11H – Final Review Packet
I – Limits
Evaluate the following limits.
1. limx→∞
x 2  5x 3  9
3x 3  5 x  1
2. limx→3
x3
x2  9
3. limx→0
sin x
x
4. limx→0
cos x  1
sin x
5. limx→4
2
x4
6. limx→2-
2
2x
(2 x  1)(3  x)
( x  1)( x  3)
(A) -3 (B) -2 (C) 2 (D) 3 (E) nonexistent
7. limx→∞
8. limx→0
(A)
5x 4  8x 2
is
3x 4  16 x 2
1
5
(B) 0 (C) 1 (D)
(E) nonexistent
2
3
0 x2
 ln x
9. If f (x) =  2
then limx→2 f (x) is
 x ln 2 2  x  4,
(A) ln 2
(B) ln 8
(C) ln 16
(D) 4
(E) nonexistent
10. For x ≥ 0, the horizontal line y=2 is an asymptote for the graph of the function f. Which
of the following statements must be true?
(A) f (0) =2
(B) f (x) ≠ 2 for all x ≥ 0
(C) f (2) is undefined.
(D) limx→2 f (x) = ∞
(E) limx→∞ f (x) = 2
II - Continuity and Differentiability
1. Let f be the function defined below. Which of the following statements about f are true?
I.
f has a limit at x = 2.
II.
f is continuous at x = 2.
III.
f is differentiable at x = 2.
2
x  4

f(x) =  x  2 x  2
 1
x2
(A) I only (B) II only (C) III only (D) I and II only (E) I, II, and III
2. Let f be the function given below. Which of the following statements are true about f ?
f (x) =
 x2 x3

4 x  7 x  3
I. limx→3 f (x) exists.
II. f is continuous at x = 3
III. f is differentiable at x = 3
(A) None
(B) I only
(C) II only
(D) I and II only
(E) I, II, and III
3. Let f be the function given by f(x) = | x |. Which of the following statements about f are true?
I. f is continuous at x = 0.
II. f is differentiable at x = 0.
III. f has an absolute minimum at x = 0.
(A) I only
(B) II only
(C) III only
(D) I and III only
(E) II and III only
III – Derivatives
Find
dy
for the following functions.
dx
1. y = sin(3x4)
3. y =
2. y = e3x(tan-1x)
7x  2
2x  1
4. y = ln(cosx)
5. 3x4 – 2xy + y3 – 4y = 20
Find
6. y = sec(lnx)
d2y
for the following.
dx 2
7. Y = xex
8. y = sin(5x)
9. x2 + y2 = 16
10. y = ln(4x – 3)

)=
9
3 3
3
 3
3 3
3
(A)
(B)
(C)
(D)
(E)
2
2
2
2
2
11. If f(x) = cos(3x), then f ′(
2
x
12. If f(x) = e , then f ′(x) =
2
2
2
(A) 2e x ln x (B) e x (C) e x
13. If y =
(A)
2
2 x
e (E) -2x2 e x
2
x
2
(D)
2
2x  3
dy
, then
=
3x  2
dx
12 x  13
(3x  2) 2
(B)
12 x  13
(3x  2) 2
(C)
5
(3x  2) 2
(D)
5
(3x  2) 2
14. If f(x) = x3 – 2 and f -1(x) = g(x), what is the value of g ′(-10)?
(E)
2
3
IV – Position, Velocity, Acceleration
1.
A particle moves along the x-axis so that its position at time t is given by
x(t) = t2 – 6t + 5. For what value of t is the velocity of the particle zero?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
2. A particle moves along the x-axis so that at any time t ≥ 0, its velocity is given by
v(t) = 3 + 4.1cos(0.9t). What is the acceleration of the particle at time t = 4?
(A) -2.016 (B) -0.677 (C) 1.633 (D) 1.814 (E) 2.978
(Calculator)
V – Derivative Tests
For the following functions find:
a) local max/min b) intervals of increasing/decreasing c) points of inflection
d) intervals of concave up/down
1. f(x) = 3x3 – 9x2
2. f(x) = -2x3 + 6x2 – 3
3. Let f be a function with a second derivative given by f ″(x) = x2(x – 3)(x – 6). What are
the x-coordinates of the points of inflection of the graph of f?
(A) 0 only (B) 3 only (C) 0 and 6 only (D) 3 and 6 only (E) 0, 3, and 6
4. The first derivative of the function f is defined by f ′(x) = sin(x3 – x) for 0 ≤ x ≤ 2. On
what intervals is f increasing?
(A) 1 ≤ x ≤ 1.445 only (B) 1 ≤ x ≤ 1.691 (C) 1.445 ≤ x ≤ 1.875
(D) 0.577 ≤ x ≤ 1.445 and 1.875 ≤ x ≤ 2 (E) 0 ≤ x ≤ 1 and 1.691 ≤ x ≤ 2
(Calculator)
5. The derivative of the function f is given by f ′(x) = x2cos(x2). How many points of
inflection does the graph of f have on the open interval (-2, 2)?
(A) One (B) Two (C) Three (D) Four (E) Five
(Calculator)
6. Let f be the function given by f(x) = 2xex. The graph of f is concave down when
(A) x < -2 (B) x > -2 (C) x < -1 (D) x > -1 (E) x < 0
7. What is the x-coordinate of the point of inflection on the graph of
1
y = x 3  5 x 2  24 ?
3
 10
(A) 5
(B) 0
(C)
(D) -5
(E) -10
3
8. The function f is given by f(x) = x4 + x2 - 2. On which of the following intervals is f
increasing?
1
1
(A) (
(C) (0, ∞)
(E) (,
, )
)
2
2
1 1
(B) (
(D) (-∞, 0)
,
)
2 2
9. If g is a differentiable function such that g(x) < 0 for all real numbers x and if
f ′(x) = (x2 - 4)g(x), which of the following is true?
(A) f has a relative maximum at x = -2 and a relative minimum at x = 2
(B) f has a relative minimum at x= -2 and a relative maximum at x = 2.
(C) f has relative minima at x = -2 and at x = 2.
(D) f has relative maxima at x = -2 and at x = 2
(E) It cannot be determined if f has any relative extrema.
VI – Derivative Graphs
1. The graph of f ′, the derivative of f, is shown below for -2 ≤ x ≤ 5. On what intervals is f
increasing?
(A) [-2, 1] only (B) [-2, 3] (C) [3, 5] only (D) [0, 1.5] and [3, 5]
(E) [-2, -1], [1, 2], and [4, 5]
2. The graph of the derivative of a function f is shown in the figure below. The graph has
horizontal tangent lines at x = -1, x = 1, and x = 3. At which of the following values of x
does f have a relative maximum?
(A) -2 only (B) 1 only (C) 4 only (D) -1 and 3 only (E) -2, 1, and 4
3. The table gives selected values of the velocity, v(t), of a particle moving along the x-axis.
At time t = 0, the particle is at the origin. Which of the following could be the graph of
the position, x(t), of the particle for 0 ≤ t ≤ 4?
t
v(t)
0
-1
1
2
2
3
3
0
4
-4
4.
5. The figure below shows the graph of f ‘, the derivative of a function f. The domain of f is
the interval [-4, 4]. Which of the following are true about the graph of f?
I.
II.
III.
At the points where x = -3 and x = 2 there are horizontal tangents.
At the point where x = 1 there is a relative minimum point.
At the point where x = -3 there is an inflection point.
(A) None (B) II only (C) III only (D) II and III only (E) I, II, and III
6. The graph of f ‘, the derivative of a function f, is shown below. Which of the following
statements are true about the function f?
I. f is increasing on the interval (-2, -1).
II. f has an inflection point at x = 0.
III. f is concave up on the interval (-1, 0).
(A) I only (B) II only (C) III only (D) I and II only (E) I, II, and III
7. The graph of the derivative of f is shown in the figure.
I. Suppose that f(3) = 1. Find an equation of the line tangent to
the graph of f at the point (3, 1).
II. Where does f have a local minimum? Explain briefly.
III. Estimate f “(2).
IV. Where does f have an inflection point? Explain briefly.
V. Where does f achieve its maximum on the interval [1, 4]?
VII – Theorems
1. The function f is continuous for -2 ≤ x ≤ 1 and differentiable for -2 < x < 1. If f(-2) = -5
and f(1) = 4, which of the following statements could be false?
(A) There exists c, where -2 < c < 1, such that f(c) = 0.
(B) There exists c, where -2 < c < 1, such that f ′(c) = 0.
(C) There exists c, where -2 < c < 1, such that f(c) = 3.
(D) There exists c, where -2 < c < 1, such that f ′(c) = 3.
(E) There exists c, where -2 ≤ x ≤ 1, such that f(c) ≥ f(x) for all x on the closed
interval -2 ≤ x ≤ 1.
2. The number c satisfying the Mean Value Theorem for f(x) = sin(x) on the interval [1, 1.5]
is: (Calculator)
(A) .995 (B) 1.058 (C) 1.239 (D) 1.253 (E) 1.399
3. The function f is continuous on the closed interval [-5, 5], and f(-2) = 6, f(1) = -3, and
f(4) = 6. Which of the following statements must be true?
(A) The equation f(x) = 0 has at least two solutions on the closed interval [-5, 5].
(B)
(C)
(D)
(E)
The equation f(x) = 0 has exactly two solutions on the closed interval [-5, 5].
The equation f ‘(x) = 0 has at least one solution on the closed interval [-5, 5].
The equation f ‘(x) = 3 has at least one solution on the open interval (1, 4).
The graph of f has at least one point of inflection on the closed interval [-5, 5].
VIII – Equation of Tangent Lines
1. What is the slope of the line tangent to the curve y = arctan(4x) at the point at which
1
x= ?
4
1
1
(A) 2 (B)
(C) 0 (D)
(E) -2
2
2
2. Let f be the function defined by f (x) = 4x3 - 5x + 3. Which of the following is an
equation of the line tangent to the graph of f at the point where x = -1?
(A)
(B)
(C)
(D)
(E)
y = 7x -3
y = 7x + 7
y = 7x + 11
y = -5x -1
y = -5x – 5
3. Let f be the function given by f(x) = 3e2x and let g be the function given by g(x) = 6x3.
At what value of x do the graphs of f and g have parallel tangent lines? (Calculator)
(A) -0.701
(C) -0.391
(E) -0.258
(B) -0.567
(D) -0.302
4. The function f is twice differentiable with f(2) = 1, f ′(2) = 4, and f ″(2) = 3. What is the
value of the approximation of f(1.9) using the line tangent to the graph of f at x = 2?
(A) 0.4 (B) 0.6 (C) 0.7 (D) 1.3 (E) 1.4
5. Let f(x) = 3x2 – 5x + 1. Use the tangent line at x = 2 to approximate the value of f(1.9).
IX – Related Rates
1. The radius of a sphere is decreasing at a rate of 2 centimeters per second. At the instant
when the radius of the sphere is 3 centimeters, what is the rate of change, in square
centimeters per second, of the surface area of the sphere? (The surface area S of a sphere
with radius r is S = 4  r2.)
(A) -108  (B) -72  (C) -48  (D) -24  (E) -16 
2. The edge of a cube is increasing at the uniform rate of 0.2 inches per second. At the
instant when the total surface area becomes 150 square inches, what is the rate of increase,
in cubic inches per second, of the volume of the cube?
(A) 5 in3/sec (B) 10 in3/sec (C) 15 in3/sec (D) 20 in3/sec (E) 25 in3/sec
3. Two cars start moving from the same point. One travels south at 60 mi/hr and the other
travels west at 25 mi/hr. At what rate is the distance between the cars increasing two
hours later?
X – Optimization
1. The sum of x and y is 8. What is the maximum value of xy – x2?
2. A farmer has 2400 ft of fencing and wants to fence off a rectangular field that borders a
straight river. He needs no fence along the river. What are the dimensions of the field that
has the largest area?
3. If y = 2x – 8, what is the minimum value of the product xy?
(A) -16 (B) -8 (C) -4 (D) 0 (E) 2
XI – Difference Quotient
 cos( x  h)  cos x 
1. limh→0 
=
h


(A) sin x (B) -sin x (C) cos x (D) -cos x (E) does not exist
f ( 6  h )  f ( 6)
= -2 which of the following must be true?
h
I. f ‘(6) exists.
II. f(x) is continuous at x = 6.
III. f(6) < 0
2. Given limh→0
(A) None (B) I and II only (C) I and III only (D) II and III only (E) I, II, and III
sin(
3. limh→0

6
 h)  sin(
h

6
)