Math107 – QUIZ I - Amazon S3



NAME: ____________________________ Math151 – Practice Problems (Exam 2 Part I)

M. Wallace

Put answers in spaces provided unless otherwise stated. SHOW ALL WORK. (No Work = No Credit)

Any answers without detailed, supporting work will not receive full credit – even if the answer is correct.

Remember that you lose points for sloppy work, not following directions, and unclear answers.

Simplify answers completely. NO CALCULATOR. NO NOTES. NO BOOKS.

Exam1, Part I: COVERS: HW22 (#27-30 only), HW23, HW24, HW25, HW26

1. Consider the function: [pic].

ANSWER THE QUESTION TO EACH PART IN ITS GIVEN SPACE. Write NONE if appropriate.

(a) Find the first and second derivatives of f(x).

(b) Identify the increasing and decreasing intervals and any local extrema of f(x) with the First Derivative Test.

(c) Identify the concavity intervals of f(x) and any inflection points? Show work.

(d) Sketch a graph the function on the axes below.

(a) (b) Work:

(d) Appropriately label points.

2. Consider the function: [pic].

ANSWER THE QUESTION TO EACH PART IN ITS GIVEN SPACE. Write NONE if appropriate.

(a) Find the first and second derivatives of f(x).

(b) Identify the increasing and decreasing intervals and any local extrema of f(x) with the First Derivative Test.

(c) Identify the concavity intervals of f(x) and any inflection points? Show work.

(d) Sketch a graph the function on the axes below.

(a) (b) Work:

(d) Appropriately label points.

3. The following is a graph of [pic]

(a) For what values of x does the graph of f have a horizontal tangent?

Justify (explain) your answer.

(b) For what values of x in the interval (-4, 4) does f have a relative minimum?

Justify(explain) your answer.

(c) For what values of x in the interval (-4, 4) does f have a relative maximum?

Justify your answer.

(d) For what value of x is the graph of f concave downward?

Justify your answer.

(e) For what value of x is the graph of f concave upward?

Justify your answer.

(f) For what values of x does the graph of f have an inflection point?

Justify your answer.

4. The DERIVATIVE FUNCTION f ((x) is given in the form of the graph shown. Suppose f(x) is defined over [-1,6].

(i) Over which x-intervals is f(x) increasing? _____ _______________.

(ii) Over which x-intervals is f(x) decreasing? _____ _______________.

(iii) At which x-values is there a potential absolute maximum: ______.

(iv) At which x-values is there a potential absolute minimum: ______.

5. Find the absolute maximum and minimum values of the function [pic] on the interval [0, 2π]. Show/explain all work for full credit.

Part of the exercise here is building, training, and honing your skills of estimation, and certainly showing those skills via examination. You should be able to show your reasoning to find the solution without the calculator.

6. Find the exact slope of the tangent line to the parametric curve { x = 7cos(t) , y = 5 sin(t) } at the point where t = π /6 .

Give an exact answer; do not use a decimal.

7. The cost of producing x units of stuffed alligator toys is C(x) = 0.002x2 + 9x + 8000. Find the marginal cost at the production level of 1000 units. Write a sentence thoroughly describing the answer, without using the words, “marginal cost.”

8. Find the exact x-coordinate where there is a horizontal tangent line to the graph of f(x) = x9x.

9. Find (dz)/(dt) in terms of t, given: z = x e3y, x = t 3, y = 4 + 5t. (dz)/(dt) = __________________

10. Consider the function f(x) = 6x + 6x−1. Find the critical values of f(x).

Find the intervals of increasing and decreasing for the function. Show work.

(a) Critical Values:? ________________. (b) Over which x-intervals is f(x) increasing? __________________.

(c) Over which x-intervals is f(x) decreasing? _____ _______________.

11. Let f(x) = x / (x + 5). Find the values of x where f ′(x) = 4. Give exact answers (not decimal approximations).

12. The height of a triangle is increasing at a rate of 4 cm per minute, while the area of the triangle is increasing at a rate of 10 square centimeters per minute. At what rate is the base of the triangle changing when the height is 6 cm and the area is 33 square centimeters? Is it increasing or decreasing?

Answer: __________________

13. A customer wants to create a rectangular outside patio that is bordering a pond, where the total area of the patio is 400 square feet. The customer wants to have shrubs along the sides of the patio and have parallel strips of shrubs breaking the garden into 3 equally spaced sections. The customer wants the front of the patio to be lined with flowers and does NOT need flowers or shrubs along the back of the patio as there is the pond there. The landscape designer charges $25 per linear foot for the shrubs and $16 per linear foot for the flowers.

What will be the minimum cost that the customer will need to spend?

14. Rectilinear Motion

The displacement in centimeters of a particle moving along a straight line in t seconds is given by the position function

a) Compute the particle’s velocity function and acceleration function at time t.

b) During which time intervals is the particle speeding up? Show work/explain intervals.

c) What is the position of the particle along line at time t = 0 seconds?

d) What is the total distance that the particle travels between time 0 and time 20? Show/Explain work.

15. The surface area of a circular oil spill is increasing at a rate of 100 square feet per hour. How quickly is the radius increasing when the radius is 72 feet?

Leave answer exact (no decimals). Show work.

16. A company has determined that its demand for selling q units of an item is given as q = −5p + 80 at price p per item. The company decides to decrease the price by $2 per month. Find and describe how the rate at which the revenue is changing per month when the items are priced $10 each.

Answer: _______________

17. As a marketing consultant for a particular book, you decide to simplify work and assume a linear demand equation for the book, q(p), where q is the demand function in annual sales in terms of p, the price per book.

(a) Your market studies reveal the following sales figures:

When the price is set at $50 per book, the sales amount to 10,000 per year, and the sales drop to 1000 per year when the price is set at $80 per book. Use this data to calculate the linear demand equation, q(p).

Linear Demand Function: _______________________

(b) What level of production (quantity sold) maximizes the revenue?

(c) What is the maximum revenue?

Answer: __________________

Label Units.

Answer: __________________

Label Units.

18. As a television manufacturer has been selling 1500 televisions a week at $360 each. A market survey indicates that for each $21 rebate offered to a buyer, the number of television sets sold will increase by 280 per week.

REBATE: PRICE DROP

(a) Find the linear demand function p(q), for the price p as a function in terms of ticket sales q.

Linear Demand Function: _______________________

(b) What rebate should be set in order to maximize revenue?

Answer: __________________

Label Units.

19. The Acme Widget Company has found that if widgets are priced at $220, then 500 widgets will be sold. They have also found that for every decrease of $18, there will be 900 more widgets sold.

(a) Find the linear demand function p(q) and the revenue function R(q).

Linear Demand Equation: _______________________

Revenue Function: _______________________

(b) If the marginal cost of widgets is $80 per widget with the fixed costs for the Acme Widget Company being $20,000, then what price should be set in order to maximize profit?

Answer: __________________

Label Units.

20. At 10am, ship A is 80 nautical miles due west of ship B. Ship A is sailing east at 10 knots, and ship B is sailing north at 40 knots. How fast (in knots) is the distance between the ships changing at 1pm?

(Note: 1 knot is a speed of 1 nautical mile per hour.)

21. Find the positive value of the parameter t, corresponding to a point on the curve parametrized by the curve

[pic] , for which the tangent line passes through the origin.

22. An object is launched into the air vertically from a 20-foot tall roof top with an initial velocity 160 feet/second.

What is the object's maximum height in meters?

23. An object is launched into the air vertically from ground level and hits the ground after 12 seconds. What was the maximum height of the object in meters?

24. Find the critical values for each function. (Do not need to classify them.)

a) [pic] b) [pic] c) [pic] over [0,2π) only

-----------------------

Relative Maxima Relative Minima

(c) Work

[pic]

Inflection Points

Relative Maxima Relative Minima

(c) Work

Inflection Points

[pic]

[pic]

shrubs

脘脛脜腆腇膍膎膏膣膤膥膦膧臠臢臧臩臬臭臮舡舣舯舸舾艁艒艔艖艚艡艣艦芖芡芢

shrubs

shrubs

shrubs

flowers

pond

[pic]

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