Calculus BC Review for 9 weeks Exam
CHAPTER 1
1. Given: lim f  x  
x c
3
1
and lim g  x  
x

c
2
2
a) Find lim  4 f  x  
x c
b) Find lim  f  x  g  x  
x c
c) Find lim  f  x   g  x  
x c
d) Find lim
x c
f  x
g  x
2 a. Explain (be specific) what is meant by the equation:.
lim f ( x)  4
x2
b. Is it possible for the above statement to be true and yet f (2)  1 ? Explain.
3. Write the three conditions that must be met for a function to be continuous at a point c:
1.
2.
3.
4. Given: lim 𝑓(𝑥) = ∞
𝑥→3
and lim ℎ(𝑥) = −∞ and lim 𝑔(𝑥) = 5 and lim 𝑘(𝑥) =
𝑥→3
𝑥→3
−4
Find
a) lim[𝑘(𝑥)𝑓(𝑥)] =
b) lim [𝑘(𝑥)ℎ(𝑥)] =
𝑥→3
𝑥→3
c) lim[𝑓(𝑥) + 𝑔(𝑥)] =
d) lim[𝑘(𝑥)𝑓(𝑥)] =
c) lim[ℎ(𝑥) − 𝑔(𝑥)] =
d) lim[ 𝑓(𝑥)] =
𝑥→3
𝑥→3
𝑔(𝑥)
𝑥→3
𝑥→3
5. For the function f whose graph is shown, state the following:
f(x)
a.
lim 𝑓(𝑥)
𝑥→2+
b. lim− 𝑓(𝑥) c. lim 𝑓(𝑥)
𝑥→2
𝑥→2
𝑥→3
d. 𝑓(2)
x2  6 x  9
6. Describe the type of discontinuities for f ( x)  2
x  3x  18
.
if x=1
4,
 2
7. Let f ( x)   x -1
. Describe where f(x) is continuous. (3 step process) If
, if x  1

 x-1
there are any discontinuities describe the type.
8. Use the three step process to determine if f(x) is continuous at x=2:
 4  x, x  2
f ( x)  
2x+2, x>2
9. Find the constant, a, so that the function is continuous on the entire real number
line:
5a  2 x, x  0

g ( x)  10sin x
 x , x<0
10. Verify the Intermediate Value Theorem holds true for the given information:
f ( x)  x2  6x  8 on 3,5 and f (c)  0
Find the value, c, guaranteed by the Intermediate Value Theorem
Show step by step work
11. 6 + 𝑥 2 − 3𝑥 ≤ 𝑓(𝑥) ≤ 3𝑥 − 2 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 lim 𝑓(𝑥)
𝑥→2
Find the limit. Show all necessary work. Some of the problems you may want to sketch a
graph.
12.
13.
14.
1
1
x
lim

2
lim

lim
x  2 x  2


x


x 3 x  3
cos x
15. lim x2  4
x 3
17.
lim
x 0
16. lim
x 5
5 x
x 2  25
18.
x4 2
x
1
1

lim x  3 3
x 0
x
sin 7 x
x 0
3x
19.
20. lim
 x  x 
lim
x 0
2
 x2
x
cos 𝑥
21. lim
𝜋
𝑐𝑜𝑡𝑥
𝑥→
2
sin 2𝑥
22. lim sin 3𝑥
𝑥→0
CHAPTER 2
1. f ( x)  2 x 2  1
a) Find the derivative by the LIMIT PROCESS.
b) Find 𝒇′ (𝒙) at c=2 using the alternate definition of the derivative.
Find the y for numbers 2 through 15.
2. y  x3  3x 2  7 x  2 x 3
3. y 
cos t
t3
x5  4 x3  2 x
x4
4. y 
3
4x2
5. y 
6. y 
2 x2  1
x 1
7. y  x 2 tan x
8. y  sin 2 5x
3
9. y  cot  2 x 
10. y  csc x
11.
y  sec3 x
12. y  3 4 x3  5
13. Evaluate the derivative of f ( x)  4sin x  x at the point  0,0

14. Find the equation of the tangent line for f ( x)  tan x at the point  ,1
4


15. Determine the points at which y  x 4  8x 2  2 has a horizontal
tangent line. (4pts)
multiple choice
1
x
16. Let f  x   x  . Find f   x  .
A. 1 
1
x2
B. 
17. y  cos4 (6 x  5)
2
x3
C.
2
x3
D. 1 
1
x2
E. Does Not Exist
Sections : 2.5, 2.6, 5.1, 5.3, 5.4, 5.5
dy
Find
by implicit differentiation:
dx
1. y 3  y 2  x 2  36
2. x 2  xy  y 2  4
3. sin x  x 1  tan y 
4. y  sin  xy 
dy
dy
by implicit differentiation and evaluate
at the given point:
dx
dx
5. x2  y3  0, 1,1
Find
Find an equation of the tangent line to the graph at the given point:
2
2
6.  x  1   y  2   20,  3,4 
d2y
Find
by implicit differentiation: (5pts)
dx 2
7. y 2  x 2  36
Identify the quantities for the “given and find” Show all work.
8. A spherical balloon is inflated with gas at the rate of 35 cubic ft per minute. How fast
4


is the radius of the balloon increasing at the instant the radius is 3 ft?  V   r 3 
3


Given:
Find:
________________________________________________________________________
9. A paper cup, which is in the shape of a right circular cone, is 16 cm deep. Water is
poured into the cup at a constant rate of 2 cm3 / sec . At the instant the radius is 3 cm,
1


what is the rate of change of the radius?  V   r 2 h 
3


Given:
Find:
10. A 8 meter ladder is leaning against a house. The foot of the ladder is pulled away
from the house at a rate of 0.2 m/sec. (Draw a picture and determine an equation)
a. Determine how fast the top of the ladder is descending when the foot of the ladder is 3
meters from the house.
Given:
Find:
b. Find the rate at which the angle between the ladder and the wall of the house is
changing when the base of the ladder is 3 meters from the house.
11. f  x   x5  3x3  2x  1, find (f 1 )'(1).
12. f  x   sin x, -

2
x

1
find (f 1 ) '   .
2
2
For numbers 13 – 22 do the following
Find the derivative of the following functions.
Factor out the common factors.
Get common denominator unless otherwise instructed
Use proper notation.
Show all work.
13. y  log 7 9 x
14. y  x  ln x 
15. y  2 x  7 3 x 
16. y  ln  7e3 x 
 x 
17. y  ln  2 
 x 1 
18. y  e x
3
2
3
19. y  x 2e x
20. y  log 2

21. y   e2t  e2t 
3
x
x 1
22. y  ln x x 2  1

Circle the letter of the choice that represents the derivative:
 ex 
23. y  ln  x 
 e 1 
ex
A. x  x
e 1
1
x
e 1
C.
1
x
e 1
C. tan x 
sec2 x
tan x
D.
B. 
D. 0
E.
ex  2
ex 1
24. y  ln  sec x  tan x 
A.
1
sec x
B. sec x
1
sec x  tan x
E. 
1
sec x  tan x
25. y  e x cos 2 x
A. e x  cos 2 x  sin 2 x 
B. e x  sin 2 x  cos 2 x 
D. e x  cos 2 x  2sin 2 x 
C. 2e  x sin 2 x
E. e x sin 2 x
26. If f  x   ln x3 , then f   3 is
A. 3
B. 1
C. 
1
3
D. 1
E. None of these
27. Using implicit differentiation, find the derivative of sin x  cos y  2  0
cos x
2  cos x
A.  cot x
B.  cot y
C.
D.  csc y cos x
E.
sin y
sin y
28. Find the derivative of: y 
e x  e x
(MUST SHOW WORK)
e x  e x
A. 0
B. 1
C.
e
4
x
e

x 2
D.
e
2
x
e

x 2
E.
1
e  e 2 x
2x