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# REVIEW TEST 1 - PowerPoint PPT Presentation

REVIEW TEST 1. 1. Find the slope of the line determined by (3, -1) and (5, 7). A. - 4. B. 4. D. - 1/2. C. 1/2. E. None of the above. Sorry, that answer is incorrect. Remember slope is defined by Where x 1 and y 1 are the coordinates of the first point. Please try again.

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REVIEW TEST 1

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### 1. Find the slope of the line determined by (3, -1) and (5, 7).

A. - 4

B. 4

D. - 1/2

C. 1/2

E. None of the above

Remember slope is defined by

Where x1 and y1 are the coordinates of the first point.

Great, the correct answer is 4, since -

### 2. Find the slope of 3x + 2y = 18.

A. 3

B. - 3

D. 2/3

C. -3/2

E. None of the above

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

Hey great, the correct answer is - 3/2, since

y = - 3/2 x + 9

And the coefficient of x is the slope.

### 3. Find the y-intercept of 3x + 2y = 18.

A. 6

B. 18

D. 9

C. - 6

E. None of the above

Remember to find the y-intercept you solve the equation for y and the y-intercept is the b term –

y = mx + b

OK, the correct answer is since 9 since

y = - 3/2 x + 9

And the constant term is 9.

### 4. Find the domain of

A. x  7

B. x > 7

D. x  7

C. x ≠ 7

E. None of the above

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

Terrrrriffffic, the correct answer is x  7,

Since x – 7 ≥ 0 means

x  7

### 5. Find the domain of

A. x ≠ 4

B. x > 4

D. x < 4

C. x ≠ - 4

E. None of the above

That answer is incorrect. It is a hard one!

Remember you may have a zero in the denominator. Hence x – 4 ≠ 0.

Terrrrriffffic, the correct answer is x ≠ 4 ,

since the denominator may not be 0 and it is when x = 4.

### 6. Considerf (x) = x 2 + 2x – 1and find the x-intercepts.

A. x = 0 and x = 2

• x = - 0.41 & x = 2.41

C. x = 0.41 and x = – 2.41

D. x = - 2 & x = 1

E. None of the above

There are at least three ways to find the x-intercepts:

1. Factor the quadratic and solve the two factors. (This one does not factor easily.)

3. Graph it on our calculator and use ZERO under the CALC menu.

Yes, the correct answers are –2.41 and 0.41.

I used my calculator and ZERO under the CALC menu.

### 7. Considerf (x) = x 2 + 2x – 1and find the y-intercept.

A. y = 0

• y = 0.41

C. y = – 2.41

D. y = - 1

E. None of the above

There are at least three ways to find the y-intercepts:

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.

OK, the correct answer y = - 1.

I used the fact that C = - 1 in the original equation.

### 8. Considerf (x) = x 2 + 2x – 1and find the vertex.

A. x = - 1 and y = - 2

• x = - 1 & y = 2

C. x = - 2 and y = – 1

D. x = - 2 & y = 1

E. None of the above

To find the vertex:

1. Use the formulas:

2. Graph it on our calculator and use MINIMUM under the CALC menu.

OR

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.

### 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

• A maximum of 1.

C. A minimum of – 2.

D. A minimum of 1.

E. None of the above

To find the maximum or minimum use the results of the previous question.

Yes, the answer is a minimum of - 2.

See the results of the previous question #13.

### 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

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

That answer is correct but you forgot to simplify!

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.

### 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

• 2x + h – 5

D. 2xh + h 2 – 5h

C. 3

E. None of the above

That answer is incorrect. It is a hard one! This is an important question.

Remember that the difference quotion is,

i.e. you need to divide the answer for question 11 by h.

That answer is incorrect. It is a hard one! This is an important question.

Remember that the difference function is,

You forgot to do the final step and divide the difference by h.

WOW, the correct answer is indeed 2x + h – 5.

NOTE: You will need this answer for the next question.

### 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

• m = - 1

D. m = 1

C. m = 3

E. None of the above

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

OR2. Substitute x = 0 and h = 3 into the result of question 12.

Great, Substitute x = 0 and h = 3 into

2x + h – 5

yields 0 + 3 – 5 = - 2

### 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

Yes Yes Yes, the correct answer is y = - x – 1.

I used my calculator and the DRAW menu using TANGENT.

### 15. Find the following limit.

A. 2

• 3

C. 1

D. - 1

E. None of the above

Most of the time to find the limit you can just substitute the value x approaches. In this case substitution will work. Try it.

Yes, the correct answer is 1.

Substituting 1 for x yields

### 16. Find the following limit.

A. 1

• - 1

C. Undefined

D. 0

E. None of the above

Most of the time to find the limit you can just substitute the value x approaches. In this case substitution will work. Try it.

OK, the correct answer is 0.

Substituting 2 for x yields

### 17. Find the following limit.

A. 3

• Undefined

C. 0

D. 1

E. None of the above

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.

After factoring and canceling the expression becomes

And substituting 3 for x yields

### 18. Find the following limit.

A. 0

B. undefined

C. x 2

D. x 2 - x

E. None of the above

Try factoring out an h and cancelling

Yes, the answer is x 2

= x 2

### 19. Is the following function continuous at x = 3?

A. Yes

B. No

C. None of the above

Find the limits as x approaches 3 from the left and from the right.

### 20. In the previous problem you found that the following function was not continuous at x = 3?Why is that true?

The Limit does not exist so it is not continuous!

### 21. Find the derivative for

A. y’ = - 2 x- 3

B. y’ = 2 x

D. y’ = x + 3 - 5

C. y’ = 2x + 3

E. None of the above

Not quite, that answer is incorrect.

Use the definition for derivative – the five step procedure – or use the power rule.

Hey great, the correct answer is y’ = 2x + 3

Nice work on the power rule.

### APPLICATIONS

Now, let’s try some application problems.

CLICK CLICK CLICK

CLICK

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

• \$4,800 per day.

C. \$240 per item.

D. \$260 per item.

E. None of the above

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

Yes, the answer \$4,800 per day or divide that by the 20 items produced to get \$240 per item.

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

• C (x) = 4800 x + 400.

C. C (x) = 5600 x + 400

D. C (x) = 240 x + 400

E. None of the above

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

You may remember that the slope is the per unit variable cost and the y-intercept is the fixed cost and y = mx + b.

C (x) = 240 x + 400.

NOTE: You will need this equation for the next question.

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

• C (x) = \$4800

C. C (x) = \$266.67

D. C (x) = \$240

E. None of the above

All you need to do is substitute 15 into the correct total cost equation from the previous problem.

C (15) = (240)(15) + 400

= \$4,000.

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

• R (x) = 75x – 3 x 2

C. R (x) = 200 + 13x

D. R (x) = 125x + 16 x 2

E. None of the above

The revenue equation is formed by multiplying the price equation by x. That is, revenue is the price times the number sold. R = px

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.

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

• 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

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

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

### 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)

• (17.25, 401.02)

C. (3.58, 201.55)

D. (8.56, 334.87)

E. None of the above

Graph both equations on your calculator and find the point of intersection using CALC and INTERSECT.

You have magic fingers on your calculator.

The correct answers are (2.42, 163.64) and

(17.25, 401.02)

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

• 468.75

C. 12.5

D. 333.33

E. None of the above

Graph the equation on your calculator and find the point of maximum using CALC and MAXIMUM.

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.

Great calculator work.

The correct answers are 12.5 thousand tee shirts.

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

• 468.75

C. 12.5

D. 333.33

E. None of the above

Sorry incorrect. Try the following hint.

Graph the equation on your calculator and find the point of maximum using CALC and MAXIMUM.

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.

Great calculator work.

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

• \$468.75

C. \$12.5

D. \$37.50

E. None of the above.

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.

WOW, yes a tough one!

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.

### 31. The number of bacteria in a culture x hours after treatment is given byf (x) = - x 2 + 12 x + 1000.A. Find f ’ (x).

A. x + 2

• 2x + 12

C. - 2x + 12

D. - x 2 + 2x + 2000)

E. None of the above

Use the power rule to find the derivative.

If y = x n , then y’ = n x n – 1 .

Well done!

The correct answer is - 2x + 12

You will need this answer for the next question.

31. The number of bacteria in a culture x hours after treatment is given byf (x) = - x 2 + 12 x + 1000.B. Find The instantaneous rate of change after 2 hours.

A. - 8

• 1020

C. 8

D. 1012

E. None of the above

Sorry that is incorrect.

Use the results of the previous question and substitute x = 2.

Sorry that is not the correct answer, but it might be close.

Use the results of the previous question and substitute x = 2.

Well done!

Substitute 2 into the answer of the previous question giving

- 2 (2) + 12 = - 4 = 12 = 8