Greetings AP Calculus AB Class,
I am ecstatic to have the opportunity to teach you all next school year. In fact, I am so excited, I
have already compiled a homework assignment for you. Enclosed is said assignment. It consists of a
few review problems to keep you fresh for next school year. They should be fairly easy for you;
however, do not fret if there is something you do not understand. I will be using this to gauge our
starting point for the course. Since this is primarily for feedback purposes, I have included the answers
at the end of the document. By all means, check your work, but please do not look at the answers
before attempting the problems. You may check your graphs using Desmos (also available as an app).
Feel free to work together if you would like, but everyone make their own answers. Please attempt all
the problems showing your work and have ready a list of any problems that gave you any issue, and I
will have some homework credit ready for you come August.
I look forward to our epic mathematic journey and hope you do as well. Enjoy your summer,
and feel free to contact me with any questions, comments, or concerns.
Sincerely,
--Dr. Hamilton
[email protected]
Chapter 1
1. (AB/BC, non-calculator)
The function f is defined as follows:
f ( x)
x2 5x 6
.
2x2 7 x 3
(a) State the value(s) of x for which f is not continuous.
(b) Evaluate lim f ( x) . Show the work that leads to your answer.
x
3
(c) State the equation(s) for the vertical asymptote(s) for the graph of y
(d) State the equation(s) for the horizontal asymptote(s) for the graph of y
work that leads to your answer.
f ( x).
f ( x) . Show the
2. (AB/BC, calculator neutral)
y
y
x
x
Graph of f
Graph of g
The graphs of functions f and g are shown above. Evaluate each limit using the graphs provided.
Show the computations that lead to your answer.
(a) lim f ( x) 4
x 1
(b) lim
x
3
5
g ( x)
(c) lim f ( x) g ( x)
x
2
(d) lim
x
3
f ( x)
(Assume that f and g are linear on the interval
g ( x) 1
[2,3]).
3. (AB/BC, calculator neutral)
A hot cup of tea is placed on a counter and left to cool. The temperature of the tea, in degrees
Fahrenheit (correct to the nearest degree), x minutes after the cup is placed on the counter is
modeled by a continuous function T ( x) for 0 x 10 . Values of T ( x) at various times x are
shown in the table below.
x
0
3
4
6
8
9
T ( x)
180
174
172
168
164
162
(a) Evaluate: lim T ( x) . Justify your answer.
x
4
(b) Using the data in the table, find the average rate of change in the temperature of the tea for
3 x 8 . Include units on your final answer.
(c) Identify, using the times listed in the table, the shortest interval during which there must exist
a time x for which the temperature of the tea is 166.5 ? Justify your answer.
(d) Use the data in the table to find the best estimate of the slope of the line tangent to the graph
of T at x 8 .
4. (AB/BC, non-calculator)
Find the value of each of these limits, or else explain why the limit does not exist. Show the
computations which lead to your answers.
(a) lim
x
3x 4 6 x3 x 2 x 1
2 x3 9 x 4 5
(b) lim
x
0
x 5
5x
2 2 cos 2 x
0
x sin x
(c) lim
x
5 5
(d) lim x 2
x
2 x
2
5
5. (AB/BC, non-calculator)
The position function s (t )
4.9t 2 396.9 , gives the height (in meters) of an object that has
fallen from a height of 396.9 meters after t seconds.
(a) Explain why there must exist a time t , 1 t
382 meters above the ground.
2 , at which the height of the object must be
(b) Find the time at which the object hits the ground.
(c) Find the average rate of change in s over the interval t
8,9 . Include units of measure.
Explain why this is a good estimate of the velocity at which the object hits the ground. How can
this estimate be improved?
(d) Evaluate: lim
t
3
s(t ) s (3)
. Show the work that leads to your answer. Include units.
t 3
6. (AB/BC, calculator neutral)
Let a and b represent real numbers. Define
ax 2
f ( x)
x b if x
ax b if 2
2
x 5 .
2ax 7 if x 5
(a) Find the values of a and b such that f is continuous everywhere.
(b) Evaluate: lim f ( x) .
x
(c) Let g ( x)
3
f ( x)
.
x 1
Evaluate: lim g ( x) .
x 1
7. (AB/BC, calculator neutral)
y
x
The graph of function g is shown above. Which of the following is true? I. lim g ( x) 1
x
2
II. lim g ( x)
x
2
g (2)
III. g is continuous at x 3 .
(a) I only
(b) I and II only
(c) I and III only
(d) III only
(e) I, II and III
8. (AB/BC, calculator neutral)
y
x
The graph of the function f is shown above. The line x 1 is a vertical asymptote. Which of the
following statements about f is true?
(a) lim
x 1
(b) lim 1
x
3
(c) lim f ( x)
x
3
lim f ( x)
x
3
(d) lim f ( x) does not exist
x
4
(e) lim f ( x)
x
0
lim f ( x)
x
3
9. (AB/BC, non-calculator)
Define f ( x)
x 2 4 x 32
if x
x2 2x 8
8
if x
2, 4
.
4
Which of the following statements about f are true?
I. f is not continuous at x
II. lim f ( x)
x
III. x
(b) I only
(c) I and II only
(d) I and III only
4
4 is a vertical asymptote of the graph of y
(a) None
(e) I, II and III
4.
f ( x) .
10. (AB/BC, calculator neutral)
y
x
The figure above shows three rectangles each with a vertex on the graph of y 16 x 2 . The sum
of the areas of these rectangles is
(a) 42 sq. units
(b) 40 sq. units
(c) 34 sq. units
(d) 33 sq. units
(e) 29 sq. units
Chapter 1 (Solutions)
Question 1
The function f is defined as follows: f ( x)
x2 5x 6
.
2 x3 7 x 2 3x
(a) State the value(s) of x at which f is not continuous.
(b) Evaluate lim f ( x) . Show the work that leads to your answer.
x
3
(c) State the equation(s) for the vertical asymptote(s) for the graph of y
f ( x).
(d) State the equation(s) for the horizontal asymptote(s) for the graph of y
f ( x) . Show the
work that leads to your answer.
(a) f ( x)
x 3 x 2
x 2x 1 x 3
f is discontinuous at x
x 2
3 x (2 x 1)
1
15
(b) lim
x
(c) x
1
; x
2
0
1
; x
2
3 and x
0
3: 1 per answer
2 : 1: reduced fraction
1: answer
2: 1 per answer
Question 1 (cont.)
x2 5x 6
(d) lim 2
x
2x 7x 3
1
2
2 : 1: limit expression
1: answer
Question 2
y
y
x
x
Graph of f
Graph of g
The graphs of functions f and g are shown above. Evaluate each limit using the graphs
provided. Show the computations that lead to your answer.
(a) lim f ( x) 4
x 1
(b) lim
x
3
5
g ( x)
(c) lim f ( x) g ( x)
x
2
(d) lim
x
3
f ( x)
(Assume that f and g are linear on the interval [2,3]).
g ( x) 1
(a) lim f ( x) 4 = 3 4 7
x 1
(b) lim
x
3
5
5
=
g ( x) 1
5
2 : 1: breaking up limit
1: answer
2 : 1: breaking up limit
1: answer
Question 2 (cont.)
2 : 1: breaking up limit
1: answer
(c) lim f ( x) g ( x) = (2)(0) 0
x
2
lim
(d)
x
3
f ( x)
g ( x) 1
2x 6
3 ( x 2) 1
x 3
2 lim
x 3 x 3
lim
x
2
3 : 2: algebraic representations 1: factoring and answer
Question 3
A hot cup of tea is placed on a counter and left to cool. The temperature of the tea, in degrees
Fahrenheit (correct to the nearest degree), x minutes after the cup is placed on the counter is
modeled by a continuous function T ( x) for 0 x 10 . Values of T ( x) at various times x are
shown in the table below.
x
0
3
4
6
8
9
T ( x)
180
174
172
168
164
162
(a) Evaluate: lim T ( x) . Justify your answer.
x
4
(b) Using the data in the table, find the average rate of change in the temperature of the tea for
3 x 8 . Include units on your final answer.
(c) Identify, using the times listed in the table, the shortest interval during which there must exist
a time x for which the temperature of the tea is 166.5 ? Justify your answer.
(d) Use the data in the table to find the best estimate of the slope of the line tangent to the graph
of T at x 8 .
(a) Since T is continuous for 0
x
4
2 : 1: justification
1: answer
lim T ( x) 172
x
x 10 , lim T ( x) T (4).
4
(b)
T (8) T (3)
8 3
2
F
min
1: setup
3 : 1: answer
1: units
Question 3 (cont.)
(c) By the Intermediate Value Theorem, the
temperature must be 166.5 for some time 6
since T (6) 166.5
(d)
T (9) T (8)
9 8
x 8
2 : 1: answer
1: justification
T (8) .
2
2 : 1: setup
1: answer
Question 4
Find the value of each of these limits, or else explain why the limit does not exist. Show the
computations which lead to your answers.
(a) lim
x
3x 4 6 x3 x 2 x 1
2 x3 9 x 4 5
x 5
5x
(b) lim
x
0
5
2 2 cos 2 x
(c) lim
x 0
x sin x
5 5
(d) lim x 2
x
2 x
2
1
3
(a)
(b) lim
x
0
1: answer
x
5x
(c) 2 lim
x
0
sin x
x
5
x
0
1
5
x 5
5
1
10 5
3 : 2 : simplify
1: answer
2 : 1: simplify
1: answer
2
2 x
2 2 x( x
2)
(d) 5 lim
x
x 5
lim
5
1
lim
2x 2x
5
4
3 : 2 : simplify
1: answer
Question 5
The position function s (t )
4.9t 2 396.9 , gives the height (in meters) of an object that has
fallen from a height of 396.9 meters after t seconds.
(a) Explain why there must exist a time t, 1 t
meters above the ground.
2 , at which the height of the object must be 382
(b) Find the time at which the object hits the ground.
(c) Find the average rate of change in s over the interval t
8,9 . Include units of measure.
Explain why this is a good estimate of the velocity at which the object hits the ground. How can
this estimate be improved?
(d) Evaluate: lim
t
3
s (t ) s (3)
. Show the work that leads to your answer. Include units.
t 3
(a) s (1) 382 s (2) . The function s (t ) is a polynomial, and is
therefore continuous. Therefore by the Intermediate Value Theorem,
there exists a value t, 1 t
(b) 0
(c)
4.9t 2 396.9
s (9) s (8)
9 8
the interval 8 t
t
83.3
1: explanation
2 , for which s (t ) 382 .
9s
1: answer
m
. This is the average velocity on
sec
9 . Since the object hits the ground at t
9,
1: answer
4 : 1: units
2: 1 per explanation
this average velocity is close to the instantaneous velocity
at t = 9. The estimate
s (9) s(t )
can be improved by allowing the value of t to approach 9.
9 t
Question 5 (cont.)
(d) lim
t
3
s (t ) s (3)
t 3
4.9(t 3)( t 3)
t 3
29.4
m
sec
1: work
3 : 1: answer
1: units
Question 6
Let a and b represent real numbers.
ax 2
Define f ( x)
x b if x
ax b if 2
2
x 5 .
2ax 7 if x 5
(a) Find the values of a and b such that f is continuous everywhere.
(b) Evaluate: lim f ( x) .
x
(c) Let g ( x)
3
f ( x)
.
x 1
Evaluate: lim g ( x) .
x 1
(a)
4a 2 b 2a b
5a b 10a 7
2; b 3
1: limits
4 : 1: equations
2: answers
2 : 1: correct interval
1: answer
(b) 9
2x2 x 3
(c) lim
x 1
x 1
a
5
2 : 1: correct interval 1: answer
Questions 7-10
7. c
g (2) 3 lim g ( x) 1
8. e
both of these limits equal 2
9. b
x
lim f ( x)
x
4
2
2
lim f ( x) 1 so the horizontal asymptote is y 1
x
10. c
15 + 12 + 7 = 34