Name _____________
Period ____
Honors Brief Calc.
Semester 2 Final Exam Review #3
1. Which of the following is false?
a) lim  2
b) lim  4
c) lim  3
d) lim  1
x 1
x 1
x 0
x 1
2. If, for all values of x, f ' ( x)  0 and f ' ' ( x)  0 , which of the following curves could
represent a part of the graph of f?
a)
b)
c)
d)
3. Find the slope of the tangent line to the graph of f ( x)  x 5  7 x at x = 1.
a) 1
b) -6
c) -2
d) 5
4. Find the area enclosed by f ( x)  1  x 3 , g ( x)  x 2  1 , and x = 1.
a) 3.33
b) 4
c) 2
d) 4.67
5. The relationship between profit P of a firm and the selling price x of its goods is
P  25 x  x 2 . What selling price is required to generate a maximum profit?
a) $25.00
b) $50.00
c) $75.00
d) $12.50
6. Find the relative(local) maximum of y  2 x 3  15x 2  36 x
a) (1, 23)
b) (2, 28)
c) (3, 27)
d) (2, 28) and (3, 27)
7. If f ' ( x)  x 3  x , find f (x)
3
a)
c)
1 4 2 2
x  x C
4
3
b)
x4
2

C
4
x
2
8. Evaluate
1 4 1
x 
x C
3
2
d) 3x 2 
1
2 x
C
1
 7 x dx
1
b) ln 14  ln 7
a) 7 ln 2
c)
1
ln 2
7
9. Integrate by parts
a)
d) ln
 xe
1 2 1 4x
x  e C
2
4
c) 1  4e 4 x  C
4x
2
1
 ln
7
7
dx
b) xe4 x  e 4 x  C
d)
1 4x 1 4x
xe  e  C
4
16
10. If y  5 x 2  2 x  5 then
dy
at x = 3 is
dx
a) -44
b) 32
c) -28
d) -51
11. If f ( x)  (5 x  5) 7 then find f ' ' ( x)
a) 210(5 x  5) 6
b) 1050(5 x  5) 5
c) 35(5 x  5) 6
d) 210(5 x  5) 5
12. Evaluate  (4 x 3  8 x  15)dx
a) 4 x 4  4 x 2  15 x  C
b) x 4  4 x 2  15 x  C
c) 12 x 2  8  C
d) 4 x 4  8 x 2  15 x  C
13. Write (but do not solve) the integral that expresses the area of the region bounded by the
graphs of y  0 , x  0 , x  4 , and y  e x
4
4
x
 e dx
a)
b)
0
 x  e dx
x
0
4
4
c)  e x 1dx
d)  (e x  x)dx
0
0
14. Find the value of lim
x2
x2  4
x 2  2x
a) The value is undefined.
b) 0
c) 4
d) 2
15. Find
d x 3 5
e
dx
a) 3  e x
3
5
c) 3  x 2  e x
b) 3  x  e x
3
5
16. Solve the differential equation
x  3.
a) y 
1 3
x  2 x 2  4 x  14
3
c) y  2 x  4
d) x  e x
3
5
dy
 x 2  4 x  4 using the boundary condition y  1 and
dx
b) y 
1 3
x  2 x 2  4 x  16
3
d) y 
1 3
x  2x 2  4x  9
3
c
17. For n  1,  x n dx equals
a
a) n  (c  a)
c)
c n1  a n1
n 1
n 1
c n1  a n1
b)
n 1
d)
c n1  a n1
n 1
18. For which one of the following does the limit not exist?
a) lim
b) lim
c) lim
d) lim
x4
x  1
x0
x  1
19. Use implicit differentiation to find
a)
2 y
2y  x
dy
if y 2  2 x  xy
dx
2y
b)
2 x
d) 2  y
c) x  (2  2 y )
x2  x  6
x  3 x 2  2 x  3
20. Find the limit: lim
a) The limit does not exist
b)
5
4
c) 0
d)
1
2
21. Let C(x) be the cost function in dollars of producing x units. Let R(x) be the revenue
function for selling x units. Write the equation that would be needed to determine how
many units should be sold to maximize profit?
a) R' ( x)  0
b) C ' ( x)  0
c) R ( x)  C ( x)
d) R' ( x)  C ' ( x)
22. Evaluate
a)
x
3
( x 4  2) 3 dx
1 4
( x  2) 4  C
16
b)
1 4
( x  2) 4  C
4
d)
1
 x 4  ( x 4  2) 4  C
64
3
1
1

c)  x 4   x 5  2 x   C
4
5

( x  h) 5  x 5
at the point x = -3
h 0
h
23. Find lim
a) 0
b) 405
c) -3
d) -243
24. If y  sin( 2 x 3 ) , find y '
a) 6 x 2 cos( 2 x 3 )
b)  6 x 2 cos( 2 x 3 )
c) cos(6 x 2 )
d)  cos(6 x 2 )
25. If f ( x) 
a)
c)
8x  2
, find f ' ( x)
x3 1
(8 x  2)(3x 2 )  ( x 3  1)(8)
( x 3  1) 2
( x 3  1)(8)  (8 x  2)(3x 2 )
( x 3  1) 2
b)
d)
8
3x 2
4x 2  2x
x4
x
4