Download Calculus BC Review Book

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Calculus BC Review Book
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Chapters 8 through 10
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5/20/2013
Candice Johnson & Destini Torres
About the AUTHOR
Destini Torres was born and raised in New York
City. She is going to college in the fall of 2013.
Candice Johnson was born in New York City and
raised in the Bronx. She is going to college in the
fall of 2013.
TABLE of CONTENTS
Chapter Eight
Page 2 through three
 Integration techniques
 L’ Hopital’s Rule
 Improper Integrals
Chapter Nine
Page three through seven
 Infinite Series (Converges verses Diverges)
Chapter Ten
Page eight through ten
 Parametric Equations
 Polar coordinates
1
Chapter 8: IntegratIon technIques, l’hopItal’s rule, and
improper integrals
Basic Integration and Differentiation Rules
Here is a list of integrals and derivatives that you should know:
Using Integration by Parts to Solve Integrals
To use integration by parts, follow the formula:
Identify u, v, du and dv within the integral. Plug in the vales and evaluate.
Example:
u= x
du= 1 dx
dv= ex dx
v= ex
2
Integrating Improper Integrals
Important Steps:
1.
2.
3.
4.
5.
Set a limit approaching to the upper end of the integral (for example, if the
integral is from 0 to infinity, set your limit as t approaching infinity).
Integrate your integrand as normal.
Substitute your upper limit and t.
Evaluate the limit for your answer.
If you get a constant L, the integral converges. If your answer is infinity, the
integral diverges.
Example:
L’ Hopital Rule
0
∞
0
∞
It is quite simple. Now, anytime a limit has a fraction of and
just take the
derivative of the top function alone and the bottom function. (DO NOT DO THE
QUOTIENT RULE)
EXAMPLE#1
EXAMPLE#2
=
=
=
=
= 4(-1) = -4
=
1
∞
=0
3
CHAPTER 9: Infinite series
9.1- Sequences
Theorem 9.1: If { } is a sequence such that f(n)= f
then
or every positive integer n,
Theorem 9.2: Properties of Limits of Sequences
and
c is any real number
1.
2.
3.
4.
Theorem 9.4: Absolute Value Theorem
If
, then
EXAMPLE
1, 4, 7, 10 The expression is 3n-2
.
lim 3𝑛 − 2 = ∞
𝑛→∞
DIVERGES
9.2- Convergent and Divergent Series
Sn = a1 + a2 +…+ an
If the sequence of partial sums {Sn} converges to S, then the series
converges. (the nth partial sum).
If {Sn} diverges, then the series diverges. (sum of the series)
Theorem 9.6: Convergence of a Geometric Series
Diverges:
Converges: 0
EXAMPLE
a= 3
r= 1/2
1
≥1
= 3( 2 )
1
4
Theorem 9.7: Properties of Infinite Series
If
and
then…
1.
2.
3.
Theorem 9.8 & Theorem 9.9: Limit of nth of a Convergent and a Divergent Series
If
converges, then
If
, then
.
.
9.3- Integral Test and P-series Test
Theorem 9.10: Integral Test
If f is positive, continuous, and decreasing for x ≥ 1 and
either both converge or both diverge.
Theorem 9.11: Convergence of p-series
=
1. converges: p › 1
2. diverges: 0 ‹ p ≤ 1
5
EXAMPLE
9.4- Direct Comparison Test and Limit Comparison Test
Theorem 9.12: Direct Comparison Test
If
converges, then
resembles
converges
=
If
diverges, then
diverges.
=n≥1
≤
DIVERGENT
Theorem 9.13: Limit Comparison Test
Suppose that
› 0,
› 0, and
Where L is finite and positive, then the two series and either both converge or
both diverge.
9.5 Alternating Series Test
If one or none of the
conditions are met, then the
series DIVERGES!
9.6 Ratio Test
6
9.7 Taylor Polynomials and Approximations
9.10 Taylor and Maclaurin Series
7
CHAPTER 10: parametric equations and polar coordinates
What are parametric functions?
If f and g are continuous functions of t on an interval t, then the equations:
𝒙 = 𝒇(𝒕) 𝒂𝒏𝒅 𝒚 = 𝒈(𝒕)
are called parametric functions.
A set of parametric equations on a graph is known as a plane curve, or C.
Examples:
x=2t3+2
y=4t2
x=3t2+ 5t
y=t4+7t2+9
How do you find the derivative of a parametric
function?
The slope of the curve C at point (x,y) is
dy = dy/dt
dx dx/dt
when dx/dt ≠ 0
This means that we must take the derivative of x=f(t) and y=g(t) with respect to t.
Examples:
1. What is the derivative of the set of parametric equations, x=2t3+2 and y=4t2
dy = dy/dt = 8t
dx dx/dt 6t2
2. Find dy/dx for the set of parametric equations, x=3t+5 and y=t2
dy = dy/dt = 2t
dx dx/dt
3
8
How do you find the second derivative of a
parametric function?
The second derivative of the curve C at point (x,y) is given by
d2y = d/dt [dy/dx]
dx2
dx/dt
We must take the derivative of dy/dx and the derivative of x=f(t).
Example:
1. Find d2y/dx2 for the set of parametric functions, x=3t+5 and y=t2
d2y = d/dt [dy/dx] = d/dt [2t/3] = 2/3 = 2
dx2
dx/dt
3
3
How do we find the tangent line to a parametric curve?
1.
2.
3.
4.
Find dy/dx for the set of parametric equations
Evaluate dy/dx at the given t value
Find your x and y values by plugging t into your parametric equations separately
Use the point slope formula or equation formula to find the tangent line
Example: Find the equation of the tangent line at the point x=t2+ 1 and y=t3+6t+5 at t=2
1. dy = dy/dt = 3t2+6
dx dx/dt
2t
3. x=t2+1 x=22+1 x=5
y=t3+6t+5  y=23+6(2)+5 y=25
2. dy = 3(2)2 + 6 = 18 = 9
dx
2(2)
4
4.
2
y-y1= m(x-x1)
y-25=9/2 (x-5)
y=9x – 95
2 2
How do we find the arc length of a parametric curve?
To find arc length, use the formula:
s=
dt
Example: Find the arc length of the curve, x=t2 and y=6t on the
interval, 0≤t≤1
S=
dt = 6.109
9
How do we graph a polar equation?
To graph a polar equation, use the POL mode on your calculator: Go to mode Pol
enter (adjust the window to fit the graph)
Example: r= 5cos(Ө)
How do we find the area of a polar equation?
The area of a polar graph is given by:
Where α= alpha, β=beta, and r=polar equation
Example: Find the area of the polar equation r=4sinӨ on the interval 0 ≤ Ө ≤ π
10
1. Ms. Zhao.com: “AP Calculus AB.” AP Calculus AB.
N.p.,n.d. Web. 18 May 2013
2. Larson, Ron, Robert P. Hostetler, and Bruce H. Edwards.
Calculus of a Single Variable. Boston: Houghton Mifflin,
2006. Print.
3. Paul’s Online Math Notes: “Paul’s Online Notes: Calculus
2- Parametric Equations and Polar Coordinates. “Paul’s
Online Notes: Calculus 2- Parametric Equations and Polar
Coordinates. N.p., n.d. Web. 21 May 2013.
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