Math 166 - Study Guide: Chapter 10
• Polar Coordinates:
1. Know how to change from Polar coordinates and equations to Cartesian coordinates and equations
by using:
* r 2 = x2 + y 2
y
* tan(θ) =
x
* x = r cos(θ)
* y = r sin(θ)
2. Make sure you understand how to perform algebraic manipulations in order to get the given
equation into a recognizable form.
3. Example: r = cos(θ) ⇒ r2 = r cos(θ) ⇒ x2 + y 2 = x
• Graphs and Area:
1. Tests for Symmetry:
– Symmetric wrt the Polar Axis:
Plug in (r, −θ) or (−r, π − θ) for (r, θ)
– Symmetric wrt to the Line θ = π2 :
Plug in (−r, −θ) or (r, π − θ) for (r, θ)
– Symmetric wrt the Pole (Origin):
Plug in (−r, θ) or (r, π + θ) for (r, θ)
2. Graphing:
– Know how to make a table to graph the curves. You choose different values for θ and plug
them into the equation to find r.
– All intersection points of two curves may not be found by setting the two equations equal to
each other. The only way to ensure that you find all of them is to graph both curves.
3. Calculus of Polar Coordinates:
– Integration: Remember slice, approximate, integrate.
∗ If the region is defined by one curve, then the approximation is given by:
∆A =
1
[radius]2 ∆θ
2
∗ If the region is defined by more than once curve, then the approximation is given by:
∆A =
1
(outer radius)2 − (inner radius)2 ∆θ
2
∗ This is similar to the defference between washers and disks.
∗ Also, remember to take advantage of symmetry if you can.
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– Differentiation: If given r = f (θ), know how to find the slope of the tangent line.
∗ If f 0 (θ) 6= 0 and the tangent line is not passing through the pole, then the slope is given
by:
m=
dr
dθ
dr
dθ
0
sin(θ) + r cos(θ)
cos(θ) − r sin(θ)
f (θ) sin(θ) + r cos(θ)
= 0
f (θ) cos(θ) − r sin(θ)
∗ If f 0 (θ) 6= 0 and the tangent line passes through the pole, then the slope is given by:
m=
dr
dθ
dr
dθ
0
sin(θ) + r cos(θ)
cos(θ) − r sin(θ)
f (θ) sin(θ) + r cos(θ)
= 0
f (θ) cos(θ) − r sin(θ)
• Power Series:
– Know the following Series:
1
1.
= 1 + x + x2 + x3 + . . .
1−x
x3
x4
x5
x2
+
+
+
+ ...
2. ex = 1 + x +
2!
3!
4!
5!
x3
x5
x7
3. sin (x) = x −
+
−
+ ...
3!
5!
7!
2
4
6
x
x
x
4. cos (x) = 1 −
+
−
+ ...
2!
4!
6!
– Alterations of Power Series to obtain different Power Series:
∗
∗
∗
∗
Add/Subtract polynomials/constant to a power series in order to obtain a new power series.
Mult/Divide a power series by polynomials/constants in order to obtain a new power series.
Mult/Divide a two power series in order to obtain a new power series.
Plug in (+/−)xc , for some constant c, into the x of a power series in order to obtain a new
powers.
∗ Integrate/Differentiata a power series to obtain a new power series. (NOTE: Alterations to
1
the power series expansion of
in order to obtain new power series expansions of rational
1−x
functions and logarithmic functions are especially important.)
• Other Series:
1. Taylor Series: Yields an approximation to the function f (x) about a value a.
∞
X
f (k) (a)
f 00 (a)
f 000 (a)
k
– f (x) =
(x − a) = f (a) + f 0 (a)(x − a) +
(x − a)2 +
(x − a)3 + . . .
k!
2!
3!
k=0
– You should also know how to find the nth order Taylor expansion for f (x) and estimate the
associated error term.
– Also, know how to solve for n in a general nth order Taylor expansion for f (x) in order to
find the value for n that yields a desired error bound.
2. Maclaurin Series: Just a Taylor Series with a = 0.
2
– f (x) =
∞
X
f (k) (0)
k=0
k!
xk = f (0) + f 0 (0)x +
f 00 (0) 2 f 000 (0) 3
x +
x + ...
2!
3!
– The rest is just identical to the Taylor Series.
• Convergence Sets and the Radius on Convergence:
1. Convergence Sets: The is the set of all values of x where the series converges.
– Generally, you find the convergence set using the absolute ratio test and then find the x values
that yield ρ < 1.
– Examples:
|x|
= 0 then |x| is a constant and the limit is
∗ If simplified absolute ratio test yields lim
k→∞ k
less than one ∀x ∈ (−∞, ∞).
3k |x|
∗ If simplified absolute ratio test yields lim
, then 3 |x| < 1 if −1 < 3x < 1 which
k→∞
k
yields the convergence set x ∈ (−1/3, 1/3).
∗ If simplified absolute ratio test yields lim 3k |x|, then lim 3k |x| < 1 if x = 0 which
k→∞
k→∞
yields the convergence set x = 0.
– Do NOT forget to check the endpoints of the interval individually to determine if they are in
the convergence set.
2. Radius of Convergence: It is always half the length of the interval on convergence.
• Other Notes:
– I will expect that you know the integrals/derivatives of the basic (polynomial, rational, logarithmic, exponential, roots, trig, and inverse trig) functions.
– You may want to know the identities
cosh(x) =
ex + e−x
2
sinh(x) =
ex − e−x
2
– Then,
d (cosh (x))
= sinh(x)
dx
d (sinh (x))
= cosh(x).
dx
– As always, it is very important that you READ THE DIRECTIONS for every question.
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