1 Math 115 Off-Line Homework Section 3.4/3.5/3.6 WebAssign Section 3.4/3.5/3.6 #1 (ebook Section 3.4#26) The following table shows the percentage of the U.S. Discretionary Budget allocated to education in 2003, 2005 and 2009 (t = 0 represents 2000). Compute and interpret the average rate of change of P(t) over the given periods: Year t 3 5 9 Percentage P(t) 6.8 7 6.2 a. The period from 2003 to 2009. Answer in a sentence and include units with your answer. b. The period from 2005 to 2009. Answer in a sentence and include units with your answer. WebAssign Section 3.4/3.5/3.6 #2 (ebook Section 3.4#10) Calculate the average rate of change of the given function over the given interval. Use correct units with your answer. The y-axis is in dollars. Here a = 20, b = 24, and c = 28. Interval [1, 5]. Average rate of change (include units!):
2 Math 115 Off-Line Homework Section 3.4/3.5/3.6 WebAssign Section 3.4/3.5/3.6 #4 (ebook Section 3.4#13) Calculate the average rate of change of the function f (x) = x 3 3 over the interval [1, 3]. WebAssign Section 3.4/3.5/3.6 #6 (ebook Section 3.5#7) Consider the function as representing the value of an ounce of palladium in U.S. dollars as a function of the time t in days. R(t) = 270 + 20t 3 a. Find the average rate of change of R over the intervals [t, t + h], where t = 1 and h = 1, 0.1, and 0.01 days. Record your answers in the table. The average rate of change of R for t in the h Interval interval [1, 1 + h] units: 1 [1, 2] 0.1 [1, 1.1] 0.01 b. Use your answers to estimate the instantaneous rate of change of R at t = 1. Include units with your answer.
3 Math 115 Off-Line Homework Section 3.4/3.5/3.6 Definition The definition of the derivative of a function f (x) is given by WebAssign Section 3.6 #9 (ebook Section 3.6#15) Compute the derivative f (x) of the given function using the definition. f (x) = x 2 + 1 WebAssign Section 3.4/3.5/3.6 #11 (ebook Section 3.6#20) Compute the derivative f (x) algebraically, using the definition. f (x) = 2x 2 + x
4 Math 115 Off-Line Homework Section 3.4/3.5/3.6 WebAssign Section 3.4/3.5/3.6 #12 (ebook Section 3.6#25) Compute the derivative f (x) algebraically, using the definition. f (x) = 1 x
1 Math 115 Off-Line Homework Section 4.1 WebAssign Section 4.1 #3 (ebook Section 4.1#7) Calculate the following. d ( dx 2x4 + 3x 3 1) WebAssign Section 4.1 #4 (ebook Section 4.1#20) Calculate the following. Express your answer without negative exponents. d ( dx x0.5 + 2x 0.5 ) WebAssign Section 4.1 #5 (ebook Section 4.1#28) Find the derivative of the function h(x) = 2 x 2 x 3 + 1 x 4 WebAssign Section 4.1 #6 (ebook Section 4.1#37) Find the derivative of the function s(x) = x x 2 1 x WebAssign Section 4.1 #7 (ebook Section 4.1#40) 2x + x2 Find the derivative of the function t(x) = x
2 Math 115 Off-Line Homework Section 4.1 WebAssign Section 4.1 #10 (ebook Section 4.1#61) Find the equation of the tangent line to the graph of the function f (x) = x + 2 x at x = 3. Add the graph of the tangent line to the graph of f shown below. WebAssign Section 4.1 #15 (ebook Section 4.1#97) Increasing numbers of manatees ("sea sirens") have been killed by boats off the Florida coast. The following graph shows the relationship between the number of boats registered in Florida and the number of manatees killed each year. The regression curve shown is given by f (x) = 3.55x 2 30.2x + 81 where x is the number of boats (in units of 100,000) registered in Florida in a particular year and f (x) is the number of manatees killed by boats in Florida that year. a. Find f '(x). 10 8 6 4 2 0-10 -9-8 -7-6 -5-4 -3-2 -2-1 0 1 2 3 4 5 6 7 8 9 10-4 -6-8 -10 b. What are the units of measurement of f '(x)? c. At a level of 800,000 boats, at what rate is the number of manatee deaths increasing? Include units with your answer. d. Is f (x) increasing or decreasing with x? How do you know? e. Is f '(x) increasing or decreasing with x? How do you know?
1 Math 115 Off-Line Homework Section 4.2 WebAssign Section 4.2 #5 (ebook Section 4.2#11) Your college newspaper, The Collegiate Investigator, sells for 90 per copy. The cost in dollars of producing x copies of an edition is given by C(x) = 70 + 0.1x + 0.001x 2 a. Calculate the revenue, marginal revenue, profit and marginal profit functions. b. Compute the revenue, marginal revenue, profit and marginal profit, if you have produced and sold 500 copies of the latest edition. Interpret the results (include units on your answers!). R(500) = means: R'(500) = means: P(500) = means: P'(500) = means: c. For which values of x is the marginal profit zero? Interpret your answer. Since P(x) is a quadratic function, P '( ) = 0 means P(x) has its vertex at x =. What does that say about the profit from newspaper sales?
2 Math 115 Off-Line Homework Section 4.2 WebAssign Section 4.2 #8 (ebook Section 4.2#13) Suppose P(x) represents the profit in dollars on the sale of x DVDs. Suppose P(1,000) = 3,000 and P'(1,000) = 3. Interpret P(1,000) = 3,000 in terms of profit. The units on these quantities are: 1,000 3,000 3 Interpret P'(1,000) = 3 in terms of profit. WebAssign Section 4.2 #9 (ebook Section 4.2#16) Your monthly profit (in dollars) from selling magazines is given by P(x) = 5x + x where x is the number of magazine you sell in a month. If you are currently selling 50 magazines per month, find your profit and your marginal profit. Interpret your results: The current profit is (include units), and the profit is (circle one) increasing / decreasing at a rate of (include units).
3 Math 115 Off-Line Homework Section 4.2 WebAssign Section 4.2 #11 (ebook Section 4.2#17) Assume the demand equation for tuna in a small coastal town is given by p = 20,000 q 1.5 when 200 q 800 where p is the price (in dollars) per pound of tuna, and q is the weight (in pounds) of tuna that can be sold at the price p in one month. a. Calculate the price that the town's fishery should charge for tuna in order to produce a demand of 400 pounds of tuna per month. b. Find the monthly revenue R as a function of he number of pounds of tuna q. c. Find the marginal revenue function. d. Calculate the revenue and marginal revenue at a demand level of 400 pounds per month and interpret the result. e. If the town fishery's monthly tuna catch amounted to 400 pounds of tuna, and the price is at the level in part (a), would you recommend that the fishery raise or lower the price of tuna in order to increase its revenue? Hint: Since R'(400) =, the revenue is increasing / decreasing when the demand is for 400 pounds of tuna. So to increase revenue, you should make demand higher than 400 / lower than 400. What should you do to the price to make that happen? Revenue (dollars) 1600 1400 1200 1000 800 600 400 200 0 0 200 400 600 800 1000 Quantity of Tuna (pounds)
4 Math 115 Off-Line Homework Section 4.2 WebAssign Section 4.2 #13 (ebook Section 4.2#23) Your company is planning to air a number of television commercials during the ABC Television Network's presentation of the Academy Awards. ABC is charging your company $1.6 million per 30-second spot. Additional fixed costs (development and personnel costs) amount to $500,000, and the network has agreed to provide a discount of $10,000 x for x television spots. a. Find the cost function C, the marginal cost function C', and the average cost function C. b. Compute C'(3) and C (3). (Round all answers to 3 significant figures. Include units.) c. Use your answers in part (b) to say whether the average cost is increasing or decreasing as x increases from a starting point of 3 advertising spots. (See Example 4 Section 4.2, in the text or lecture notes.)
1 Math 115 Off-Line Homework Section 4.5 Find the derivative of each of the following functions. WebAssign Section 4.5 #7 WebAssign Section 4.5 #3 (ebook Section 4.5#7) (ebook Section 4.5#11) f (x) = e x + 3 g(x) = 4 x WebAssign Section 4.5 #4 (ebook Section 4.5#14) h(x) = 3 x2 x Extra practice (ebook Section 4.5#9) f (x) = e x Extra practice (ebook Section 4.5#13 g(x) = 2 x2 1 Extra pratice (ebook Section 4.5#59 h(x) = 3 2 x 4 WebAssign Section 4.5 #6 (ebook Section 4.5#54) Find the derivative. h(x) = e 2 x2 x+1/ x WebAssign Section 4.5 #8 Find the derivative. h(x) = 5 x2 + 2 6 x WebAssign Section 4.5 #9 Find the derivative. h(x) = 7e 3x + x 60 + 60
2 Math 115 Off-Line Homework Section 4.5 WebAssign Section 4.5 #12 (ebook Section 4.5#98) A few weeks into the SARS epidemic in 2003, the number of cases was increasing by about 4% each day. On April 1, 2003, there were 1,804 cases. a. Find an exponential model that predicts the number A(t) of people infected t days after April 1, 2003. b. How fast was the epidemic spreading on April 30, 2003? Round your answer to the nearest whole number of new cases per day.)
1 Math 115 Off-Line Homework Section 4.3 WebAssign Section 4.3 #1 (ebook Section 4.3#95) Suppose f and g are functions of time, and at time t = 3, f equals 5 and is rising at a rate of 2 units per second, and g equals 4 and is rising at a rate of 5 units per second. Fill in the blanks: the product fg equals and is rising at a rate of units per second. WebAssign Section 4.3 #3 (ebook Section 4.3#97) Suppose f and g are functions of time, and at time t = 3, f equals 5 and is rising at a rate of 2 units per second, and g equals 4 and is rising at a rate of 5 units per second. Fill in the blanks: the product f/g equals and is rising at a rate of units per second. WebAssign Section 4.3 #5 (ebook Section 4.3#81) Dorothy Wagner is currently selling 20 "I heart Calculus" T-shirts per day, but sales are dropping at a rate of 3 shirts per day. She is currently charging $7 per T-shirt, but to compensate for dwindling sales, she is increasing the unit price by $1 per day. How fast and in what direction is her daily revenue currently changing? WebAssign Section 4.3 #6 (ebook Section 4.3#21) Let y = 2x + 4 dy. Calculate. Simplify your answers. 3x 1 dx WebAssign Section 4.3 #8 (ebook Section 4.3#30) Let y = (4x 2 + x)(x x 2 ). Calculate dy. Practice the product rule, but don t simplify. dx
2 Math 115 Off-Line Homework Section 4.3 WebAssign Section 4.3 #10 (ebook Section 4.5#66) 1 Find the derivative of the function g(x) = e x + e. x WebAssign Section 4.3 #11 (ebook Section 4.5#64) Find the derivative of the function g(x) = (x 2 + 1)4 x2 1. WebAssign Section 4.3 #12 (ebook Section 4.5#62) Find the derivative of the function g(x) = e 2 x 4 2 x.
3 Math 115 Off-Line Homework Section 4.3 WebAssign Section 4.3 #15 (ebook Section 4.3#77) The monthly sales of Sunny Electronics' new sound system, t months after the system's introduction, is given by q(t) = 2,000t 100t 2 units. The price Sunny charges t months after the system's introduction is p(t) = 1000 t 2 dollars per sound system. Find the rate of change of monthly sales, the rate of change of the price and the rate of change of the revenue 5 months after the introduction of the sound system. Interpret each answer. Interpretations: q'(5) =, which means months after its introduction, the is increasing / decreasing at a rate of (include units). p'(5) =, which means months after its introduction, the is increasing / decreasing at a rate of (include units). R'(5) =, which means months after its introduction, the is increasing / decreasing at a rate of (include units).