Name ________________________________ Period _______
Score _______ / 10
AP Calculus AB QUIZ 5 - Topics 2.1-2.4
Definition of the Derivative
Part I. Multiple Choice
Choose the best answer for each problem. Each problem is worth 2 points.
You have 20 minutes.
f ( x  x)  f ( x)
is sometimes
x 0
x
f ( a  h)  f ( a )
presented as lim
.
h 0
h
1. The expression lim
Give the meaning of the expressions (i – iv) in
Question 1 by choosing from the possibilities (A-I)
below and writing the answer in the blank provided.
Each expression has only one correct answer. Refer to
the diagram to help you. (Each blank is worth ½ point.)
i. f ( a  h)
iii.
f ( a  h)  f ( a )
h
_____B____
____H____
ii. f (a  h)  f (a)
iv. lim
h 0
f ( a  h)  f ( a )
h
____D____
____I_____
(A) the y-value of Point P
(B) the y-value of Point Q
(C) the (horizontal) distance from a to a + h
(D) the (vertical) distance from f (a) to f (a + h)
(E) the (slant) distance from P to Q
(F) the slope of the secant line at P
(G) the slope of the secant line at Q
(H) the slope of the secant line joining P and Q
(I) the slope of the tangent line at P
(1  x) 2  (1)
equals f (c ) for some function, f ( x) , and some constant, c.
x 0
x
Determine f ( x) and c.
2. The limit lim
(A) f ( x)  x 2  1; c  1
(B) f ( x)  x; c  1
(C) f ( x)  x 2 ; c  1
(D) f ( x)  x 2 ; c  1
3. In the figure to the right, the graph of the function f ( x) is shown.
Arrange these values from smallest to largest.
1
f (1)  f  
4
I.
3
4
II. lim
f ( x)  f 1
x 1
x 1
(A) I, II, III
(B)
II, I, III
(D) III, II, I
(E)
II, III, I
III.
f (2)  f (1)
2 1
(C) III, I, II
4. Let f ( x)  x  2 . Which of the following gives the correct limit definition of the derivative of f ( x)
evaluated at x  6 ?
5.
x2 2 2
x6
x2 2 2
f '(6)  lim
x 6
x6
(A) f '(6) 
(B)
(C)
(D)
x2 2 2
x 2
x2
x2  6
f '(6)  lim
x 6
x6
f '(6)  lim
The function shown at the right is differentiable at which of the
following?
(A) x  1
(D) x  2 and x  3
(B) x  2
(C) x  3
(E) x  0 , x  3 , and x  4
Name ________________________________ Period _______
Score _______ / 10
AP Calculus AB QUIZ 5 - Topics 2.1-2.4
Definition of the Derivative
Part I. Multiple Choice
Choose the best answer for each problem. Each problem is worth 2 points.
You have 20 minutes.
f ( x  x)  f ( x)
is sometimes
x 0
x
f ( a  h)  f ( a )
presented as lim
.
h 0
h
1. The expression lim
Give the meaning of the expressions (i – iv) in Question
1 by choosing from the possibilities (A-I) below and
writing the answer in the blank provided. Each
expression has only one corrct answer. Refer to the
diagram to help you. (Each blank is worth ½ point.)
i. f ( a  h)
iii.
ii. f (a  h)  f (a)
____A_____
f ( a  h)  f ( a )
h
iv. lim
____I_____
h 0
f ( a  h)  f ( a )
h
____C____
_____H____
(A) the y-value of Point Q
(B) the y-value of Point P
(C) the (vertical) distance from f (a) to f (a + h)
(D) the (horizontal) distance from a to a + h
(E) the (slant) distance from P to Q
(F) the slope of the secant line at Q
(G) the slope of the secant line at P
(H) the slope of the tangent line at P
(I) the slope of the secant line joining P and Q
2. Let f ( x)  x  2 . Which of the following gives the correct limit definition of the derivative of f ( x)
evaluated at x  6 ?
(A) f '(6)  lim
x2 2 2
x6
(B)
f '(6)  lim
f '(6)  lim
x2 2 2
x2
(D)
f '(6) 
x 6
(C)
x 2
x 6
x2  6
x6
x2 2 2
x6
3. In the figure to the right, the graph of the function f ( x) is shown.
Arrange these values from largest to smallest.
1
f (1)  f  
4
I.
3
4
4.
II. lim
x 1
f ( x)  f 1
x 1
(A) III, I, II
(B)
I, II, III
(D) II, I, III
(E)
III, II, I
III.
f (2)  f (1)
2 1
(C) II, III, I
The function shown at the right is differentiable at which of the
following?
(A) x  0 , x  3 , and x  4
(B) x  2 and x  3
(C) x  3
(E) x  1
(D) x  2
(1  x) 2  (1)
5. The limit lim
equals f (c ) for some function, f ( x) , and some constant, c.
x 0
x
Determine f ( x) and c.
(A) f ( x)  x 2  1; c  1
(B) f ( x)  x 2 ; c  1
(C) f ( x)  x; c  1
(D) f ( x)  x 2 ; c  1