Math107 – QUIZ I - Amazon S3



NAME: ____________________________ Math151 – Practice (Exam 1 Part II)

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. (GRAPHING OR SCIENTIFIC) CALCULATOR. NO NOTES. NO BOOKS.

Exam1, Part II: COVERS HW12-HW22

1. (a) Write the limit definition of the derivative and use it to find the derivative of[pic]. (long way)

Use proper notation throughout for full credit.

(b) Write the limit definition of the derivative and use it to find the derivative of[pic]. (long way)

Use proper notation throughout for full credit.

2. Identify the kind of discontinuity of each function (removable, jump, or infinite). If more than one kind, identify all.

Tell at which value of x, the discontinuity occurs. If no discontinuity occurs, state NONE.

(a) [pic] (b) [pic] (c) [pic]

(d)[pic] (e) [pic] (f) [pic]

(g) [pic] (h) [pic]

3. Determine if the following limits exist, and if so, compute the limit. Provide Reasoning FOR CREDIT.

a) b) c) d)

e) f) g) h)

4. Determine for what values of a and b is the line [pic] tangent to the curve[pic] when [pic].

5. The curve 4x2 + 32x + 9y2 + 72y = −172  has a horizontal tangent line at the point (−4,−2) and at one other point. Find the coordinates of the second point where the curve has a horizontal tangent line.

6. Find the exact x-coordinate where there is a horizontal tangent line to the graph of f(x) = (3x + 5)9x.

7. Find ALL x-values, where the graph of the function f(x) = 4sin(6x) + 3x + 1 has a horizontal tangent line. (No Decimals)

8. Find the point, where the tangent line to y = 10x8  at x = 0.35 crosses the x-axis. Show all work without a calculator.

9. Find the equation of the tangent line to the parabola y = (x + 1)2 + 8, whose slope is negative, and that passes through the point (0,−7).

10. The graphs of   f(x) = x3 − 9x2 + 24x − 16    and   g(x) = 3x2 − 21x + 34   intersect at two points. At one of those intersections, the graphs are tangent to each other (meaning they have a common tangent line). Find the coordinates of that point.

11. Given that [pic], [pic], and [pic], find [pic]. No decimals.

12. Given that[pic], [pic], and [pic], find [pic].

13. Given that[pic], [pic], and [pic], [pic], [pic], [pic],

find [pic]and[pic] .

14. Use the following chart. No decimals.

(a) Find [pic] if [pic].

(b) Find [pic] if [pic].4

(c) Find [pic] if [pic].

15. Verify that d/dx(cot x) = -csc2x. (Show ALL of your steps for credit.)

16. Verify that d/dx(csc x) = -csc x cot x. (Show ALL of your steps for credit.)

17. A particle moves along a straight line and its position s at time t is given [pic], where s is measured in feet and t in seconds.

(a) Find the velocity (in ft/sec) of the particle at time [pic]: _______________

(b) The particle stops moving (i.e. is in a rest) twice. At what two times does this happen? ____________________

18. The population of a slowly growing bacterial colony after [pic] hours is given by [pic].

Find the growth rate after 4 hours. Write a sentence with the meaning of the answer.

19. The A ball is thrown upward in the air, and its height above the ground after t seconds is H(t) = 67t − 16t 2  feet.

Find the time t, when the ball will be traveling downward at 33.5 feet per second.

20. The demand equation for a particular of toy is given by price p = –2q + 65, where q is the quantity of toys sold in thousands and p is the price per toy in dollars. The cost to produce q thousand toys is C(q) = 5q + 4000 thousand dollars.

(a) Find (in terms of the quantity q sold) the revenue function R, profit function ( , marginal revenue function and the marginal profit function. (Use ( for profit, since we are using p for price.)

Revenue: ________________________ Profit: _________________________

Marginal Revenue: _____________________ Marginal Profit: ________________________

(b) If the company is currently producing 12,000 toys per year, should it increase or decrease its toy production?

Find and write a sentence analyzing [pic] to help answer the question.

21. A widget that is to be sold at a small specialty store has $200 in fixed costs and $8 per item in variable costs.

The linear demand of the widget from the store owners’ information is that they can sell 9 widgets, if they charge $151 each, and 5 widgets at a price of $163 each.

(A) Find how many units are sold/produced when the marginal profit is $110 per widget?

(B) Write a sentence describing what your answer represents (without using “marginal profit” in the sentence).

22. An object is shot upward in the air with an initial velocity of 122 meters per second, and its height above the ground after t seconds is H(t) = 67t − 16t 2  feet.

Find the time t, when the ball will be traveling downward at 33.5 feet per second.

23. A certain material sold in pounds at a small specialty store has $300 in fixed costs and $8 per pound in variable costs. The owners assume they can sell 6 pounds of the material, if they charge $82 per pound, and 10 pounds at a price of $70 per pound. (You may and should assume that demand is a linear function for this problem.)

(a) How much profit is made when 20 pounds are produced and sold?

(b) For how many pounds produced/sold is the marginal profit -28 dollars per pound? Label Units.

(c) Write a sentence describing what your answer represents (without using “marginal profit” in the sentence).

24. 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).

(b) If the marginal (variable) cost of widgets is $80 per widget with the fixed costs for the Acme Widget Company being $20,000, then find the marginal profit function.

25. Differentiate each of the following.

(a) [pic] (b) [pic] (c) [pic]

(d) [pic] (e) [pic] (f) [pic]

(g) [pic] (h) [pic] (i) [pic]

26. Find dy/dx.

(a) [pic] (b) [pic] (c) [pic]

(d) [pic] (e) [pic] (f) [pic]

27. Find the EQUATION of the tangent line in Slope-Intercept Form to the curve [pic]

at x = 2 in Quadrant IV. (No Decimals)

28. Find the equation for the line tangent to 2y2 + 9y = –x2 y + 5x + 7 at x = 1 in Quadrant I.

29. Find [pic]given: [pic]. Your answer should only be in terms of the variable t.

[pic]= ___________  

30. 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: __________________

31. 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.)

Answer: __________________

32. 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? Give answer exact (no decimals) and give answer estimated to two decimals.

Answer: _______________

33. 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: _______________

34. Water is leaking out of an inverted conical tank at a rate of 9600 cubic centimeters per min at the same time that water is being pumped into the tank at a constant rate. The tank has height 14 meters and the diameter at the top is 6 meters. If the water level is rising at a rate of 15 centimeters per minute, when the height of the water is 300 centimeters, then find the rate at which water is being pumped into the tank in cubic centimeters per minute.

Answer: _______________

35. A particular lamp that is to be sold at a small specialty store has $1000 in fixed costs and $3 per lamp in variable costs. The owners have spoken with a marketing expert that found that the linear demand of the lamp has a demand selling price of p = -6q + 305, where q is the demand quantity of lamps being sold.

The owners decide to decrease the price by $.84 each week.

Find and describe how the rate at which the Profit П is changing per week, when the lamps are priced $59 each.

(Write a sentence.)

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[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

|[pic] |1 |2 |3 |4 |

|[pic] |1 |8 |4 |3 |

|[pic] |5 |4 |6 |2 |

|[pic] |3 |1 |2 |4 |

|[pic] |2 |3 |1 |7 |

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