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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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REVIEWTEST 1

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.

A. 3

B. - 3

D. 2/3

C. -3/2

E. None of the above

Remember to find the slope you solve the equation for y and the slope is the coefficient of the x term –

y = mx + b

y = - 3/2 x + 9

And the coefficient of x is the slope.

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

y = - 3/2 x + 9

And the constant term is 9.

A. x  7

B. x > 7

D. x  7

C. x ≠ 7

E. None of the above

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

Since x – 7 ≥ 0 means

x  7

A. x ≠ 4

B. x > 4

D. x < 4

C. x ≠ - 4

E. None of the above

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

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

6. Consider 7).f (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.

I used my calculator and ZERO under the CALC menu.

7. Consider 7).f (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.

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

8. Consider 7).f (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

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 7). 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.

See the results of the previous question #13.

10. A bit of pre-calculus now! 7).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! substitute correctly.

Grrrrreat, the correct answer is substitute correctly.

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. substitute correctly.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. important question.

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

13. If f (x) = x important question. 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

Sorry that answer is incorrect. important question.

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

OR 2. Substitute x = 0 and h = 3 into the result of question 12.

Great, Substitute x = 0 and h = 3 into important question.

2x + h – 5

yields 0 + 3 – 5 = - 2

14. If f (x) = x important question. 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

No that answer is incorrect. important question.

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

I used my calculator and the DRAW menu using TANGENT.

15. Find the following limit. important question.

A. 2

• 3

C. 1

D. - 1

E. None of the above

Sorry that answer is incorrect. important question.

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. important question.

Substituting 1 for x yields

16. Find the following limit. important question.

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. important question.

Substituting 2 for x yields

17. Find the following limit. important question.

A. 3

• Undefined

C. 0

D. 1

E. None of the above

No, that answer is incorrect. important question.

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.

Correctamundo, the answer is 3. important question.

After factoring and canceling the expression becomes

And substituting 3 for x yields

18. Find the following limit. important question.

A. 0

B. undefined

C. x 2

D. x 2 - x

E. None of the above

No that answer is incorrect. important question.

Try factoring out an h and cancelling

Yes, the answer is important question.x 2

= x 2

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

A. Yes

B. No

C. None of the above

No that answer is incorrect. important question.

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

Yes, the answer is important question.No!

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! function was not continuous at x = 3?

21. Find the derivative for function was not continuous at x = 3?

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. function was not continuous at x = 3?

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

Hey great, the correct answer is function was not continuous at x = 3?y’ = 2x + 3

Nice work on the power rule.

APPLICATIONS function was not continuous at x = 3?

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

Sorry that answer is incorrect. and total cost of \$5,200 per day at an output of 20 items. .

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

No that answer is incorrect. and total cost of \$5,200 per day at an output of 20 items.

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.

Yes, the answer is and total cost of \$5,200 per day at an output of 20 items.

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

Too bad that answer is incorrect. and total cost of \$5,200 per day at an output of 20 items.

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

Yes, the answer is and total cost of \$5,200 per day at an output of 20 items.

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

No that answer is incorrect. bicycles are:

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

Yesaroonie, the answer is bicycles are:

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

No that answer is incorrect. bicycles are:

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

Perfect, the answer is bicycles are:

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

Sorry that answer is incorrect. one break-even point.

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

You have magic fingers on your calculator. one break-even point.

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

Not the correct answer. x is in thousands, for making tee shirts with domain 0 to 20 thousand. R (x) = x(75 – 3x).

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. x is in thousands, for making tee shirts with domain 0 to 20 thousand. R (x) = x(75 – 3x).

You were asked for the output. You want the x value of the maximum not the y value.

Great calculator work. x is in thousands, for making tee shirts with domain 0 to 20 thousand. R (x) = x(75 – 3x).

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. x is in thousands, for making tee shirts with domain 0 to 20 thousand. R (x) = x(75 – 3x).

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. x is in thousands, for making tee shirts with domain 0 to 20 thousand. R (x) = x(75 – 3x).

You were asked for the output. You want the y value of the maximum not the x value.

Great calculator work. x is in thousands, for making tee shirts with domain 0 to 20 thousand. R (x) = x(75 – 3x).

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.

Sorry that answer is wrong. x is in thousands, for making tee shirts with domain 0 to 20 thousand. R (x) = x(75 – 3x).

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! x is in thousands, for making tee shirts with domain 0 to 20 thousand. R (x) = x(75 – 3x).

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 by f (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

Too bad, that is incorrect. treatment is given by

Use the power rule to find the derivative.

If y = x n , then y’ = n x n – 1 .

Well done! treatment is given by

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 by f (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. treatment is given by

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

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

Substitute 2 into the answer of the previous question giving

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