AP CALCULUS AB
CHAPTER 1 AND 2
REVIEW
NO CALCULATOR
lim f (x)
x→2 +
lim f (x)
lim f (x)
x→2 −
lim f (x)
x→0 +
x→2
2
1
1
lim f (x)
x→1+
2
lim f (x)
x→1−
3
4
lim f (x)
x→3
lim f (x)
x→1
NO CALCULATOR
lim f (x) = 1
x→2 +
lim f (x) = 1
lim f (x) = 1
x→2 −
lim f (x) = 1
x→0 +
x→2
ANSWERS
2
1
1
lim f (x) = 1
x→1+
2
lim f (x) = 0
x→1−
3
4
lim f (x) = 2
x→3
lim f (x) = does not exist
x→1
NO CALCULATOR
Find the value of a making the piecewise function continuous
3
2x
x ≤1
f (x) =
ax − 1 x > 1
NO CALCULATOR
Find the value of a making the piecewise function continuous
3
2x
x ≤1
f (x) =
ax − 1 x > 1
Let: 2x3=ax-1 at x=1 so 2(1)3=a1-1
ANSWER: a=3
NO CALCULATOR
f(x)= x2+2x
a. f ‘ (x)=
b. What is the slope of the curve at (1,3)?
c. Write the equation of the tangent line at (1,3).
d. Write the equation of the normal line at (1,3).
e. For what value of x is the tangent horizontal?
NO CALCULATOR
f(x)= x2+2x
ANSWERS
a. f ‘ (x)=2x+2
b. What is the slope of the curve at (1,3)?
f ‘ (1)=4
c. Write the equation of the tangent line at (1,3).
y-3=4(x-1)
d. Write the equation of the normal line at (1,3).
y-3=-1/4(x-1)
e. For what value of x is the tangent horizontal?
x=-1
NO CALCULATOR
f ‘ (x)
a. For what value(s) of x does f(x) have a relative minimum? Justify your answer.
b. For what values of x is f(x) concave up? Justify your answer.
NO CALCULATOR
f ‘ (x)
ANSWERS
a.  For what value(s) of x does f(x) have a relative minimum?
At x=-3 and x=4.
where f ‘ (x)=0 and f ‘ (x) changes from negative to
positive.
b. For what values of x is f(x) concave up?
Where f ‘(x) is increasing.
(−∞,−1.2) (3,∞)
NO CALCULATOR
EVALUATE THE LIMITS
−3x 2
lim
=
x→−∞ 4 x + 5
x 4 + 3x
lim 2
=
x→∞ x + 1
x 2 + 3x
lim
=
x→−∞ 3x 2 + 1
3x 4 + 3x
lim 6
=
x→∞ x + 1
NO CALCULATOR
EVALUATE THE LIMITS
ANSWERS
−3x 2
lim
=∞
x→−∞ 4 x + 5
x 4 + 3x
lim 2
=∞
x→∞ x + 1
x 2 + 3x 1
lim
=
x→−∞ 3x 2 + 1
3
3x 4 + 3x
lim 6
=0
x→∞ x + 1
CALCULATOR ALLOWED
ex + 3
f (x) =
x+3
lim f (x) =
x→∞
lim f (x) =
x→−∞
Write the equation of any horizontal asymptotes.
Write the equation of any vertical asymptotes.
CALCULATOR ALLOWED
lim f (x) = ∞
ex + 3
f (x) =
x+3
ANSWERS
x→∞
lim f (x) = 0
x→−∞
Write the equation of any horizontal asymptotes.
Write the equation of any vertical asymptotes.
y=0
x=-3
Graph of f(x)
CALCULATOR ALLOWED
Find the average rate of change for x2-3x over the interval [-1,5].
What is the instantaneous rate of change for f(x) at x=-2?
CALCULATOR ALLOWED
ANSWERS
Find the average rate of change for x2-3x over the interval [-1,5].
f (5) − f (−1) 6
= =1
5 − −1
6
What is the instantaneous rate of change for f(x) at x=-2?
SINCE f ‘ (x)= 2x-3
f ‘ (x)=-7
NO CALCULATOR
g(x) has a local max at x=-.5 and a local min at x=1.9
g(x)
Where is g ‘ (x) positive?
Where is g ‘ (x) decreasing? (estimate your answer)
Rank from least to greatest: g(-.7), g ‘ (-.5), g ‘ (3)
NO CALCULATOR
g(x) has a local max at x=-.5 and a local min at x=1.9
g(x)
ANSWERS
(−∞,−.5) (2,∞)
Where is g ‘ (x) decreasing? (estimate your answer) (−∞,0)
Where g is concave up.
Where is g ‘ (x) positive?
Rank from least to greatest: g(-2), g ‘ (-.5), g ‘ (3)
g(-2)=-, g ‘ (-.5)=0, g’ (3)=+
NO CALCULATOR
Given a graph of f, sketch a graph of f’
NO CALCULATOR
Given a graph of f, sketch a graph of f’
f (x)
ANSWER
f ‘ (x)
NO CALCULATOR
Given a graph of g ‘, sketch a graph of g
g ‘ (x)
NO CALCULATOR
Given a graph of g ‘, sketch a graph of g
ANSWER
g ‘ (x)
g (x)