AP Calculus AB
Chapter 2 Review #2
Name:
Topic: Limits
1)
A)
2)
A)
3)
A)
4)
⎪⎧2x − 3 x ≤ 2
2
⎪⎩ x − 1 x > 2
f (x ) = ⎨
lim f (x ) =
B)
x →2+
f (x ) =
C)
lim f (x ) =
lim f (x ) =
C)
Conclusion:
x →2
−10x + 2
x2 +5
lim f (x ) =
B)
x →+ ∞
f (x ) =
lim f (x ) =
x →2−
x →− ∞
sin(2x )
x
lim f (x ) =
B)
x →+ ∞
lim f (x ) =
x →0
Use the values in the table to find lim f (x ) =
x →−1.8
x
f(x)
-1.83
-22.51
-1.82
-22.54
-1.81
-22.57
-1.8
-1.79
-22.63
-1.78
-22.66
-1.77
-22.69
5)
The graph of which of the following equations has y = 1 as an asymptote?
A)
y = cos x
6)
If lim f (x ) = L , where L is a real number, which of the following must be true?
I.
f (a ) = L
B)
y = ex
C)
y =
x2
x2 −5
x →a
A) I only
B) I and II
C) I and III II.
D) II and III
E) I, II, and III
lim f (x ) = L
x →a −
III.
lim f (x ) = L
x →a +
Topic: Sandwich or Squeeze Theorem
7)
⎛ π π⎞
, ⎟.
⎝ 2 2⎠
Suppose that 1 − x 4 ≤ f (x ) ≤ sec x for x ∈ ⎜ −
Find lim f (x ) and explain your reasoning.
x →0
Topic: Continuity
8)
⎧−x 2 + 8 x < 2
⎪
continuous at x = 2?
Is h (x ) = ⎨ 1
⎪ x +3 x ≥2
⎩2
9)
Let f be the function defined as follows:
10)
A)
⎪⎧ x − 3 + 1 x < 3
f (x ) = ⎨ 2
⎪⎩ax + bx x ≥ 3
If a = 3 and b = 2, is f continuous for all x? Justify your answer.
B)
Describe all values of a and b for which f is a continuous function.
x 3 − 2x 2 − 29x − 42
Let f (x ) =
. Which of the following statements is true?
x2 −9
B)
f (x ) has a removable discontinuity at x = -3
f (x ) has a jump discontinuity at x = 3
C)
If f (3) = −
D)
f (x ) has infinite discontinuities at x = -3 and x = 3
lim f (x ) = ∞
A)
E)
11)
5
, then f (x ) is continuous at x = 3
3
x →−3
What point would fix the removable discontinuity found in f (x ) =
4x − 8
?
x +x −6
2
Topic: Average vs Instantaneous Rate of Change
12)
Find the average rate of change for f (x ) = 6 − x 2 when x ∈ ⎡⎣ −2, 4 ⎤⎦ .
13)
Find the instantaneous rate of change for f (x ) = 6 − x 2 when x = −2 .
14)
Find the equation of the tangent line to f (x ) = 6 − x 2 when x = −2 .
15)
Find the equation of the normal line to f (x ) = 6 − x 2 when x = −2 .
16)
Find all values of x at which the slope of the function f (x ) =
Topic: Intermediate Value Theorem
17)
Is there a root for f (x ) = x 3 − 2x + 4 when x ∈ ⎡⎣ −1,2⎤⎦ ?
1
x
is equal to −
1
.
4