Review Assignment II

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1 MATH Intuitive Calculus KSU Name:. Review Assignment II 1. Let C(x) be the cost, in dollars, of manufacturing x widgets. Fill in the table with a mathematical expression and appropriate units corresponding to each description. Description Mathematical Expression Units cost of manufacturing 100 widgets cost of manufacturing the 101 st widget average rate of change of cost from a production level of 100 widgets to a production level of 101 widgets instantaneous rate of change of cost at a production level of 100 widgets instantaneous rate of change of cost at a production level of 100 widgets (another expression) 2. Let f(x) be a function. Fill in the table with a mathematical expression corresponding to each description. Description Mathematical Expression y-coordinate of the point on the graph y = f(x) where x = 27 height above the x-axis of the point on the graph y = f(x) where x = 27 slope of the secant line through the points on the graph y = f(x) where x = 27 and where x = 30 slope of the tangent line to the graph y = f(x) at the point where x = 27 slope of the tangent line to the graph y = f(x) at the point where x = 27 (another expression) 1

2 3. Let T (t) be temperature, in degrees Fahrenheit, at time t. For each scenario, fill in the banks with one of the symbols <, >, =, or? (if the sign cannot be determined). (a) The temperature held steady at 70 all afternoon. T (t) 0 T (t) 0 T (t) 0 (b) The temperature increased from the low in the 60 s at a slow but steady rate. T (t) 0 T (t) 0 T (t) 0 (c) At midnight, the temperature was 0 and it has been falling more and more rapidly ever since. T (t) 0 T (t) 0 T (t) 0 (d) As the sun came out, the temperature increased more and more quickly. T (t) 0 T (t) 0 T (t) 0 (e) The temperature is still falling, although not as rapidly as earlier in the evening. T (t) 0 T (t) 0 T (t) 0 4. Find the absolute extreme values of the function f(x) = 3x 4 16x x 2 on the closed, bounded interval [ 1, 4]. You must show all your steps carefully so that I know you are using calculus rather than relying on your grapher. The absolute minimum value of f on [ 1, 4] is which occurs at x =. The absolute maximum value of f on [ 1, 4] is which occurs at x =. 2

3 5. Let f(x) = 1 3 x x2 2x 1. Find f (x) = and f (x) =. (a) Find the domain of f. (f) Find the x-coordinates of all possible inflection points of f. (b) Give the ordered pairs for all y-intercepts of f. (g) Construct a sign chart for f indicating the sign of f and the corresponding behavior of f. Justify your answer. (c) Find the x-coordinates of all critical points of f. (d) Find the y-coordinates of all critical points of f. (h) Sketch the graph of the function f. Indicate all intercepts, critical points, and inflection points as well as other important behavior very clearly. (e) Construct a sign chart for f indicating the sign of f and the corresponding behavior of f. Justify your answer

4 6. A store can sell 20 HDTV sets per week at a price of $400 each. The manager estimates that for each $10 price reduction, she will sell 2 more HDTV s each week. The HDTV s cost the store $200 each. Let x represent the number of $10 price reductions. (All of the following questions refer to weekly sales of HDTV s.) (g) Use calculus to find the value of x for which the store s profits are maximal. Show your reasoning carefully. Verify that you have indeed found the absolute maximum point of the function. (a) Express the price p of an HDTV as a function of x. (b) Express the quantity q of HDTV s sold weekly as a function of x. (c) Express revenue R as a function of x. (d) Express cost C as a function of x. (h) What price should the store set to maximize profits? (e) Express profit P as a function of x. (i) What quantity will the store sell at this price? (f) Find P (x). (j) What is the maximal profit? 4

5 7. Using calculus, show that of all rectangles whose area is 1000 ft 2, the one with minimal perimeter is a square. Show your reasoning. Be sure to: Introduce all variables with Let statements. Include the units. Draw and label a diagram. Verify that you have indeed found the maximum or minimum point (on the appropriate domain). Answer the question posed in the problem in a complete sentence, using appropriate units. 5

6 8. Do this without a calculator. Let f(x) = 4 x. (a) Give the ordered pairs for all intercepts of f. Indicate none if there aren t any. Show the algebra explicitly. i. y-intercepts: (c) Evaluate each limit. Additional work not needed here. i. lim x 4x = ii. lim x 4x = ii. x-intercepts: (d) Give the equations of all asymptotes of f. Indicate none if there aren t any. Additional work not needed here. i. vertical asymptotes: (b) Complete the table of values. x 2 f(x) ii. horizontal asymptotes: 1 1/2 0 (e) Sketch the graph y = f(x). Label all points computed above with their ordered pairs and all asymptotes with their equations. 1/2 1 2 (f) Sketch the graph y = log 4 x on the same coordinate system. Label at least six points with their ordered pairs and all asymptotes with their equations. 6

7 9. A bank offers money market accounts at 2.75% annual interest compounded continuously. (a) Give the formula for the amount F V in the account after t years when P V dollars are invested. (b) How soon will a deposit triple in value? Show your reasoning. Round to the nearest year. 10. A bank offers money market accounts at 5.25% annual interest. Rounded to the nearest cent, what is the present value of $1,000 ten years from now... (a)... if interest is compounded continuously? (b)... if interest is compounded weekly? (c)... if interest is compounded quarterly? 7

8 11. Evaluate each indefinite integral. Try simplifying the integrand algebraically instead of or in addition to using a substitution. Show all steps. Check your answer by differentiating. (a) e 3x ( 2 e 3x) 5 dx Check: (b) ( x ) 2 x 3 dx Check: (c) ln(1 x) 1 x dx Check: 8

9 12. (a) If w (t) is the rate of growth of a child in pounds per year, what does 10 5 w (t) dt represent? (b) If a honeybee population starts with 100 bees and increases at a rate n (t) bees per week, what does represent? 15 0 n (t) dt (c) If oil leaks from a tank at a rate of r (t) gallons per minute, what does r (t) dt represent? 13. The marginal cost of manufacturing x yards of a certain fabric is x x 2 (in dollars per yard). Find the increase in cost if the production level is raised from 2000 yards to 4000 yards. Introduce your function(s) with a Let statement. 9

10 14. Find each definite integral. Give exact answers, simplified. Show all steps for full credit. (a) 2 1 ( 2x + x x + 1 ) dx 2x (b) a 0 x a 2 x 2 dx (c) e 9 e 1 x ln x dx 10