Exam Review
Name:_____________________________
Period: _________
Teacher: Timme, 1/09
Course: Calculus
I. Multiple Choice – Work these out no choices given on review
1) What are the domain and range of y  16  x 2  3 ?
 2, x  2
 1  x ,  2  x  0

f
(
x
)

Graph the function,
 2
 x  1, 0  x  2
 4 x  5, x  2
and answer the following questions.
2)
3)
lim f ( x)
4) lim f ( x)
6) lim f ( x )
lim f ( x)
5) lim f ( x )
7)
x 2 
x 2
x 2 
x 0
x2
f (2)
2 x 2  5x  3
x  3
x3
17) Find the derivative of
f ( x)  e2 x 1  ln x 2
2x 2  x  3
x  5 x 3  4 x 2  5 x7
18) Find
9) Evaluate lim
10) Evaluate lim
2 x2  x  2
11) Evaluate lim
x 1
x 1
12) Find the derivative of
4
y  3x5  4 x 4  x3  x 2  4 x  5
3
13) Find the derivative of f ( x) 
6x  1
5 x
14) Find the derivative of y  x 2  6 x
15) Find the derivative of
f ( x)  3ln  4 x 2 
16) Find the derivative of
f ( x)  x2 cos  x   
8) Is the
function
continuous
? Why?
dy
: x 2  3xy  y 2  5
dx
19) The function,
2 x  3, x  1
f ( x)   2
is …
 x  2, x  1
a) differentiable, but discontinuous
b) continuous, but not
differentiable
c) discontinuous, and not
differentiable d) continuous, and
differentiable
20) An object is moving along a
horizontal line, its position at each
time is represented by the function:
s(t )  5t 2  30t  50 . When time =
0, it is moving to the left. When
does the object turn around and start
moving to the right?
Course: Calculus, Timme 1/09
page 2
21) An explosion throws a rock straight
up into the air. The rock’s height
with respect to time is
h(t )  192t  16t 2 . What is the
average velocity of the rock from t =
1 to 2 seconds?
22) An explosion throws a rock straight
up into the air. The rock’s height
with respect to time is
h(t )  192t  16t 2 . When is the
instantaneous velocity of the rock
zero?
23) A function is decreasing and concave
down on an interval when
?
24) To maximize the area of a rectangle
inscribed between the x-axis and the
function y  cos(x) , give the
function for the area in terms of the
x-coordinate.
25) If the dimensions of a sheet, of
metal, is 12x18 with 4 congruent
squares removed from the corners.
Write an equation of the volume of
this box
II. Free Response - Show all of your work on the test, place a box around your answer.
26) A discussion question.
27) Graph the function below and answer the questions:
 21 x  1 , x  2

y   x 2  4 , 2  x  2
4 x  8 , x  2

a) Is the function differentiable at x = -2? Explain.
b) Is the function differentiable at x = 2? Explain.
28) Find the first derivative of the function y  5 x 2  4 x  3 using the formal (limit)
definition of a derivative.
29) Use analytic methods to graph the function y  x3  27 x  25 and list the local maxima
and minima and point(s) of inflection.
Local Max: ___________
Local Min: ___________
Point of Inflection: _____________
Course: Calculus, Timme 1/09
page 3
30) Sketch a function with the following properties:
x
f
f
1
3
0
1
2
2
1
f 
5
4
3
2
1
0
dne dne
x  1


1  x  1


1 x  2


x2


-5 -4 -3 -2 -1
-1
-2
-3
-4
-5
y
x
1
2
3
4
5
31) Find the first derivative of y  tan 3  x 2  5 x  using the definition of a derivative.
32) You are planning to make an open rectangular box from a piece of cardboard that is 12in
by 20 in. by cutting a square (of side = x in.) from each corner and folding the sides up.
96
36
a) What is the formula for the volume of the box, in terms of x?
b) What value of x will generate the box with the largest volume?
c) What are the dimensions of the largest box possible?
33) What two numbers have the largest product and a sum of 168?
34) If the cost function for a product is c( x)  2 x3  27 x 2  84 x  230 , where x represents
thousands of units produced, is there a production level that minimizes average cost, and
if so, what is it??
Exam Review
Name:_____________________________
Period: _________
Teacher: Timme, 1/09
Course: Calculus
I. Multiple Choice – Work these out no choices given on review
1) What are the domain and range of y  16  x 2  3 ?
 2, x  2
 1  x ,  2  x  0

f
(
x
)

Graph the function,
 2
 x  1, 0  x  2
 4 x  5, x  2
and answer the following questions.
2) lim f ( x)
x 2 
4) lim f ( x)
=2
x 2
= dne
6) lim f ( x )
x2
8) No
lim f ( x)
=3
x 2
3) lim f ( x)
x 2 
5) lim f ( x )
=1
x 0
= -1
2 x 2  5x  3
9) Evaluate lim
x  3
x3
= -7
2x  x  3
x  5 x  4 x 2  5 x7
=0
2
10) Evaluate lim
7)
f (2)
f (2) =1
≠
15) Find the derivative of
f ( x)  3ln  4 x 2 
f ( x) 
6
x
3
2 x2  x  2
x 1
x 1
= +∞
11) Evaluate lim
12) Find the derivative of
4
y  3x5  4 x 4  x3  x 2  4 x  5
3
4
3
y  15 x  16 x  4 x 2  2 x  4
13) Find the derivative of
6x 1
f ( x) 
5 x
31
f ( x) 
2
5  x 
f ( x)   x 2 sin  x     2 x cos  x   
17) Find the derivative of
f ( x)  e2 x 1  ln x 2
f ( x)  2e2 x 1 
18) Find
y  x2  6x
14) Find the derivative of
16) Find the derivative of
f ( x)  x 2 cos  x   
y 
x3
x2  6x
dy
:
dx
2
x
dy 2 x  3 y

dx 3x  2 y
19) The function,
2 x  3, x  1
f ( x)   2
is …
x

2
,
x

1

d) continuous, and differentiable
Course: Calculus, Timme 1/09
page 5
20) An object is moving along a
horizontal line, its position at each
time is represented by the function:
s(t )  5t 2  30t  50 . When time =
0, it is moving to the left. When
does the object turn around and start
moving to the right? t=3
21) An explosion throws a rock straight
up into the air. The rock’s height
with respect to time is
h(t )  192t  16t 2 . What is the
average velocity of the rock from t =
1 to 2 seconds? Avg=144
23) A function is decreasing and concave
down on an interval when
?
f ( x)  0 and f ( x)  0
24) To maximize the area of a rectangle
inscribed between the x-axis and the
function y  cos(x) , give the
function for the area in terms of the
x-coordinate. A( x)  2 x cos( x)
25) If the dimensions of a sheet, of
metal, is 12x18 with 4 congruent
squares removed from the corners.
Write an equation of the volume of
this box
V= (12-2x)(18-2x)x
22) An explosion throws a rock straight
up into the air. The rock’s height
with respect to time is
h(t )  192t  16t 2 . When is the
instantaneous velocity of the rock
zero? t=6
II. Free Response - Show all of your work on the test, place a box around your answer.
26) A discussion question.
27) Graph the function below and answer the questions:
 21 x  1 , x  2

y   x 2  4 , 2  x  2
4 x  8 , x  2

a) Is the function differentiable at x = -2? Explain.
No slope from left not equal to the slope from the right
b) Is the function differentiable at x = 2? Explain.
Yes slope from left equals the slope from the right
28) Find the first derivative of the function y  5 x 2  4 x  3 using the formal (limit)
definition of a derivative.
Course: Calculus, Timme 1/09
page 6
5  x  x   4  x  x   3   5 x 2  4 x  3
2
lim
x  0
x
5  x  2 xx  x   4  x  x   3   5 x 2  4 x  3
2
lim
2
x
5 x  10 xx  5x  4 x  4x  3  5 x 2  4 x  3
lim
x  0
x
2
10 xx  5x  4x
lim
x  0
x
x 10 x  5x  4 
lim
 lim 10 x  5x  4  10 x  4
x  0
x  0
x
x  0
2
2
29) Use analytic methods to graph the function y  x3  27 x  25 and list the local maxima
and minima and point(s) of inflection.
Local Max: __(-3,79) ____
Local Min:
(3,-29)
.
Point of Inflection: _____(0,25)___
30) Sketch a function with the following properties:
x
f
f
1
3
0
1
2
2
1
f 
0
dne dne
x  1


1  x  1


1 x  2


x2


31) Find the first derivative of
of a derivative.
y  tan 3  x 2  5 x 
y   6 x  15  tan 2  x 2  5 x  sec 2  x 2  5 x 
using the definition
Course: Calculus, Timme 1/09
page 7
32) You are planning to make an open rectangular box from a piece of cardboard that is 12in
by 20 in. by cutting a square (of side = x in.) from each corner and folding the sides up.
96
36
a) What is the formula for the volume of the box, in terms of x?
V ( x)   96  2 x 36  2 x  x
b) What value of x will generate the box with the largest volume?
V ( x)  3456  528 x  12 x 2  0
288  44 x  x 2  0
x  8 or x  36
c) What are the dimensions of the largest box possible?
80 x 20 x 8
33) What two numbers have the largest product and a sum of 168?
84 and 84
34) If the cost function for a product is c( x)  2 x3  27 x 2  84 x  230 , where x represents
thousands of units produced, is there a production level that minimizes average cost, and
if so, what is it??
230
Average  4 x  27  2  0
x
x  7.7158