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Mr. Visca – AP Calculus ABUnit 2: Limits and Continuity Unit Review Notes

Average Speed

ratio: =

Instantaneous Speed

Definition of a Limit

Find the limits:

1 sided vs 2 sided limits

Limits as x approaches +/- infinity

Continuity

Given the graph below, is f(x) continuous at x = 1? x = ½?

Types of Discontinuity

Hole, jump, infinite (asymptote)

Intermediate Value Theorem for Continuous Functions

If a function is continuous from [a,b], then there is a value in-between, x = c, where f(c) exists

Average Rate of Change:

Instantaneous Rate of Change:

lim h→ 0

f (x+h )−f (x)h

Equation of a line:

Slope-intercept form:

Point-slope form:

Normal line:

Find the equation of the tangent line and normal line of the equation f(x) = 4 - x2, at the point x = 1.

Meh…

Part 1: No Calculator Allowed. Unit 2 REVIEW

Part A: A calculator may not be used on this part of the test.

1. limx→−3

( x5−x+5 )=

2. If c≠0 , then find limx→ c

x−cx2−c2

3. limx→2

2x−2

=

4. Find the horizontal asymptote of

4 x2+6 xx2−9 .

5. limx→0

sin x4 x2−8 x

=

6. What is limx→∞

x3−2 x2+3 x−4−1+2 x−3 x2−9x3

?

7. The graph of a function f is shown below. At which values of x (a,b,c,d,e) is f continuous?

8.

The graph of the function f is shown above. Which of the following statements is true?

(A) limx→af ( x )= lim

x→bf ( x )

(B) limx→af ( x )=2

(C) limx→bf ( x )=2

(D) limx→bf ( x )=1

(E) limx→af ( x ) does not exist .

9. Let f ( x )=¿ {3 x+6 if x≤5 ¿¿¿¿

Find limx→5f ( x )

Is f continuous at x = 5?

10. If f ( x )=¿ {ex for 0<x≤3 ¿ ¿¿¿

find limx→3f ( x )

11. What is the average rate of change of the function f given by f ( x )=2 x2+6 x+10 on the closed

interval [1, 5]?

12. Given the function f ( x )=x2−4 x+8 , use the intermediate value theorem to show that the equation

f ( x )=5 must have at least two solutions in the interval [-1, 4].

Part 2: A graphing calculator is required for some of the questions on this part of the test.

13. Let f be the function given by f ( x )=x6 3x . Find

limx→∞

f ( x ) and

limx→−∞

f ( x ).

14. Consider y=2x2+6 x when x = –4.

a) Find the slope of the curve at that point.

b) Write the equation of the tangent line at that point.

c) Write the equation of the normal to the curve at that point.