Note: I gave a few examples of nearly each of these. eg. #17 and #18 are the same type of problem.

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1 Study Guide for Exam 3 Sections covered: 3.6, Ch 5 and Ch 7 Exam highlights 1 implicit differentiation 3 plain derivatives 3 plain antiderivatives (1 with substitution) 1 Find and interpret Partial Derivatives 1 Find minimum and maximum values using the D test 1 integrate a rate Note: I gave a few examples of nearly each of these. eg. #17 and #18 are the same type of problem. Topics: Implicit differentiation, related rates, area under a graph, area between 2 graphs, exponential and logarithmic functions, derivatives of exponential and logarithmic functions, antiderivatives of exponential functions and 1/x, applications, recover a function from a given rate, finding the accumulated value over an interval, functions of several variables, partial derivatives and their interpretation, and optimizing a function of 2 variables. dy 1. a) Find dx using implicit differentiation if y is given by the equation 10x 3 y + 4xy 4 = e 2x - 1 b) Find the equation of the tangent line at the point (1,0). m = e 2 / 5 Tangent line: y =

2 c) 3x 2 e 4x (3 + 4x) d) 4x 4 (5lnx + 1) e) + f) 2. The number x pair of sneakers that a manufacturer will supply per week and their price p (in dollars) are related by the equation 2x 2 = 10, p 2. If the price is rising at the rate of $3 per week, find how the supply will change if the current price is $50. dx/dt = 2.41 pair of sneakers per week. 3. Omit 4. Evaluate each integral: a) b) c) a) 10ln x + d) Let u = x 2 + 5, du = 2xdx with new limits for u being 5 and 9.

3 e) = f) = g) u = 2/x, du =(- 2/x 2 )dx with new limits for u being 2 and 1. h) u =e 3x + 1, du =3e 3x dx with new limits for u being e and e A company's marginal revenue function is MR = x 1/4, where x is the number of units. Find the Revenue function. (use R(0) = 0) R(x) = =

4 Since R(0) = 0, R(x) = 6. A company's marginal cost function is MC = 21x 4/3-6x 1/2 + 50, where x is the number of units and the company's fixed costs are $3,000. Find the Cost function MC(x) = 21x 4/3-6x 1/2 + 50, C(x) = C (x) = 7. In 2013, Amazon's annual revenue was 74 billion dollars and growing at the rate of 3.2x billion dollars per year, where x is the number of years since Find a formula to predict Amazon's annual revenue at any time x and use your formula to predict their revenue in (source: The New York Times) R(x) = And R(0) = 74 billion So, R(x) = R(7) = billion 8. The divorce rate in the US has been declining in recent years. The number of divorces per year is predicted to be decreasing at the rate of 0.94e -0.02t million where t is the number of years since a) Find a formula for the total number of divorces within t years of b) Use your formula to find the total number of divorces from 2014 to (Accumulation of a function) D(t) = = 47e -0.02t + 47 #Divorce from 2014 to 2020 =

5 9. A real estate office is selling condominiums at a rate of 100e -x/4 per week after x weeks. How many condominiums will be sold during the first 8 weeks? 10. An experimental drug lowers a patient's blood serum cholesterol at a rate of units per day, where t is the number of days since the drug was administered between day 0 and day 5. Find the total change in the first 3 days. units 11. Use a definite integral to find the area under the graph of from x = 1 to x = 4. Area = 12. Find the area bounded by the pair of functions y = 12x - 3x 2 and y = 6x Sketch the region and find the points of intersection. Done in class. 13. Given f(x,y) = 10x 2 y 2 + e x + lny, find

6 14. A company manufactures washing machines and dryers. The cost to manufacture a washing machine is $210 and a dryer is $180. The fixed costs are $4000. a) Find the cost function letting x represent the number of washing machines and y the number of dryers. C(x,y) = 210x + 180y A study found that a business person with a master's degree in business administration (MBA) earned an average salary of S(x,y) = 48, x y dollars in 2005 where x is the number of years of work experience before the MBA, and y the number of years of work experience after the MBA. Find and interpret S x and S y. S x = The average salary of a person with an MBA will increase by $4930 for each additional year of work experience prior to earning an MBA. S y = The average salary of a person with an MBA will increase by $3840 for each additional year of work experience after the MBA. Another question: which is more advantageous to one's salary, a year of work experience prior to earning an MBA or after? 16. Find the relative extreme values of f(x,y) = y 3 - x 2-2x - 12y. CPTs:

7 f x = -2x - 2 = 0 and f y = 3y 2-12 = 0 x = -1 and y = 2, -2 CPTs (-1, 2) & (-1, -2). f xx = -2 & f xy = 0 f yy = 6y D = -12y At (-1,2), D<0 f(-1,2) is a Saddle Point. At (-1,-2), D>0 and f xx = -2 f(-1,-2) = 17 is a Relative Maximum. 17. A company manufactures two products. The price function for product A is p = 16 - x for x in [0, 16] and the price function for product B is q = y for y in [0, 38], both in thousands of dollars, where x and y are the amounts of product A and B, respectively. If the cost function is C(x,y) = 10x + 12y - xy + 6 thousand dollars, find a) the Revenue Function, b) the Profit function, and c) the quantities and the prices of the two products that maximize profit. Also find the maximum profit. Verify that this is a maximum. a) R(x,y) = x(p) + y(q) = x(16-x) + y(19-0.5y) = 16x - x y - 0.5y 2 b) P(x,y) = -x 2-0.5y 2 + xy + 6x + 7y - 6 c) Use the D-test :) P x = -2x + y + 6 = 0 and P y = -y + x + 7 = 0 Solving the system: -2x + y = - 6 x - y = - 7 yields the Critical Point (13, 20) Now apply the D-test.

8 P xx = -2 & P xy = 1 P yy = -1 D = 2-1 = 1 At (13,20), D>0 and P xx = -2 P(13,20) = 103 is a Relative Maximum. 18. The number of office workers near a beach resort who call in "sick" on a warm sunny day is f(x,y) = xy - x 2 - y x + 50y where x is the air temperature with x in [70, 100], and y is the water temperature with y in [60, 80]. Find the air and water temperatures that maximize the number of absentees. CPTs: f x = y - 2x = 0 and f y = x - 2y + 50 = 0 CPT (90, 70) Apply the D-Test f xx = -2 & f xy = 1 f yy = -2 D = 3 At (90,70), D>0 and f xx = -2 f(90,70) = 1500 is a Relative Maximum. This means that when the air temperature is 90⁰ and the water temperature is 70⁰, the number of office workers near a beach resort who call in sick is maximized(a relative max) and that the maximum # of office workers calling in "sick" will be 1500!