
where
Pn (x) = f (a) + f 0 (a)(x − a) +
and
Rn (x) =
f (n) (a)
f 00 (a)
(x − a)2 + · · ·
(x − a)n
2!
n!
f (n+1) (c)
(x − a)n+1
(n + 1)!
where c is some number between x and a.
• Remainder estimation: if there is a constant M such that |f (n+1) (t)| ≤
M for all t between x and a, then
|Rn (x)| ≤
M
|x − a|n+1 .
(n + 1)!
Exercises
1. Consider the function f (x) = cos x and its second order Taylor polynomial
P2 (x) = 1 − x2 /2.
(a) Compute P2 (π/6)
(b) Estimate the error in approximating f (π/6) by P2 (π/6). That is,
give an upper bound for |f (π/6) − P2 (π/6)|.
(c) For approximately what values of x can you replace cos x by P2 (x)
with an error of magnitude no greater that 10−4 .
4
2. Consider the function f (x) = ex and its fourth order Taylor polynomial
P4 (x) = 1 + x + x2 /2 + x3 /6 + x4 /24.
(a) Compute P4 (1/2)
(b) Estimate the error in approximating f (1/2) by P2 (1/2). That is, give
an upper bound for |f (1/2) − P4 (1/2)|.
(c) For approximately what values of x in [0, 2] can you replace ex by
P4 (x) with an error of magnitude no greater that 10−4 .
10.10 Binomial series
Topics
• When considering functions of the form (1 + x)m where m is some given
real number, the Maclaurin series is also known as the binomial series.
The technique for determining terms of the binomial series is to simply
use the formula
∞
X
f (n) (0) n
x .
n!
n=0
Exercises
Find the first four terms of the Maclaurin series (aka the binomial series) for
the following functions
1. (1 + x)1/2
2. (1 + x)1/3
3. (1 − x)−1/2
11.1 Parameterizations of plane curves
Topics
• Convert from parametric equation to Cartesian equation
• Find parametric equation for a curve given its Cartesian equation or a
geometric description
5
Exercises
1. Find the Cartesian equation defined by the following parametric curves:
x = 4 sec t
(a)
, 0 ≤ t ≤ 2π
y = 2 tan t
√
x = √t + 1
, t≥0
(b)
y= t
2. Find parametric equations for the following:
(a) the motion of a particle that starts at (a, 0) and traces two times
counter-clockwise along the ellipse
x2
y2
+
= 1.
a2
b2
(b) the line segment with endpoints (−1, 3) and (3, −2).
11.2 Calculus with parametric curves
Topics
• Find the equation of tangent line to a parametric curve
• Find the area of a region bounded by a parametric curve and the x-axis
• Find the arc length of a parametric curve
• Find the area of a surface formed by revolving a parametric curve about
the x-axis
Formulas
•
d
dy
dx
/dt
dy
dy/dt
d y
=
,
=
2
dx
dx/dt
dx
dx/dt
Z x=b
Z t=t2
dx
•
y dx =
y
dt
dt
x=a
t=t1
s
Z t2 2 2
dx
dy
• L=
+
dt
dt
dt
t1
s 2
Z t2
2
dx
dy
• A=
2πy
+
dt
dt
dt
t1
2
6
Exercises
1. Consider the curve
x = 2t3 + 3
y = t4
(a) Find the equation of the tangent line of the curve at t = −1.
(b) Determine the concavity of the curve at t = −1.
x = tan t
2. Find where the curve
has zero slope.
y = sec t
3. Find the area of the region enclosed by the x-axis and the curve
4. Find the length of the curve
x = 8 cos t + 8t sin t
,
y = 8 sin t − 8t cos t
x = 1 + e−t
.
y = t − t2
0 ≤ t ≤ π/2.
11.3 Polar coordinates
Topics
• Convert from polar to Cartesian coordinates and vice versa
• Convert from polar to Cartesian equations and vice versa
• Graph some simple polar curves and regions
Key relations
• x = r cos θ,
• r 2 = x2 + y 2 ,
y = r sin θ
tan θ = y/x
Exercises
1. Find Cartesian coordinates for the following points given in polar coordinates
√
(a) ( 2, π/4)
(b) (1, 0)
(c) (0, π/2)
√
(d) (− 2, π/4)
7
2. Find polar coordinates for the following points given in Cartesian coordinates
(a) (−2, 2)
(b) (0, 3)
√
(c) (− 3, 1)
(d) (5, −12)
3. Sketch the curve or region given in polar coordinates
(a) 0 ≤ θ ≤ π/6,
(b) θ = 2π/3,
r≥0
r ≤ −2
(c) −π/4 ≤ θ ≤ π/4,
−1 ≤ r ≤ 1
4. Convert the following polar equations to Cartesian equations
(a) r = csc θer cos θ
(b) r2 sin 2θ = 2
(c) r2 + 2r2 cos θ sin θ = 1
(d) cos2 θ = sin2 θ
5. Convert the following Cartesian equations to polar equations
(a) x − y = 3
(b) x2 + (y − 2)2 = 4
11.4 Graphing in polar coordinates
Topics
• Graph more complicated curves such as cardioids, roses, and lemniscates
• Determine symmetries of polar curves
• A polar curve of the form r = f (θ) has parametric equations
x = f (θ) cos θ
.
y = f (θ) sin θ
The slope of the curve at a given point θ is then
dy
dy/dθ
f 0 (θ) sin θ + f (θ) cos θ
=
= 0
dx
dx/dθ
f (θ) cos θ − f (θ) sin θ
8
Key technique for graphing
In order to graph a polar equation of the form r = f (θ) in the xy-plane:
1. Graph the curve r = f (θ) in the θr-plane
2. Each point on the previous graph gives the polar coordinates for a point
on the desired graph. Use the previous graph as a guide for plotting these
points in the xy-plane.
Exercises
1. Graph each of the following polar equations in the xy-plane and identify
symmetries
(a) r = 2 + 2 sin θ
(b) r = cos 3θ
(c) r = cos 2θ
(d) r2 = 4 cos 2θ
2. Find the polar coordinates of the points on the rose curve r = sin 2θ where
the slope is 1.
3. Find the polar coordinates of the points on the cardioid r = 1 − cos θ
where the tangent line is vertical.
11.5 Areas and lengths in polar coordinates
Formulas
• The area of the region between the origin and the curve r = f (θ) for
α ≤ θ ≤ β is given by
Z β
1
f (θ)2 dθ.
2
α
• The area of the region inside the curve r = f (θ) and outside r = g(θ) for
α ≤ θ ≤ β is given by
Z
β
α
1
f (θ)2 − g(θ)2 dθ.
2
9
• The length of the polar curve r = f (θ) for α ≤ θ ≤ β is given by
Z
β
p
f (θ)2 + f 0 (θ)2 dθ.
α
Exercises
1. Find the area inside one loop of r = cos 2θ.
2. Find the area of the region bounded by r = 2 and r = 2(1 − cos θ).
3. Find the area inside r = 3 cos θ and outside r = 1 + cos θ.
√
4. Find the length of the curve r = θ2 for 0 ≤ θ ≤ 5.
√
5. Find the length of the curve r = 1 + sin 2θ for 0 ≤ θ ≤ π/2.
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