Final Exam Sample Problems

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1 MATH 00 Sec. Final Exam Sample Problems Please READ this! We will have the final exam on Monday, May rd from 0:0 a.m. to 2:0 p.m.. Here are sample problems for the new materials and the problems from the past semesters. On the exam, you can use a calculator as long as it is NOT a graphing calculator or any device that has more functions other than calculation (i.e., cell phones, PDAs). You will have to show all your work in order to obtain full credit. Please write CLEARLY and NEATLY as much as possible. I highly recommend that you will use a pencil rather than a pen. Please read Student Code (especially SECTION V) before coming to the class ( You can bring either THREE 4 by 6 note cards or ONE 8.5 by 8.5 sheet that contains formulas, theorems, and rules of your choice. You CANNOT put any examples or solutions to problems. Chapter 0. Find the value of (6i 2 + 5i). i= 2. Find the expression of the sum without using the summation symbol:. Use rectangles to find the area under the graph of y = x 2 from x = 0 to x =. Use n equal subintervals. 4. Check your answer of # by using a definite integral. 5. Evaluate the integrals. (a) 6. If 2 4 (x x 2 + 4x + 2) dx (b) f(x) dx = and 4 7. Find the area between the curves. f(x) dx = 7, find (x + 4) 2 dx (c) 2f(x) dx. (a) y = x 2 x + 2 and y = x from x = 0 to x = 5 (b) y = x 2 and y = 4x + 5 (c) y = (x ) and y = 2x 8. The cost of producing x units of a certain item is C(x) = x x (a) Use C(x) to find the average cost of producing 000 units. n k=2 k n 2 e 2x dx (b) Find the average value of the cost function C(x) over the interval from 0 to Find the total income over the next 0 years from a continuous income stream that has an annual flow rate at time t given by f(t) = 25e 0.05t in thousands of dollars per year.

2 0. Suppose that a machine s production is considered a continuous income stream with an annual rate of flow at time t given by f(t) = 50e 0.2t in thousands of dollars per year. Money is worth 8%, compounded continuously. (a) Find the present value of the machine s production over the next 5 years. (b) Find the future value of the production 5 years from now. (c) Find the capital value of the stream.. (#5 (HW) on page 90) A 58-year-old couple are considering opening a business of their own. They will either purchase an established Gift and Card Shoppe or open a new Video Rental Palace. The Gift Shoppe has a continuous income stream with an annual rate of flow at time t given by G(t) = 0, 000 (dollars per year) and the Video Palace has a continuous income stream with a projected annual rate of flow at time t given by V (t) = 2, 600e 0.08t (dollars per year) The initial investment is the same for both businesses, and money is worth 0% compounded continuously. Find the present value of each business over the next 7 years (until the couple reach age 65) to see which is the better buy. 2. (#6 on page 90) If the couple in Problem 5 plan to keep the business until age 70 (for the next 2 years), find each present value to see which business is the better buy in this case.. The demand function for a product is p = 64 4x, and the supply function is p = x, where x is the number of units and p is in dollars. (a) Find the market equilibrium. (b) Find the consumer s surplus at market equilibrium. (c) Find the producer s surplus at market equilibrium. 4. (#25 (HW) on page 9) A monopoly has a total cost function C = x + 6x 2 for its product, which has demand function p = 60 x 2x 2. Find the consumer s surplus at the point where the monopoly has maximum profit. Chapter 4 (sections 4.2). Find z x and z y. (a) z = 4x 2 y + x y (e) z = e xy + y ln x (b) z = x 2 + 2y 2 (c) z = (xy + ) 2 (d) z = e x2 y ln xy (f) z = e 2. Find the partial derivative of f(x, y) = 4x 5xy 2 + y with respect to x at the point (, 2, 8). 2

3 More problems. Find the vertical and horizontal asymptotes of the function 2. Find the limit f(x) = 2x2 5x 2 x 2 x 2 x 2 4x + lim x x 2 + 2x If it does not exist, explain the reason with computations to verify your argument.. Determine the differentiability and continuity of the function { f(x) = x2 if x 0 0 if x < 0 at x = Find the derivatives of the following functions. (a) y = 2 (x4 2) 6 (b) p(q) = q2 + 2q + 2q 5q (c) f(x) = 2x(x + ) / 5. Find dy of the following functions. Simplify completely. dx (a) y = e2x (b) x + y = e x+y (c) y = x log 2(x + ) (d) y = log (2xy) (e) y = x(2x + ) /2 (f) ln(x + y) = (xy) 2 6. The price of a type of PDA is $00 per unit for the first 0,000 units of sales. After that, for each,000 units of sales, the price will be reduced by $0 per unit. How many units of sales will maximize the total revenue? 7. For the following functions, answer parts (a)-(i) with proper reasoning. () y = 2x2 8 (x ) (2) y = 2x2 (x + 2) 2, (a) Find the domain of the function. (b) Find the vertical and horizontal asymptotes if they exist. (c) Find the relative maxima, relative minima, horizontal points of inflection if they exist. (d) Find the absolute maximum and absolute minimum if they exist. (e) Find the points of inflection if they exist. (f) Find the concavity of the function in the entire domain. (g) Find the intercepts. (h) Find coordinates of two other points. (i) Sketch the graph of the function. Make sure to show all the information you found above. If your graph that shows the overall shape does not show the detail of everything well, show, in addition, zoom-in views of parts. 8. Evaluate the integrals. 9x 2 + (a) x + x dx (b) 0 xe 2 x2 dx (c) (x 5 2 x + + 4x ) 2 dx (d) x (2x + 4x) 4 dx

4 9. Suppose that the demand for a product is given by p = 00, where p is the price and q is the ln(q + ) quantity. (a) Find the elasticity η(q) as a function of q. (b) Find the type of elasticity when p = $ Suppose that the demand for a product is given by p = ln quantity and q >. ( ) q, where p is the price, q is the q (a) Find the elasticity of demand as a function of the quantity demanded q (i.e., find η(q)). (b) Find the type of elasticity when p = $0.. Suppose the supply (before taxation) and demand functions for a product are p = 5q + 20 and p + 5q = 00 respectively, where p is in dollars and q is the number of units. (a) Find the quantity q that maximizes the tax revenue T. (b) Find the tax per unit t that maximizes the tax revenue T, and find the maximum tax revenue. 2. Find the area under the curve y = x 2 + from x = 0 to x = using rectangles. Note that with n equal subintervals, the sum of the areas of rectangles with the left-hand endpoints is given by { n ( ) 2 i S L = + } n n i=0. Find the area between the curves y = x and y = x Suppose a product has a daily marginal revenue MR = 60 and a daily marginal cost MC = x, both in dollar per unit. If the total cost is $0 when 0 units are produced, how many units will give the maximum profit and what is the maximum profit? 5. Find the derivative of each of the following functions using the definition of derivative. (a) f(x) = 4 x 2 (b) f(x) = x + x 2 6. Determine the differentiability of the function { x if x 0 f(x) = x 2 if x < 0 by checking the definition of derivative with left and right limits at certain point(s). 7. Find the equation of the tangent line to the graph of the function y = slope-intercept form. (x 2) x at x = in the 8. If the total revenue function for a product is R(x) = 20x + 80x 2 x dollars and the total cost function is C(x) = x + 2x 2 dollars, where x represents the number of units produced, answer the following questions. (a) Find the marginal profit function. (b) What is the marginal profit when 0 units are produced and sold? What does this mean? (c) Producing how many units will minimize average cost? Find the minimum average cost. 4

5 9. Suppose a product has a daily marginal revenue and a daily marginal cost MR(= R (x)) = 6e 0.08x, MC(= C (x)) = x + 8, respectively, both in dollars per unit. The daily fixed cost is $00. (a) Find the daily revenue and daily cost functions. (b) At what rate per day is the profit changing if the number of units produced and sold is 00 and increasing at a rate of 0 units per day? 20. If consumption is $ billions when disposable income is $.5 billions, and if the marginal propensity to save is ds dy = 0.5, (in billions of dollars) 4y + find the national consumption function. 2. A monopoly has a daily average cost for a product, C = 5x + 0. The daily demand for x units of the product is given by p = 0x Find the consumer s surplus at the point where the monopoly has maximum profit. 22. Suppose that a business provides a continuous income stream with an annual rate of flow at time t given by f(t) = 2, 000e 0.4(t+2) in dollars per year. If the interest rate is 7%, compounded continuously, answer the following questions. (a) Find the future value of the business 8 years from now. (c) Find the capital value of the business. 2. Find the partial derivative z x of the function z = xe xy + x 2 y. 5