Name _____________________________________Per ______ Date __________________
Final Exam Review. Brief Calculus. Fall 2013
1. Let
 x 2  4, x  1
f ( x)  
.
x 1
 1,
Determine the following limit. lim f ( x)
x 1
2. Find the limit (if it exists):
lim
 x  x 
2
– 11 x  x  + 2   x 2 – 11x + 2 
x
x  0
A)
B)
C)
D)
E)
1 3 11 2
x – x + 2x
3
2
3
x – 11x 2 + 2 x
0
2 x – 11
x 2 – 11x + 2
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.
x3  2 x 2  24 x
lim 2
x 3 x  9 x  18
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
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)
6. Find the limit: lim
x 13
x+7
.
x – 13
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
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
9. Find the derivative of the function f  x  
x3  6 x
.
3
10. Differentiate the given function.
y  5 x9  9 x
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 
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
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 
E)
f ( x)  x 7 (7  6 x)3  56  6 x 
14. Find the derivative of the given function. Simplify and express the answer using
positive exponents only.
c( x)  3x x 7  5
15. Find the f 
A)
B)
C)
D)
E)
6
 x  of
f
4
 x    x 2  1
12 x 2  4
12 x 2  2
6x2  4
6x2  2
12 x 2  1
16. Find y implicitly for 6 x 9  y 9  3.
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
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)
B)
C)
D)
E)
1.50
5.33
0.75
24.00
12.00
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
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.
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)
2
22. Identify the open intervals where the function f ( x)  4 x – 3 x + 2 is increasing or decreasing.
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
2
 –2, 2 .
24. Locate the absolute extrema of the function f ( x)  –3 x – 6 x + 2 on the closed interval
25. Find the limit.
5x2
lim
x x  6
26. Find the limit.
5 x 2  3 x  14
lim
2
x 2  5 x  8 x
For #27 and #28 -
a) Write the ordered pair that represents vector AB , b) write as a column vector, c) write as
the sum of unit vectors, and d) find the magnitude of vector AB .
27. A(-8, 1), B(1, -3)
a. ________________
b. ________________
c. ________________
d._________________
28. A ( 2,-8,3), B ( -2, 0, 1)
a. ________________
b. _______________
c. _______________
d. ________________
29. Find an ordered triple that represents


 1
u  3v  2w  z
2
if



v  4,3,5 ; w  2,6,1 ; and z  3,0,4
29. ______________
30. Find the inner product (dot product) and state whether the vector is perpendicular. Write yes or no!
3,2,3  6,3,4
30. ____________
31. Find the cross product. Then verify that the resulting vector is perpendicular to the given vector. Write yes or no!
4,0,2   7,1,0
31. __________________
32. A glider is floating through the sky at 45 miles per hour with a heading of 50 degrees north of west when it meets
a head wind of 10 miles per hour blowing 45 degrees south of east. What is its resultant velocity?
32. __________________
33. Find the angle between the vectors (to the nearest tenth of a degree).
u = <-3, 10> and v = <-4, 4>
33. __________________