REVIEW
TEST 1
1
1. Find the slope of the line determined
by (3, -1) and (5, 7).
A. - 4
B. 4
C. 1/2
D. - 1/2
E. None of the above
2
Sorry, that answer is incorrect.
Remember slope is defined by
Where x1 and y1 are the coordinates of the
first point.
Please try again.
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3
Great, the correct answer is 4, since -
Please click here for the next question.
4
2. Find the slope of 3x + 2y = 18.
A. 3
B. - 3
C. -3/2
D. 2/3
E. None of the above
5
Not quite, that answer is incorrect.
Remember to find the slope you solve the
equation for y and the slope is the coefficient
of the x term –
y = mx + b
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6
Hey great, the correct answer is - 3/2, since
y = - 3/2 x + 9
And the coefficient of x is the slope.
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7
3. Find the y-intercept of 3x + 2y = 18.
A. 6
B. 18
C. - 6
D. 9
E. None of the above
8
Too bad, that answer is incorrect.
Remember to find the y-intercept you solve
the equation for y and the y-intercept is the b
term –
y = mx + b
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9
OK, the correct answer is since 9 since
y = - 3/2 x + 9
And the constant term is 9.
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10
4. Find the domain of
A. x  7
B. x > 7
C. x ≠ 7
D. x  7
E. None of the above
11
Not quite, that answer is incorrect.
Remember you may not take the square roots
of negative numbers. Hence the interior of the
square root sign must be positive. Solve for x
in
x–7≥0
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12
Terrrrriffffic, the correct answer is x  7,
Since x – 7 ≥ 0 means
x7
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13
5. Find the domain of
A. x ≠ 4
B. x > 4
C. x ≠ - 4
D. x < 4
E. None of the above
14
That answer is incorrect. It is a hard one!
Remember you may have a zero in the
denominator. Hence x – 4 ≠ 0.
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15
Terrrrriffffic, the correct answer is x ≠ 4 ,
since the denominator may not be 0 and it is
when x = 4.
Please click here for the next question.
16
6. Consider
f (x) = x 2 + 2x – 1
and find the x-intercepts.
A. x = 0 and x = 2
B. x = - 0.41 & x = 2.41
C. x = 0.41 and x = – 2.41
D. x = - 2 & x = 1
E. None of the above
17
Sorry that answer is incorrect.
There are at least three ways to find the xintercepts:
1. Factor the quadratic and solve the two
factors. (This one does not factor easily.)
2. Use the quadratic formula.
 b  b 2  4ac
x
2a
3. Graph it on our calculator and use
ZERO under the CALC menu.
Please click here to try again.
18
Yes, the correct answers are –2.41 and 0.41.
I used my calculator and
ZERO under the CALC
menu.
Please click here for the next question.
19
7. Consider
f (x) = x 2 + 2x – 1
and find the y-intercept.
A. y = 0
B. y = 0.41
C. y = – 2.41
D. y = - 1
E. None of the above
20
Too bad that answer is incorrect.
There are at least three ways to find the yintercepts:
1. Let x = 0 and solve for y.
2. Use the fact that the y-intercept = c in the
original equation.
3. Graph it on your calculator and use
VALUE under the CALC menu with x = 0.
Please click here to try again.
21
OK, the correct answer y = - 1.
I used the fact that C = - 1 in the original equation.
Please click here for the next question.
22
8. Consider
f (x) = x 2 + 2x – 1
and find the vertex.
A. x = - 1 and y = - 2
B. x = - 1 & y = 2
C. x = - 2 and y = – 1
D. x = - 2 & y = 1
E. None of the above
23
No that answer is incorrect.
To find the vertex:
1. Use the formulas:
b
 b 2  4ac
h
and k 
2a
4a
OR
2. Graph it on our calculator and use
MINIMUM under the CALC menu.
Please click here to try again.
24
Correctamundo, the answer is (- 1, - 2).
I used my calculator and
MINIMUM under the
CALC menu.
NOTE: You will need
this answer for the next
question.
Please click here for the next question.
25
9. Consider f (x) = x 2 + 2x – 1 and find the
maximum or minimum value, whichever exist.
Identify it as either a maximum of minimum.
A. A maximum of - 2
B. A maximum of 1.
C. A minimum of – 2.
D. A minimum of 1.
E. None of the above
26
No that answer is incorrect.
To find the maximum or minimum use the results
of the previous question.
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27
Yes, the answer is a minimum of - 2.
See the results of the
previous question #13.
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28
10. A bit of pre-calculus now!
If f (x) = x 2 – 5x + 3, find f (x + h).
Please simplify your answer.
A. (x + h) 2 – 5 (x + h) + 3
B. x 2 + h 2 – 5x + 5h + 3
C. x 2 + 2xh + h 2 – 5x – 5h + 3
D. 3
E. None of the above
29
That answer is incorrect. It is a hard one!
You must substitute correctly.
Remember that f (x + h) means that you are to
substitute an x + h everywhere an x appears in the
equation. In this problem substitute an x + h for x
into
x 2 – 5x + 3.
Don’t forget to simplify
Please click here to try again.
30
That answer is correct but you forgot to
simplify!
Please click here to try again.
31
Grrrrreat, the correct answer is
x 2 + 2xh + h 2 – 5x – 5h + 3,
since (x + h) 2 - (5) (x + h) + 3 =
x 2 + 2xh + h 2 – 5x – 5h + 3.
NOTE: You will need this answer for the
next question.
Please click here for the next question.
32
12. Let’s complete that idea.
If f (x) = x 2 – 5x + 3, find the difference
quotient.
Please simplify your answer.
A. x 2 + h 2 – 5x – 5h + 3
B. 2x + h – 5
C. 3
D. 2xh + h 2 – 5h
E. None of the above
33
That answer is incorrect. It is a hard one!
This is an important question.
Remember that the difference quotion is,
f ( x  h)  f ( x)
h
i.e. you need to divide the answer for question 11
by h.
Please click here to try again.
34
That answer is incorrect. It is a hard one!
This is an important question.
Remember that the difference function is,
f ( x  h)  f ( x)
h
You forgot to do the final step and divide the
difference by h.
Please click here to try again.
35
WOW, the correct answer is indeed 2x + h – 5.
f ( x + h) - f ( x) ( x 2 + 2xh + h 2
=
h
2xh + h 2
=
h
5h
5x 5h + 3)
h
= 2x + h
( x 2 5 x + 3)
5
NOTE: You will need this answer for the
next question.
Please click here for the next question.
36
13. If f (x) = x 2 – 5x + 3, find the
slope of the secant line joining
(0, f(0)) and (3, f(3)).
A. m = - 2
B. m = - 1
C. m = 3
D. m = 1
E. None of the above
37
Sorry that answer is incorrect.
Remember to find the slope of the secant you can
do either of the following:
1. Work out the y values of the two points
given and use the slope formula m  y 2  y 1
x 2  x1
OR 2. Substitute x = 0 and h = 3 into the result
of question 12.
Please click here to try again.
38
Great, Substitute x = 0 and h = 3 into
2x + h – 5
yields 0 + 3 – 5 = - 2
Please click here for the next question.
39
14. If f (x) = x 2 – 5x + 3, find the
equation of the tangent line at (2, f(2)).
A. y = x + 1
B. y = - 2x - 1
C. y = - x - 1
D. y = - 2x + 1
E. None of the above
40
No that answer is incorrect.
Use your calculator and the DRAW menu using
TANGENT.
Please click here to try again.
41
Yes Yes Yes, the correct answer is y = - x – 1.
I used my calculator and the DRAW menu using TANGENT.
Please click here for the next question.
42
15. Find the following limit.
lim 2x  4
x  1 x 2  3x
A. 2
B. 3
C. 1
D. - 1
E. None of the above
43
Sorry that answer is incorrect.
Most of the time to find the limit you can just
substitute the value x approaches. In this case
substitution will work. Try it.
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44
Yes, the correct answer is 1.
Substituting 1 for x yields
24 2

1
13 2
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45
16. Find the following limit.
lim x  2
x  2 x2  x
A. 1
B. - 1
C. Undefined
D. 0
E. None of the above
46
Too bad that answer is incorrect.
Most of the time to find the limit you can just
substitute the value x approaches. In this case
substitution will work. Try it.
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47
OK, the correct answer is 0.
Substituting 2 for x yields
22 0
 0
42 2
Please click here for the next question.
48
17. Find the following limit.
lim 5x 2  15x
2
x3 x  x  6
A. 3
B. Undefined
C. 0
D. 1
E. None of the above
49
No, that answer is incorrect.
Most of the time to find the limit you can just
substitute the value x approaches. In this case
substitution will not work since you get 0/0 for
your answer. You must factor the numerator and
denominator and cancel common factors before
you substitute 3 for x.
5x 2  15x
5x ( x  3 )
5x


2
x  x  6 ( x  2)( x  3) x  2
Please click here to try again.
50
Correctamundo, the answer is 3.
After factoring and canceling the expression becomes
5x 2  15x
5x ( x  3 )
5x


2
x  x  6 ( x  2)( x  3) x  2
And substituting 3 for x yields
15
3
5
Please click here for the next question.
51
18. Find the following limit.
lim x2 h  xh2  h 3
h0
h
A. 0
B. undefined
C. x 2
D. x 2 - x
E. None of the above
52
No that answer is incorrect.
Try factoring out an h and cancelling
lim x2 h  xh2  h 3
h0
h
Please click here to try again.
53
Yes, the answer is x 2
lim x2 h  xh2  h 3
lim h( x2  xh  h 2 )

h0
h
h0
h
lim 2

x  xh  h 2
h0
= x2
Please click here for the next question.
54
19. Is the following function continuous at x = 3?
5  x
f (x )  
x  2
A. Yes
if x  3
if x  3
B. No
C. None of the above
55
No that answer is incorrect.
Find the limits as x approaches 3 from the left and
from the right.
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56
Yes, the answer is No!
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57
20. In the previous problem you found that the
following function was not continuous at x = 3?
Why is that true?
Try to explain your answer and then click
here to see if you are correct.
58
lim
lim
f (x )  5  3  2 BUT
f (x )  3  2  1


x3
x3
The Limit does not exist so it is not continuous!
Please click here for the next question.
59
21. Find the derivative for
y  x2  3x  5
A. y’ = - 2 x - 3
B. y’ = 2 x
C. y’ = 2x + 3
D. y’ = x + 3 - 5
E. None of the above
60
Not quite, that answer is incorrect.
Use the definition for derivative – the five step
procedure – or use the power rule.
Please click here to try again.
61
Hey great, the correct answer is y’ = 2x + 3
Nice work on the power rule.
Please click here for the next question.
62
APPLICATIONS
Now, let’s try some application problems.
CLICK CLICK CLICK
CLICK
63
22. A manufacturing company has fixed cost of
$400 per day and total cost of $5,200 per day
at an output of 20 items. .
A. What are the variable cost? State and
identify the variable cost as per day or
per item.
A. $5,200 per day
B. $4,800 per day.
C. $240 per item.
D. $260 per item.
E. None of the above
64
Sorry that answer is incorrect.
To find the variable cost subtract the fixed cost from
the total cost. Remember
Total Cost = Variable Cost + Fixed Cost
so
Variable Cost = Total Cost – Fixed Cost
Please click here to try again.
65
Yes, the answer $4,800 per day or divide that by the 20
items produced to get $240 per item.
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66
23. A manufacturing company has fixed cost of
$400 per day and total cost of $5,200 per day
at an output of 20 items.
B. Find the equation of the line relating the
output to the total cost. Write your final
answer in the form C (x) = mx + b.
A. C (x) = 5200 x + 400
B. C (x) = 4800 x + 400.
C. C (x) = 5600 x + 400
D. C (x) = 240 x + 400
E. None of the above
67
No that answer is incorrect.
To find the the equation you can use the two points
given (The total cost at a production level of 0 and
the total cost at a production level of 20) and
substitute into the slope/point form of an equation.
OR
y  y1  m (x  x1 )
You may remember that the slope is the per unit
variable cost and the y-intercept is the fixed cost and
y = mx + b.
Please click here to try again.
68
Yes, the answer is
C (x) = 240 x + 400.
NOTE: You will need
this equation for the next
question.
Please click here for the next question.
69
24. A manufacturing company has fixed cost of
$400 per day and total cost of $5,200 per day
at an output of 20 items.
C. What is the total cost for an output of 15
units per day?
A. C (x) = $4000
B. C (x) = $4800
C. C (x) = $266.67
D. C (x) = $240
E. None of the above
70
Too bad that answer is incorrect.
All you need to do is substitute 15 into the correct
total cost equation from the previous problem.
Please click here to try again.
71
Yes, the answer is
C (15) = (240)(15) + 400
= $4,000.
Please click here for the next question.
72
25. The price-demand and cost functions for
selling bicycles are:
p (x) = 75 – 3x and C (x) = 125 + 16x
A. What is the revenue function?
A. R (x) = 75/x - 3
B. R (x) = 75x – 3 x 2
C. R (x) = 200 + 13x
D. R (x) = 125x + 16 x 2
E. None of the above
73
No that answer is incorrect.
The revenue equation is formed by multiplying the
price equation by x. That is, revenue is the price
times the number sold. R = px
Please click here to try again.
74
Yesaroonie, the answer is
R (x) = 75x – 3 x 2
Since
R (x) = px
= (75 – 3x) x
= 75x – 3x 2
NOTE: You will need this equation for the next
question.
Please click here for the next question.
75
26. The price-demand and cost functions for
selling bicycles are:
p (x) = 75 – 3x and C (x) = 125 + 16x
B. What is the profit function?
A. P (x) = 200 + 13 x
B. P (x) = 75x – 3 x 2
C. P (x) = - 3x 2 + 91x + 125
D. P (x) = - 3x 2 + 59x - 125
E. None of the above
76
No that answer is incorrect.
The profit equation is formed by subtracting the cost
equation from the revenue equation. The revenue
equation was found in the previous problem.
P=R -C
Please click here to try again.
77
Perfect, the answer is
P (x) = - 3x 2 + 59x – 125
Since
P (x) = R – C
= (75x – 3x 2) – (125 + 16x)
= 75x – 3x 2 – 125 – 16x
= - 3x 2 + 59x - 125
Please click here for the next question.
78
27. Given the following revenue and cost
functions find one break-even point.
R (x) = x(75 – 3x) and C (x) = 125 + 16x
A. (2.42, 163.64)
B. (17.25, 401.02)
C. (3.58, 201.55)
D. (8.56, 334.87)
E. None of the above
79
Sorry that answer is incorrect.
Graph both equations on your calculator and find the
point of intersection using CALC and INTERSECT.
Please click here to try again.
80
You have magic fingers on your calculator.
The correct answers are (2.42, 163.64) and
(17.25, 401.02)
Please click here for the next question.
81
28. A company has the following revenue
function, where x is in thousands, for
making tee shirts with domain 0 to 20
thousand. R (x) = x(75 – 3x).
A. Find the output of shirts that will produce
the maximum revenue.
A. 25
B. 468.75
C. 12.5
D. 333.33
E. None of the above
82
Not the correct answer.
Graph the equation on your calculator and find the
point of maximum using CALC and MAXIMUM.
Please click here to try again.
83
Sorry that is not the correct answer, but it is
close.
You were asked for the output. You want the x value
of the maximum not the y value.
Please click here to try again.
84
Great calculator work.
The correct answers are 12.5 thousand tee shirts.
Please click here for the next question.
85
29. A company has the following revenue
function , where x is in thousands, for
making tee shirts with domain 0 to 20
thousand. R (x) = x(75 – 3x).
B. What is the maximum revenue.
A. 25
B. 468.75
C. 12.5
D. 333.33
E. None of the above
86
Sorry incorrect. Try the following hint.
Graph the equation on your calculator and find the
point of maximum using CALC and MAXIMUM.
Please click here to try again.
87
Sorry that is not the correct answer, but it is
close.
You were asked for the output. You want the y value
of the maximum not the x value.
Please click here to try again.
88
Great calculator work.
The correct answer is $468.75
Please click here for the next question.
89
30. A company has the following revenue
function , where x is in thousands, for
making tee shirts with domain 0 to 20
thousand. R (x) = x(75 – 3x).
C. What is the price per tee shirt that
maximizes the revenue.
A. $25
B. $468.75
C. $12.5
D. $37.50
E. None of the above.
90
Sorry that answer is wrong.
This problem is somewhat difficult. First, you are
looking for a price so you need a price-demand
equation. You can get that from the revenue
equation. Remember Rev = x p so p = Rev/x or
p (x) = 75 – 3x. Now use the x you found in part A
(Problem 23) to find the price.
Please click here to try again.
91
WOW, yes a tough one!
The correct answer is $37.50
You can either take the total revenue at the
maximum and divide by the x value or you can
find the price-demand equation from the given
revenue equation and substitute the x value you
found at the maximum.
Please click here for the next question.
92
31. The number of bacteria in a culture x hours
after treatment is given by
f (x) = - x 2 + 12 x + 1000.
A. Find f ’ (x).
A.
x+2
B.
2x + 12
C. - 2x + 12
D. - x 2 + 2x + 2000)
E. None of the above
93
Too bad, that is incorrect.
Use the power rule to find the derivative.
If y = x n , then y’ = n x n – 1 .
Please click here to try again.
94
Well done!
The correct answer is
- 2x + 12
You will need this answer for the next
question.
Please click here for the next question.
95
31. The number of bacteria in a culture x hours
after treatment is given by
f (x) = - x 2 + 12 x + 1000.
B. Find The instantaneous rate of change
after 2 hours.
A. - 8
B. 1020
C. 8
D. 1012
E. None of the above
96
Sorry that is incorrect.
Use the results of the previous question and
substitute x = 2.
Please click here to try again.
97
Sorry that is not the correct answer, but it
might be close.
Use the results of the previous question and
substitute x = 2.
Please click here to try again.
98
Well done!
The correct answer 8 bacteria.
Substitute 2 into the answer of the previous
question giving
- 2 (2) + 12 = - 4 = 12 = 8
Please click here for the next question.
99
31. The number of bacteria in a culture x hours
after treatment is given by
f (x) = - x 2 + 12 x + 1000.
C. Interpret the results of the previous
problem.
Answers will vary but click here to
see my answer.
100
In the next hour the number of bacterial will increase
by about 8.
Please click here for the next question.
101
That was a lot of review. I hope you found it
helpful. Good luck on the test!
102