SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 8) Decreasing Find the open interval(s) where the function is changing as requested. 1) Decreasing; f() = - + 3 2) Increasing; = 2 + 2 3) Decreasing; f() = + 4 - Identif the intervals where the function is changing as requested. 4) Decreasing Solve the problem. 9) Suppose the total cost C() to manufacture a quantit of insecticide (in hundreds of liters) is given b C() = 3-272 + 240 + 900. Where is C() decreasing? 10) The number of people P(t) (in hundreds) infected t das after an epidemic begins is approimated b P(t) = 9-35t - 5 2 t 2. When will the number of people infected start to decline? 5) Decreasing 11) Suppose a certain drug is administered to a patient, with the percent of concentration in the bloodstream t hr later given b 5t K(t) =. On what time interval is the t2 + 1 concentration of the drug increasing? Find the values of an relative etrema. 1 12) f() = 2 + 1 ) Decreasing 13) f() = 2 + 1 2 14) f() = 3-12 + 2 15) f() = 3-32 + 1 7) Increasing Find the location and value of all relative etrema for the function. 1) 1
17) 2) Find the -coordinate of the relative maimum for the function f() = ( - 8)2/3 with domain all real numbers. Find the largest open intervals where the function is concave upward. 3 27) f() = + 2 18) 28) f() = 4-82 29) f() = 3-32 - 4 + 5 30) f() = 2 + 1 Find all relative maima or minima. 19) f() = 3e - 20) = 2e- 31) f() = -32 + 18 + 1 Find the largest open intervals where the function has the indicated concavit. 32) Concave upward 21) = + ln 22) = e8 Solve the problem. 23) P() = -3 + 152-48 + 450, 3 is an approimation to the total profit (in thousands of dollars) from the sale of hundred thousand tires. Find the number of hundred thousands of tires that must be sold to maimize profit. 33) Concave upward 24) S() = -3 + 2 + 288 + 4000, 4 20 is an approimation to the number of salmon swimming upstream to spawn, where represents the water temperature in degrees Celsius. Find the temperature that produces the maimum number of salmon. 34) Concave downward Use a graphing calculator or computer graphing software to solve the problem (correct to one decimal place). 25) Find the -coordinate of the relative minimum for the function f() = ( - 4)2/3 with domain all real numbers. 2
35) Concave upward The function gives the distances (in feet) traveled in time t (in seconds) b a particle. Find the velocit and acceleration at the given time. 45) s = -2t3 + 8t2-3t + 4, t = 2 4) s = t2-5, t = 3 Decide if the given value of is a critical number for f, and if so, decide whether the point for on f is a relative minimum, relative maimum, or neither. 3) f() = (2 - )(2-3); = 1 2 Answer the question. 47) The given graph is that of the derivative of a function f. Using information obtained from the graph of f' and the fact that f(1) = 1, sketch the graph of f. Eplain how ou obtained the graph of f. 37) f() = -2-1 - 4; = 8 Find f"() for the function. 38) f() = 3-7 39) f() = 23/2-1/2-1 40) f() = 2-1 - 41) f() = 83-22 + 7 Find the requested value of the second derivative of the function. 42) f() = ln ; Find f (1). 2 Solve the problem. 43) Find the point of diminishing returns (, ) for the function R() = 5000-3 + 32 + 00, 0 20, where R() represents revenue in thousands of dollars and represents the amount spent on advertising in tens of thousands of dollars. 48) The given graph is that of the derivative of a function f. Using information obtained from the graph of f' and the fact that f(-1) = - 4 3 and f(-3) = 0, sketch the graph of f. Eplain how ou obtained the graph of f. - 44) The percent of concentration of a certain drug in the bloodstream hr after the drug is 4 administered is given b K() = 2 + 4. At what time is the concentration a maimum? - 3
49) The given graph is that of the derivative of a function f. Using information obtained from the graph of f' and the fact that f(-2) = 4, sketch the graph of f. Eplain how ou obtained the graph of f. 51) a) Continuous and differentiable for all real numbers b) f () > 0 on (-3, -1) and ( 2, ) c) f () < 0 on (-, -3) and ( -1, 2) d) f () > 0 on (-, -2) and ( 1, ) e) f () < 0 on (-2, 1) f) f (-3) = f (-1) = f (2) = 0 g) f () = 0 at (-2, 0) and (1, 1) - - Sketch a graph of a single function that has these properties. 50) a) Continuous and differentiable for all real numbers b) f () < 0 on (-, -3 ) and ( 3, ) c) f () > 0 on (-3, 3) d) f () > 0 on (-, 0 ) e) f () < 0 on ( 0, ) f) f (-3) = f (3) = 0 g) An inflection point at (0,0) 52) a) Continuous for all real numbers b) Differentiable everwhere ecept = 0 c) f () < 0 on (-, 0) d) f () > 0 on ( 0, ) e) f () < 0 on (-, 0) and (0, ) f) f(-2) = f (2) = 5 g) -intercept and -intercept at (0,0) 4
Answer Ke Testname: 1325-BLD-T3-REL-DRAW 1) (-3, ) 2) ( 0, ) 3) ( -, ), (, ) 4) (-, 3) 5) (5, 12) ) (-3, -2) 7) (0, 5) 8) (-1, 0) 9) (8, 10) 10) Da 7 11) (0, 1) 12) Relative maimum of 1 at 0. 13) No relative etrema. 14) Relative maimum of 18 at -2; Relative minimum of -14 at 2. 15) Relative maimum of 1 at 0; Relative minimum of -3 at 2. 1) Relative minimum of -1 at -3 ; Relative maimum of 2 at -1 ; Relative minimum of 1 at 2. 17) Relative minimum of 1 at 2 ; Relative maimum of -1 at -2. 18) Relative minimum of -2 at -3 ; Relative maimum of 2 at 3. 19) Relative minimum of -7.34 at -3 20) (1, 2/e), relative maimum 21) (-1, -1) relative maimum 22) (- 1/8, - 1/(8e)), relative minimum 23) 8 hundred thousand 24) 12 C 25) 4.0 2) 10.4 27) (-2, ) 28) (-, - 2 3/3), (2 3/3, ) 29) (1, ) 30) ( 3, ) 31) None 32) (0, ) 33) (0, ) 34) (-, -2) 35) (0, ) 3) Not a critical number 37) Not a critical number 9 38) - 4(3-7)3/2 39) 1.5-1/2 + 1.5-3/2 40) 2 + 2 (2-1)3 41) 48-4 42) - 3 8 43) (12, 15,5) 44) 8 hr 45) v = 5 ft/s, a = -8 ft/s2 4) v = 1.5 ft/s, a = - 5 8 ft/s 2 47) Graphs and eplanations will var to some degree. The graph of f should look similar to the following. - - 48) Graphs and eplanations will var to some degree. The graph of f should look similar to the following. - - 5
Answer Ke Testname: 1325-BLD-T3-REL-DRAW 49) Graphs and eplanations will var to some degree. The graph of f should look similar to the following. - 50) - 51) 52)