Name _____________________________________Per ______ Date __________________
Final Exam Review. Brief Calc. Fall 2014
1. Let
 x 2  4, x  1
f ( x)  
.
x 1
 1,
Determine the following limit. lim f ( x)
x 1
Ans: 5
2. Find the limit (if it exists):
lim
x  0
 x  x 
2
– 11 x  x  + 2   x 2 – 11x + 2 
x
1 3 11 2
x – x + 2x
3
2
B)
x 3 – 11x 2 + 2 x
C) 0
D)
2 x – 11
E)
x 2 – 11x + 2
Ans: D
A)
3. Graph the function with a graphing utility and use it to predict the limit. Check your
work either by using the table feature of the graphing utility or by finding the limit
algebraically.
x 3  2 x 2  24 x
lim 2
x 3 x  9 x  18
Ans: DNE
4. Find constants a and b such that the function
x  –9
24,

f ( x)  ax  b, –9  x  7
 –24,
x7

is continuous on the entire real line.
A) a = 3 , b = 0
B) a = 3 , b = –3
C) a = 3 , b = 3
D) a = –3 , b = 3
E) a = –3 , b = –3
Ans: E
5. Find the x-values (if any) at which the function f ( x) 
x–9
is not
x – 6 x – 27
2
continuous. Which of the discontinuities are removable?
A) no points of discontinuity
B)
x  9 (not removable), x  –3 (removable)
C)
x  9 (removable), x  –3 (not removable)
D) no points of continuity
E)
x  9 (not removable), x  –3 (not removable)
Ans: C
6. Find the limit: lim
x 13
x+7
.
x – 13
Ans: 
7. Determine the point(s), (if any), at which the graph of the function has a horizontal
tangent.
y ( x)  x 4  32 x  1
Ans: 2
8. The population P ( in thousands) of Japan from 1980 through 2010 can be modeled
by P  15.56t 2  802.1t  117, 001 where t is the year, with t =0 corresponding to
1980. Determine the population growth rate, dP dt .
A)
dP dt  31.12t  802.1
B)
dP dt  31.12t  802.1
C)
dP dt  31.12t  802.1
D)
dP dt  31.12t  802.1
E)
dP dt  31.12  802.1t
Ans: A
x3  6 x
9. Find the derivative of the function f  x  
.
3
Ans: f   x   x 2  2
10. Differentiate the given function.
y  5 x9  9 x
1/ 2
Ans: 1
5 x 9  9 x   45 x8  9 

2
11. Find an equation of the tangent line to the graph of f at the given point.
f ( s )  ( s  5)( s 2  6), at  3, –6 
Ans: y  –9 s + 21
12. A population of bacteria is introduced into a culture. The number of bacteria P can
4t 

be modeled by P  500 1 
where t is the time (in hours). Find the rate of
2 
 50  t 
change of the population when t = 2.
A)
31.55 bacteria/hr
B)
29.15 bacteria/hr
C)
33.65 bacteria/hr
D)
32.75 bacteria/hr
E)
30.25 bacteria/hr
Ans: A
13. Find the derivative of the function.
f ( x)  x8 (7  6 x) 4
A)
f ( x)  x 3 (7  6 x)7  56  72 x 
B)
f ( x)  6 x8 (7  6 x)3  56  72 x 
C)
f ( x)  x 7 (7  6 x) 4  56  72 x 
D)
f ( x)  x 7 (7  6 x)3  56  72 x 
f ( x)  x 7 (7  6 x)3  56  6 x 
Ans: D
E)
14. Find the derivative of the given function. Simplify and express the answer using
positive exponents only.
c( x)  3x x 7  5
Ans:

2 x
3 9 x 7  10
7
5


12
15. Find the f  6  x  of f 
A)
B)
C)
D)
E)
Ans:
4
 x    x 2  1
12 x 2  4
12 x 2  2
6x2  4
6x2  2
12 x 2  1
A
16. Find y implicitly for 6 x 9  y 9  3.
Ans:
6x8
y  8
y
2
.
17. Find the second derivative for the function f ( x) 
f ''( x)  0 .
A)
B)
C)
D)
E)
5x
and solve the equation
5x + 7
0
7
no solution
–7
1

7
Ans: C
18. Assume that x and y are differentiable functions of t. Find dx/dt given that x  2 ,
y  8 , and dy / dt  3.
y 2  x 2  60
A) 1.50
B) 5.33
C) 0.75
D) 24.00
E) 12.00
Ans: E
19. Volume and radius. Suppose that air is being pumped into a spherical balloon at a
rate of 4 in.3 / min . At what rate is the radius of the balloon increasing when the radius
is 7 in.?
A)
dr
4

dt 49
B)
dr
1

dt 7
C)
dr 49

dt 4
D)
dr
7

dt 4
E)
dr
1

dt 49
Ans: E
20. An airplane flying at an altitude of 5 miles passes directly over a radar antenna.
When the airplane is 25 miles away (s = 25), the radar detects that the distance s is
changing at a rate of 250 miles per hour. What is the speed of the airplane? Round your
answer to the nearest integer.
Ans: 255 mi/hr
21. Use the graph of y  f ( x) to identify at which of the indicated points the derivative
f '( x) changes from negative to positive.
A) (2,4)
B) (-1,2)
C) (-1,2), (5,6)
D) (5,6)
E) (2,4), (5,6)
Ans: B
22. Identify the open intervals where the function f ( x)  4 x 2 – 3 x + 2 is increasing or
decreasing.
Ans:
3

 3 
decreasing:  ,  ; increasing:  ,  
8

 8 
23. Find the x-values of all relative maxima of the given function.
y  13 x 3  4 x 2  12 x  8
A)
x0
B)
x6
C)
x4
D)
x2
E) no relative maxima
Ans: D
2
24. Locate the absolute extrema of the function f ( x)  –3 x – 6 x + 2 on the closed
 –2, 2 .
interval
Ans: absolute max: f(–1) = 5 ; absolute min: f(2) = –22
25. Medication. The number of milligrams x of a medication in the bloodstream t hours
4000t
x(t )  2
t  7 t  0 . Find the maximum value
after a dose is taken can be modeled by
of x. Round your answer to two decimal places.
A) 2.65 mg
B) 755.93 mg
C) 1663.04 mg
D) 8.20 mg
E) 1500.40 mg
Ans: B
26. Determine the open intervals on which the graph of f ( x)  8 x 2 – 7 x + 8 is concave
downward or concave upward.
Ans: concave upward on  ,  
27. Find the points of inflection and discuss the concavity of the function.
f ( x)  –5 x 3 + 4 x 2 – 3 x – 3
Ans:
4
4

inflection point at x  ; concave upward on  ,  ; concave downward on
15
15 

 4

 ,
 15 
28. Determine the dimensions of a rectangular solid (with a square base) with
maximum volume if its surface area is 361 square meters.
Ans:
19 6
19 6
square base side
; height
6
6
29. You are in a boat 2 miles from the nearest point on the coast. You are to go to point
Q located 3 miles down the coast and 1 mile inland (see figure). You can row at a rate of
1 miles per hour and you can walk at a rate of 2 miles per hour. Toward what point on
the coast should you row in order to reach point Q in the least time?
A)
B)
C)
D)
E)
Ans:
3 miles
8 miles
2 miles
1 mile
5 miles
D
30. A firm has total revenue given by R ( x)  600 x  95.5 x 2  x 3 dollars for x units of a
product. Find the maximum revenue from sales of that product.
Ans: $914
31. Find the limit.
5x2
lim
x x  6
Ans: - 
32. Find the limit.
5 x 2  3 x  14
lim
2
x 2  5 x  8 x
Ans: 5

8
33. Find the derivative of f ( x)  x 5 – 3e x .
Ans:
5
f ( x)   6 – 3e x
x
34. Find the derivative of the following function.
7
y  2  5e  x
Ans: y  35 x 6 e  x7
x
35. Find the equation of the tangent line to f ( x)  3 x  e at the point (0,1).
y  –4 x – 4
A)
y  4x – 4
B)
y  4x 1
C)
y  4x 1
D)
y  –4 x  1
E)
Ans: D
36. Find a function that satisfies the conditions f ( x)  x 4 , f (0)  8, f (0)  3 .
Ans:
1 6
f ( x) 
x 8 x  3
30