Practice Problem List II

Transcription

1 Math 46 Practice Problem List II Section 4.: 3, 3, 5, 9, 3, 9, 34, 39, 43, 53, 6-7 odd Section 4.:, 5, 7, 3, 9, 5, 35, 39, 47, 5, 55-6 odd, 65, 67 Section 4.3:, 7- odd, 5, 3, 4, 45, 53, 59, 77, 8-85 odd, 89, 9 Sinusoidal Functions d.) Find for each of the following. dt a.) = 5 sin t b.) = 4cos( t ) c.) = t sin( 3t) d.) = 6 sin( t) + 3cos(4t).) A compan s monthl sales, S (t), are seasonal and given as a function of time, t, in months, π b S( t) = + 6sin t 6 a.) Sketch a graph of S (t) for t. What is the maimum monthl sales? What is the minimum monthl sales? If t = is Januar, when during the ear are sales highest? b.) Find S () and S (), and interpret each in terms of sales. Linear Approimations Follow this path: Tet Web Site? Everthing For Calculus? Linear Approimations and Error Correction (from Chapter 4 online topics). Work through eercises dealing with linear approimations (ignore error correction). Section 5.: -5 odd, 3, 33, 37, 4, 43 Section 5.: -5 odd, 9-37 odd, odd Etras.) The graph of a cost function is given to the right. a.) At q =, estimate both average cost and marginal cost and represent our answers graphicall. b.) Approimate the value of q at which average cost is minimized. $ 4 C(q) 4 q

2 Section 5.3: -39 odd, 43, 47, 5, 55, 6-67 odd, 7, 8, 83 Etras.) For each function given below, a.) Estimate the intervals on which the derivative is positive and the intervals on which the derivative is negative. b.) Estimate the intervals on which the second derivative is positive and the intervals on which the second derivative is negative. (I) (II).) The graph of g is given to the right. At which of the marked values of is a.) g () greatest? b.) g () least? c.) g () greatest? d.) g () least? e.) g ( ) greatest f.) g ( ) least? a b c d e f 3.) Let P (t) represent the price of a share of stock of a corporation at time t. What does each of the following statements tell us about the signs of the first and second derivatives of P (t)? a.) The price of the stock is rising faster and faster. b.) The price of the stock is bottoming out. 4.) If water is flowing at a constant rate into the urn shown to the right, sketch a graph of the depth of the water against time. Mark on the graph the time at which the water reaches the widest point of the urn.

3 Section 6.: - odd, 3, 33, 4, 43, 45, 49, 5 Etras.) Find the following indefinite integrals. a.) b.) e t dt cosθ dθ c.) sin d d.) e 3r dr Section 6.3: -7 odd, 5, 9,, 7 Etras.) A car starts moving at time t = and continuousl accelerates. Its velocit is shown in the following table. Use the information to estimate how far the car travels during the seconds. t (seconds) Velocit (ft/sec) ) Coal gas is produced at a gasworks. Pollutants in the gas are removed b scrubbers, which become less efficient as time goes on. The following measurements, made at the start of each month, show the rate at which pollutants are escaping in the gas: Time (months) Rate pollutants are escaping (tons/month) a.) Make an overestimate and an underestimate of the total quantit of pollutants that escaped during the first month. b.) Make an overestimate and an underestimate of the total quantit of pollutants that escaped during the si months. Section 6.4: -9odd, 7-3 odd Etras.) Suppose f ( ) = e. a.) Is f ( ) d positive, negative, or zero? Eplain without attempting to calculate the integral. b.) Use rectangles to eplain wh < f ( ) d < 3.

4 .) The number of sales per month made b two new salespeople is shown in the figure to the right. a.) Which person has the most total sales after 6 months? After the first ear? b.) At approimatel what times (if an) have the sold roughl equal amounts? c.) Approimatel how man total sales has each person made at the end of the first ear? number of sales per month Salesperson B Salesperson A t (months) 3.) The velocit of a car (in miles per hour) is given b v( t) = t.8t, where t is in hours. a.) Write a definite integral for the distance the car travels during the first si hours. b.) Sketch a graph of velocit against time and represent the distance traveled during the first si hours as an area on our graph. c.) Find the distance traveled during the first si hours. 4.) If the graphs of f and g are in the figure to 6 the right, estimate the values of f ( ) d and 4 g() 5 g ( ) d. f() Section 6.5: -3 odd, 63, 67-8 odd, 87 Etras:.) Use our graphing calculator to calculate the following definite integrals. 5 a.) d + b.) e d 3 c.) 3 9 d

5 .) A cup of coffee at 9 o C is put into a o C room when t =. The coffee s temperature, t o f (t), is changing at a rate given b f ( t) = 7(.9) C per minute, where t is in minutes. Estimate, to one decimal place, the coffee s temperature when t =. 3.) The graph to the right shows P (t), the rate of change of the price of stock in a certain compan. a.) At what time during the five-week period was the stock at its highest value? At its lowest value? b.) If P (t) represents the price of the stock as a function of time, put the following quantities in increasing order: P ( ), P(), P(), P(3), P(4), P(5). $/share per week 5 P (t) t (weeks) 4.) A marginal cost function C (q) is given b the graph to the right. If the fied costs are $,, estimate the following. a.) The total cost to produce units. b.) The additional cost if the compan increases production from units to 4 units. c.) The value of C (5). Interpret our answer in terms of costs of production. $/unit q 5.) The figure to the right represents our velocit, v, on a biccle trip along a straight road. Suppose that ou start out 5 miles from home. Assume that positive velocit indicates movement awa from home. a.) Do ou start out going towards or awa from home? How long do ou continue in that direction and how far are ou from home when ou turn around? b.) How man times do ou change direction? c.) Do ou ever get home? Eplain. d.) Where are ou at the end of the fourhour bike ride? v (mph) - - t (hours)