CALCULUS BC
WORKSHEET ON PRECALCULUS REVIEW
Work the following on notebook paper. All work must be shown.
Find the x- and y-intercepts and domain and range, and sketch the graph. No calculator.
2. y  4  x 2
1. y  2  x
 x  12 if x  1
4. y  
3x  3 if x  1
__________________________________________________________________________________
Find the asymptotes (horizontal, vertical, and slant), symmetry, and intercepts, and sketch
the graph. No calculator.
2 x2  8
x2  2 x  3
5. y  2
6. y 
x  16
x2
x
3. y 
x
__________________________________________________________________________________
Solve. No calculator.
x 2  x  20
3
2
0
7.   x  1 x  3 3x  1  0
8.  x  1 x  3  x  2  0
9.
x2  4
__________________________________________________________________________________
Solve. Give exact answers in radians, 0  x  2 . No calculator.
10. 2sin 2 x  3sin x  2  0
11. tan 2 x  3 tan x  0
12. 2sin 2 x  cos x  1  0
13. sin  2 x   sin x
14. cos  2 x   cos x
15. 2cos  3x   3  0
__________________________________________________________________________________
Solve. Show all steps. Use your calculator, and give decimal answers correct to three decimal places.
16. e3 x1  100
17. e2 x  e x 12  0
19. log3  x  4  log3  x 1  2
20. log 2  log3  log5 x    0
18.
13
4
2  ex
21. The number of students in a school infected with the flu t days after exposure is modeled by the
function P  t  
500
.
1  e3  t
(a) How many students were infected after four days?
(b) When will 200 students be infected?
__________________________________________________________________________________
Find the limit. No calculator.
2 y  2
x3
3x 2  4 x3
22. lim 3
23. lim
24. lim
x  3 x  27
x  5 x 3  2 x
y 0
y
Answers to Precalculus Review
1. x-int.:  2, 0 

y-int.: 0, 2

D =  , 2
R = 0,  
2. x-int.:  2, 0  ,   2, 0 
y-int.:  0, 2 
D =   2, 2
R = 0, 2
3. x-int.: none
y-int.: none
D =  x : x  0
R =  1 and 1
4. x-int.: 1, 0 
y-int.:  0,  3
D =  ,  
R =  ,  
5. Horiz.: y = 2
Vert.: x  4
Slant: none
y-axis symmetry
x-int.:  2, 0  ,   2, 0 
6. Horiz.: none
Vert.: x   2
Slant: y  x  4
No symmetry
x-int.:  1, 0  ,  3, 0 


1
y-int.:  0, 
2

1 
7.  , 1   , 3
3 
7 11
10.
,
6
6

5
13. 0, ,  ,
3
3
16. 1.202
19.
13
8
21. (a) 365.529 so 366 students
22.
1
27


3
y-int.:  0,  
2

8.  , 1  3,  
4
3
3
2 4
14. 0,
,
3
3
11. 0,

, ,
17. 1.386
9.   4,  2   2, 5
5
3
3
5 7 17 19 29 31
15.
,
,
,
,
,
18 18 18 18 18 18
12.

, ,
18. 0.223
20. 125
(b) t = 2.595 days
23. 
1
2 2
24. 
4
5
CALCULUS BC
WORKSHEET ON LIMITS
Work these on notebook paper.
1. Given that lim f  x   3, lim g  x   0, lim h  x   8, find the limits that exist. If the limit
x a
x a
x a
does not exist, explain why.
 f  x   h  x  
 a  lim
x a 
 f  x  
 b  lim
x a 
3 h  x
 c  lim
x a
 d  lim
x a
 e  lim
x a
 g  lim
x a
f  x
h  x
f  x
g  x
 f  lim
x a
2
1
f  x
g  x
 h  lim
x a
f  x
2 f  x
h  x  f  x
__________________________________________________________________________________
2. The graphs of f and g are given. Use them to evaluate each limit, if it exists. If the limit does not
exist, explain why.
 f  x   g  x  
 a  lim
x 2 
 f  x   g  x  
 b  lim
x 1 
 f  x  g  x  
 c  lim
x 0 
 d  xlim
1
g  x
x3 f  x 
 e  lim
x 2
 f  lim
x 1
3  f  x
f  x
__________________________________________________________________________________
Find the following limits. Show all steps. No calculator.
sin  2 x 
sin x
3. lim
4. lim 2
x

0
x 0
2x  x
x
5. lim
x 0
7. lim
x 0
x  sin x
x
3sin  4 x 
sin  3 x 
sin 2 x
x 0
x
6. lim
x2
x  0 1  cos x
8. lim
TURN--->>
x3  8
9. lim
x 2 x  2
1
1

11. lim 3  x 3
x 0
x
10. lim
x 3
12. lim sin x
x 
13. lim  tan x
14.
x 
2
15.
x 1  2
x 3
lim csc x
lim  sec x
x 
2
16. lim cot x
x  
x 0
____________________________________________________________________________
17. A patient receives a 150-mg injection of a drug every
four hours. The graph shows the amount f  t  of the
drug in the bloodstream after t hours. Find
lim f  t  and lim f  t 
t 12
x 12
and explain the significance of these one-sided limits.
____________________________________________________________________________
18. Explain the meaning of each of the following.
(a) lim f  x   
(b) lim f  x   
(c) lim f  x   8
x 3
x 
x 4
____________________________________________________________________________
 50 
19. Sketch the graphs of y  x , y   x , and f  x   x cos 
 on the same graph over the
 x 
 50 
x-interval from 1 to 1, and use the Squeeze Theorem to find lim x cos 
 . (Use your
x 0
 x 
calculator to graph.)
20. Sketch the graphs of y  1  x2 , y  cos x, and y  f  x  , where f is any continuous
function that satisfies the inequality 1  x2  f  x   cos x for all x in the interval
   ,   . What can you say about the limit of f x as x  0? Explain your
 


 2 2
reasoning.
21. If 1  f  x   x2  2 x  2 for all x, find lim f  x  . Show the work that leads to your
x 1
answer.
22. If 3x  f  x   x3  2, evaluate lim f  x  . Show the work that leads to your answer.
x 1
Answers to Worksheet on Limits
1. (a) 5
(e) 
(b) 9
3
8
2. (a) 2
(e) 16
3. 2
4. 1
5. 2
6. 0
7. 4
8. 2
9. 12
10.
(f) 0
(g) undefined
(b) dne
(d) 
(c) 2
(c) 0
(h) 
1
3
6
11
(d) undefined
(f) 2
1
4
11. 
1
9
12. dne
13.  
14, 
15. 
16.  
17. lim f  t   150mg and lim f  t   300mg
t 12
t 12
The left limit represents the amount of the drug just before the fourth injection. The right limit
represents the amount just after the fourth injection.
18. (a) As x approaches – 3 , f  x  increases without bound.
22. (b) As x approaches 4 from the right , f  x  decreases without bound.
(c) As x increases without bound, f  x  approaches 8.
19. 0
20. lim f  x   1 by the Squeeze Theorem.
x 0
21. lim f  x   1 by the Squeeze Theorem.
x 1
22. lim f  x   3 by the Squeeze Theorem.
x 1
CALCULUS BC
WORKSHEET ON INTERMEDIATE VALUE THEOREM
Work the following on notebook paper.
On problems 1 – 4:
(a) Determine if the Intermediate Value Theorem holds for the given value of k.
(b) If the theorem holds, find a number c for which f  c   k . If the theorem does not
hold, give the reason.
(c) Draw a sketch of the curve and the line y  k .
1
1
, [a , b]   0 , 2 , k 
1. f  x  
x 1
5
1
3
, [a , b]  0 , 2 , k 
2. f  x  
x 1
4
2
3. f  x   x  3x  4 , [a , b]  2 , 3 , k   2
4. f  x   9  x 2 ,  a, b     3, 1 , k  2
________________________________________________________________________________
5. Given the function f  x   x 2  2 x  5.
(a) Does f  x   7 somewhere on the interval  1, 3 ? Use the Intermediate Value Theorem to
show why or why not.
(b) Does f  x   12 somewhere on the interval  1, 3 ? Use the Intermediate Value Theorem to
show why or why not.
6. Use the Intermediate Value Theorem to show that x3  x  0 has a root in the interval  1, 2 .
7. One night in January, the outside temperature at midnight was 42°F. At 10 AM the next morning,
the temperature was 57°F.
(a) Must there have been a time between midnight and 10 AM when the temperature was 50°F?
Explain how you know.
(b) Must there have been a time between midnight and 10 AM when the temperature was 40°F?
Explain how you know.
(c) Could there have been a time between midnight and 10 AM when the temperature was 40°F?
Explain how you know.
8. One afternoon you were driving on Hwy 290, headed for College Station. At 2 PM, you were
driving 60 miles per hour. At 3 PM, you were driving 55 miles per hour.
(a) Must there have been a time between 2 PM and 3PM when you were driving 57 miles per hour?
Explain how you know.
(b) Must there have been a time between 2 PM and 3PM when you were driving 45 miles per hour?
Explain how you know.
(c) Must there have been a time between 2 PM and 3PM when you were driving 45 miles per hour?
Explain how you know.
Answers to Worksheet on Intermediate Value Theorem
1
3
1. (a) IVT does not hold because f  0   1 and f  2   .
1
1
does not lie between and 1.
5
3
1
3
3  17
  0.562
3. (a) IVT holds. (b) c 
2
4. (a) IVT holds. (b) c   5   2.236
2. (a) IVT holds. (b) c =
5. (a) f is continuous on  1, 3. f  1  6, f  3  10. 7 lies between – 6 and 10.
 f  x   7 somewhere on  1, 3 by the IVT.
(b) Since 12 does not lie between – 6 and 10, the IVT does not guarantee a value of c on  1, 3
Where f  c   12.
6. Let f  x   x3  x. f is continuous on  1, 2. f  1  2, f  2  10. Since 0 lies between – 2 and 10,
f  x   x3  x must equal 0 for some x on the interval  1, 2 by the IVT.
7. (a) Since the temperature is a continuous function and 50° lies between 42° and 57°, the IVT guarantees
a time between midnight and 10 AM when the temperature was 50°.
(b) Since 40° does not lie between 42° and 57°, the IVT does not guarantee a time between midnight and
10 AM when the temperature was 40°.
(c) Yes, the temperature could have dropped to 40° between midnight and 10 AM.
8. (a) Since the speed is a continuous function and 57 mph lies between 55 mph and 60 mph, the IVT
guarantees a time between 2 PM and 3 PM when the speed was 57 mph.
(b) Since 45 mph does not lie between 55 mph and 60 mph, the IVT does not guarantee a time between
2 PM and 3 PM when the speed was 45 mph.
(c) Yes, you could have slowed down to 45 mph between 2 PM and 3 PM.
CALCULUS BC
WORKSHEET ON CONTINUITY AND INTERMEDIATE VALUE THEOREM
Work the following on notebook paper.
On problems 1 – 4, sketch the graph of a function f that satisfies the stated conditions.
1. f has a limit at x = 3, but it is not continuous at x = 3.
2. f is not continuous at x = 3, but if its value at x = 3 is changed from f  3  1 to f  3  0 , it becomes
continuous at x = 3.
3. f has a removable discontinuity at x = c for which f  c  is undefined.
4. f has a removable discontinuity at x = c for which f  c  is defined.
__________________________________________________________________________________________
On problems 5 – 7, use the definition of continuity to prove that the function is discontinuous at the given value
of a. Sketch the graph of the function.
5. f  x  
x2  5x  4
x 1
, a 1
 x 2  3x
if x  3

6. g  x    x 2  9
a3
 1
if x  3

e x if x  0
7. h  x   
2

 x if x  0
a0
__________________________________________________________________________________________
On problems 8 – 9, use the definition of continuity to find the values of k and/or m that will make the function
continuous everywhere.
 kx ,
8. f  x   
2
 x 2  5,
x2

9. g  x   m( x  3)  k ,  1  x  2
 3
2 x  x  7, x  1
x2
2 x  k , x  2
__________________________________________________________________________________________
On problems 10 – 12, a function f and a closed interval [a, b] are given. Show whether the conditions of the
Intermediate Value Theorem hold for the given value of k. If the conditions hold, find a number c such that
f  c   k. If the theorem does not hold, give the reason. Whether the theorem holds or not, sketch the curve
and the line y = k.
10. f  x   2  x  x 2
11. f  x   25  x 2
12. f  x  
1
 a, b   0, 3
 a, b     4.5, 3
x2
 a, b   3, 5
k 1
k 3
k
5
6
__________________________________________________________________________________________
13. Use the Intermediate Value Theorem to show that f  x   x3  x takes on the value 9 for some x in [1, 2].
__________________________________________________________________________________________
On problems 14 – 15, use the Intermediate Value Theorem to show that there is a root of the given equation in
the specified interval.
14. x 4  x  3  0
1, 2
15. cos x  x
 
0, 2 
__________________________________________________________________________________________
16. Jesse and Kay ran a 1000-m race. One minute after the race began, Jesse was running 20 km/hr, and Kay
was running 15 km/hr. Three minutes after the race began, Jesse has slowed to 17km/hr, and Kay had
speeded up to 19 km/hr. Assume that each runner’s speed is a continuous function of time. Prove that there
is a time between 1 minute and 3 minutes after the race began at which each one was running exactly the
same speed. Is it possible to tell what that speed is? Is it possible to tell when that speed occurred? Explain.
Answers to Worksheet on Continuity and Intermediate Value Theorem
1. – 4. Answers will vary.
5. f 1 is undefined. Therefore, f is not continuous at x = 1.
1
6. g  3  1. lim g  x   . lim g  x   g  3 so g is not continuous at z = 3.
x 3
2 x 3
7. lim h  x  does not exist so h is not ccontinuous at x = 0.
x 0
4
3
5
2
9. m  , k 
3
3
8. k 
1 5
2
11. IVT holds. c   4
16
12. IVT holds. c 
5
13. f is continuous on [1, 2]. f 1  2, f  2   10, and 9 lies between 2 and 10 so f  x   9 for some x
in [1, 2] by the IVT.
14. Let f  x   x 4  x  3. f is continuous on [1, 2]. f 1  1, f  2   15, and 0 lies between – 1 and 15 so
10. IVT holds. c 
f  x   0 for some x in [1, 2] by the IVT, meaning that f has a zero in [1, 2].


 
 
15. Let f  x   cos x  x. f is continuous on 0,  . f  0   1, f     . 0 lies between  and 1.
2
2
 2
2
 
Therefore, f has a root (zero) on 0,  .
 2
16. Let f  x   Jesse’s speed and g  x   Kay’s speed.
Jesse
Kay
Time Speed
Time Speed
1
20
1
15
3
17
3
19
Sketch a possible graph. Both graphs must be continuous so by the IVT they will intersect and there will be
a time when their speeds are equal.
f  x   g  x  when f  x   g  x   0. Let h  x   f  x   g  x  .
When x = 1, h  x  = 20 – 15 = 5.
When x = 3, h  x  = 17 – 19 = – 2.
Since 0 lies between – 2 and 5, there must be an x on [1, 3] for which h  x  = 0.
It is not possible to tell what that speed is or when it occurred.