MATH 11012 Intuitive Calculus Fall 2013 Circle lab day: Mon / Fri Dr. Kracht Name:. 1. Consider the function f depicted below. Final Exam Review Show all your work. y 1 1 x (a) Find each of the following (or state dne if it does not exist). x 2 x 1 x 2 x 4 x 2 + x 1 + x 2 + x 4 + x 2 x 1 x 2 x 4 f( 2) = f(1) = f(2) = f(4) = (b) Answer Yes or No. (This refers to the function depicted above.) i. Is f continuous at x = 2? ii. Is f continuous at x = 1? iii. Is f continuous at x = 2? iv. Is f continuous at x = 4? v. Is f differentiable at x = 2? vi. Is f differentiable at x = 1? vii. Is f differentiable at x = 2? viii. Is f differentiable at x = 4? (c) True or False: For a function f, the value of lim x a f(x) depends upon the value of f(a). 1
2. Find the limit if it exists, showing all work. If the limit does not exist, explain why not. (a) lim 0.94 x 0.5 (b) lim x 3 x 3 x 2 9 (c) x 2 + 2x + 1 lim x 1 x 2 1 3. Let C(x) be cost, R(x) be revenue, and P (x) be profit, all in dollars, of producing x DVD players. Suppose that C(1000) = 150, 000, C (1000) = 45, R(1000) = 175, 000, and R (1000) = 40. (a) Evaluate each of the following, showing your reasoning. Then interpret in a complete sentence in plain English (in terms of DVD players and dollars). Do not use the terms derivative or marginal in your interpretation. i. P (1000) = ii. P (1000) = (b) If the company wishes to increase its profits, should it increase or decrease the production level or let it remain at 1000? Why? 4. 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) 2
5. Find an equation of the line tangent to the graph of the function f(x) = x 3 + x 2 x + 5 at the point where x = 2 by following the given steps. (a) Find the y-coordinate of the point on the curve y = f(x) where x = 2. (b) Find the derivative f (x). You may use short-cuts. (c) Find the slope of the tangent to the curve y = f(x) at the point where x = 2. (d) Find an equation of the tangent to the curve y = f(x) at the point where x = 2. Write it in the form y = mx + b. (e) Find the points on the curve y = f(x) where the tangent line is horizontal. 6. 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) 3
7. Find and simplify the derivative, f (x), for the function f given. Use the definition of the derivative (the limit of the difference quotient). Start with the general formula and show all steps. f(x) = 3x 2 + 2x 4 f (x) = 8. Find and simplify the derivative, f (x), for the function f given. Use the definition of the derivative (the limit of the difference quotient). Start with the general formula and show all steps. f(x) = 4 5x f (x) = 4
9. Find the derivatives of the following functions. Please SIMPLIFY your answers. It might help to simplify the original function before differentiating. (a) f(x) = 5 x 4 + 1 5 x 2 f (x) = (b) f(x) = 3 x + 1 3x 1 3 f (x) = (c) f(x) = 1 x2 1 + x 2 f (x) = (d) f(x) = 1 x3 2x f (x) = Hint: Use Chain Rule (e) f(x) = x ln x x f (x) = 5
10. (12 points) A widget company s cost function is C(x) = 1000 + 300x x 2 dollars. (a) Find the exact cost of producing the 61st widget. (b) Find the marginal cost function. (c) Find the marginal cost at x = 60 and explain what it represents (in plain English). (d) For y = C(x), find a formula for the differential dy. (e) If x = 60 and dx = x = 1, find dy. (f) If x = 60 and dx = x = 1, find y. 11. (10 points) Give all possible meanings of each mathematical expression from the list below. You may use each letter any number of times. Mathematical Expression Meaning(s) (list by letter) f(a) f(a + h) f(a) h f(a + h) f(a) lim h 0 h f (a) Possible Meanings: (a) the slope of a tangent line to the graph y = f(x) (b) the y-coordinate of a point on the graph y = f(x) (c) the average rate of change of f with respect to x (d) the slope of a secant line to the graph y = f(x) (e) the instantaneous rate of change of f with respect to x (f) an output of the function f 6
12. A rocket is traveling with velocity v(t) = 3t 2 + t feet per second at time t seconds after take-off. Include appropriate units with your answers. (a) Find the velocity of the rocket after 10 seconds. (b) Find the acceleration a(t) of the rocket after t seconds. Hint: Acceleration is the rate of change of velocity with respect to time. (c) Find the acceleration of the rocket after 10 seconds. (d) If the rocket took off from a platform 12 feet above the ground, find the height h(t) of the rocket above the ground at t seconds. (e) Find the height of the rocket after 10 seconds. 13. Use the graph of f to find each of the following values. y (a) f(a) = 9 y=f(x) (b) f(a + x) = 5 (c) y = 2 (d) dy = a a+ x x 7
14. Suppose the function f has the following properties. f(0) = 5 f ( 4) = 0 f (0) = 0 f (4) = 0 f ( 2) = 0 f (2) = 0 f (4) = 0 (a) Complete the sign chart for f with the phrases increasing, decreasing, concave up, or concave down. interval (, 4) ( 4, 0) (0, 4) (4, ) f (x) + f is (b) Complete the sign chart for f with the phrases increasing, decreasing, concave up, or concave down. interval (, 2) ( 2, 2) (2, 4) (4, ) f (x) + + f is (c) Give the x-coordinates for all of the following. (Write none if there aren t any.) critical points of f: x = local maximum points of f: x = inflection points of f: x = local minimum points of f: x = (d) Sketch a possible graph of the function f. Indicate all critical points and inflection points as well as other important behavior very clearly. (You are not given the y-coordinates of most points. Label these points as (2, f(2)), for example.) 8
15. Suppose that at a price of $29.99 per widget, elasticity of demand for Super Widgets is 1.5. If the price is increased, then demand should (a) increase weakly. (b) increase strongly. (c) decrease weakly. (d) decrease strongly. (e) none of these 16. Suppose that at a price of $3,500 per widget, elasticity of demand for Tiny Widgets is 1. Then demand is (a) elastic. (b) inelastic. (c) of unit elasticity. (d) none of these 17. Suppose that at a price of $5.95 per widget, elasticity of demand for Electric Widgets is 0.67. To increase revenue, the producer should (a) raise the price. (b) lower the price. (c) leave the price alone; revenue is maximal for the current price. (d) none of these 18. (10 points) Suppose the demand function for Atomic Widgets is x = D(p) = 1000e 0.5p widgets at price p dollars per widget. (a) Find D (p). (b) Find and simplify an expression for elasticity of demand, E(p). (c) Find elasticity of demand at price $1.50 per widget. (d) Find and simplify the price p at which demand is of unit elasticity. 9
19. Let f(t) = 4t + 1, where t represents time in years. (a) Find and simplify an expression for the relative rate of change of f at time t years. (b) Find the relative rate of change of f at t = 1 year. (Express it as a percentage rate of change.) 20. Find the absolute extreme values of the function f(x) = 3x 4 16x 3 + 18x 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 =. 10
21. Let f(x) = 1 3 x3 + 1 2 x2 2x 1. Find f (x) = and f (x) =. (a) Find the domain of f. 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 3 2 1 3 2 1 1 2 1 3 2 3 (f) Find the x-coordinates of all possible inflection points 11
22. 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? 12
23. 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. 24. A bank offers money market accounts at 2.75% annual interest compounded continuously. (a) Give the formula for the amount A(t) in the account after t years when $4000 dollars are invested. (b) After how many years does the account reach $25,000 in value? Show your reasoning. Round to the nearest year. 13
25. 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? 26. 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) (x 1)e 3x2 6x dx Check: 14
( x 2 + 2 ) 2 (b) x 3 dx Check: Hint:Do some algebra. (c) ln(1 x) 1 x dx Check: 27. The marginal cost of manufacturing x yards of a certain fabric is 3 0.01x + 0.000006x 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. 15
28. Find each definite integral. Give exact answers, simplified. Show all steps for full credit. (a) 2 1 ( 2x + x 2 + 2 x + 1 ) dx 2x (b) 4 0 x 16 x 2 dx (c) e 8 e (ln x) 3 x dx 16
29. A company s marginal cost function is 1 MC(x) = dollars per unit and its fixed 2x + 625 costs are $450. Find the cost function C(x). 31. Suppose the life expectancy (in years) of a scientific calculator is a continuous random variable t with probability density function { 3 (t+3) if t 0; f(t) = 2 0 otherwise. (a) Determine the probability a randomly selected scientific calculator lasts between 6 and 12 years. (b) Determine the probability a randomly selected scientific calculator lasts 12 years or less. 30. The black squirrel population of Kent is predicted to be P (t) = 250e 0.08t, where t is the number of years after the year 1990. Find the predicted average black squirrel population between the years 2012 and 2017. (Round your answer to the nearest whole squirrel.) (c) Determine the probability a randomly selected scientific calculator lasts more than 12 years. (d) Determine the probability a randomly selected scientific calculator lasts exactly 12 years. (e) Find b so that the probability a randomly selected scientific calculator lasts b minutes or less is 0.5.. 17
32. We wish to find the area A of the region bounded by the curves y = f(x) and y = g(x) where f(x) = x 2 4 and g(x) = 2x 1. (a) Find the points of intersection of the curves algebraically. (Set up and solve an equation.) 5 1 1 (b) Determine algebraically which function is the top function on the interval determined by their points of intersection. (d) Set up and evaluate an integral representing the area A. Write out each step using proper notation. Give an exact answer, simplified. (c) Sketch the graphs, labeling the points of intersection and shading the region whose area we wish to find. 18
33. A single injection of a drug is administered to a patient. The amount Q in the body then decreases at a rate proportional to the amount present. For a particular drug, the rate is 11% per hour. Thus, the following differential equation and initial condition are true, where t is the time in hours. dq dt = 0.11Q where Q(0) = Q 0. If the initial injection is 5 milliliters, find Q = Q(t) satisfying both conditions. 36. If the Gini Index for wealth distribution for Sylvania in was 0.56 in 1925 and 0.63 in 1935, in which year was wealth distributed more equitably in Sylvania? 37. Find the consumers surplus and producers surplus at the equilibrium price level for the price-demand equation p = D(x) = 25 0.004x 2 and the price-supply equation p = S(x) = 5 + 0.004x 2. 34. Find the particular antiderivative of the derivative dx dt = 10et 8 that satisfies the condition x(0) = 50. 35. The Lorenz curve for income distribution in Freedonia in 1938 is L(x) = x 7.5. Find the Gini Index.. 19