AP Calculus AB - Chapter 2 Summary
Lesson 2.1: Rates of Change and Limits A. Vocabulary: Term
Definition
Example or graph
Average speed of a
moving body
Instantaneous speed
Limit of a function
One-sided Limits
Two-sided limits
Example:
Sandwich (Squeeze,
Pinching) Theorem
AP Calculus AB – Ch. 2 Summary
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B. Properties of Limits: If L, M, c, and k are real numbers and and
then
Sum/Difference
Rule
Product Rule
Quotient Rule
Constant Multiple
Rule
Power Rule
C. Determine the limit algebraically (by simplifying & substitution): 1.
2.
3.
4. Let f(x) =
Find
______,
______,
______, f(1) = ______
D. Determine the limit numerically (by setting up a table of values): Use a table of value to find
1)
.
AP Calculus AB – Ch. 2 Summary
2)
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E. Application of Average Speed & Instantaneous Speed: 1. A rock is dropped from the top of a building. Its height (in feet) at time t (in seconds) is described by
the function h(t) = -16t2 + 80.
a) Draw a graph of the function. Label all relevant values.
b) Find the rock’s average speed during the first 2 seconds.
c) Estimate the rock’s instantaneous speed at t = 2.
2. Mrs. Vu drove across a bridge that is one-mile long. The speed limit on the bridge was 30 mph. It took
her exactly 2 minutes to get across. When she arrived at the other end of the bridge, a highway patrol
officer was waiting for her with a ticket. Why was that?
AP Calculus AB – Ch. 2 Summary
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Lesson 2.2: Limits Involving Infinity A. Vocabulary: Term
Definition
The line y = b is a horizontal asymptote of
the graph of a function f(x) if either
Example or graph
Horizontal Asymptote
The line x = a is a vertical asymptote of the
graph of a function f(x) if either
Vertical Asymptote
End Behavior Model
Practice:
x −1
.
x2 –1
a) Use graphs, tables, and algebra to find the limits.
1. Let f(x) =
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lim
f (x) =
x → −1−
lim
f (x) =
x →1+
lim
f (x) =
x → −∞
lim
f (x) =
x →∞
b) Identify any horizontal and vertical asymptotes in the function.
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c) Find a power function end behavior model for f.
2. Let f(x) = xex. Use the graph of y = f(1/x) to find
lim
lim
f (x) and
f (x) .
x →∞
x → −∞
AP Calculus AB – Ch. 2 Summary
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3. Read Theorem 5 (pg 67) and do Exploration 1 (pg 68).
Lesson 2.3: Continuity A. Vocabulary: Term
Continuity at a Point
Definition
Interior Point: A function y = f(x) is
continuous at an interior point c of its
domain if
Example or graph
Endpoint: A function y = f(x) is continuous at
a left endpoint a or is continuous at a right
endpoint b in its domain if
B. Types of Discontinuity: Removable
Jump
Infinite
Oscillating
Do Exploration 1 on pg 77.
AP Calculus AB – Ch. 2 Summary
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C. Intermediate Value Theorem for Continuous Functions: Corollary of the Intermediate Value Theorem:
Let f be a function which is continuous on the closed interval [a, b]. Suppose that the product f(a).f(b) < 0; then
there exists c in (a, b) such that f(c) = 0. In other words, f has at least one root in the interval (a, b)
Application:
1. Explain why the equation e-x = x has at least one solution.
2. Is any real number exactly 1 less than its cube?
AP Calculus AB – Ch. 2 Summary
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Lesson 2.4: Rates of Change and Tangent Lines A. Vocabulary: Term
Definition
Example or graph
Average rate of change
of a function
Slope of a curve
Instantaneous rate of
change
Tangent line to a curve
Normal line to a curve
Practice: 1. Let f(x) = x 2 – x −1.
a) Find the average rate of change of f(x) over the interval [-2, 3]. Show
what it means graphically.
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b) Find the instantaneous rate of change (slope of the curve) at x = 1.
c) Find the equation of the tangent line to the curve at x = 1. Show the
line on the graph.
AP Calculus AB – Ch. 2 Summary
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d) Find the equation of the normal line to the curve at x = 1. Show the line
on the graph.
e) At what point is the tangent to the graph of f(x) horizontal? Show the
line on the graph
2. What is the instantaneous rate of change of the volume V of a cube with respect to its side length s when the
side length is 2 inches?
AP Calculus AB – Ch. 2 Summary
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