Find the integral. 12) 15)

Transcription

1 Find the location of the indicated absolute etremum within the specified domain. ) Minimum of f() = /- /; [0, ] 8) Maimum h() ) Minimum of f() = ; [-, ] ) Minimum of f() = ( + )/; [-, ] ) Maimum of f() = ( + )( - ); [-, ] ) Maimum of f() = ; [-, ] Find the location of the indicated absolute etremum for the function. 6) Maimum g() ) Minimum h() 9) Minimum Find the integral. h() ) 9z z - 7 dz ) te-7t dt ) (ln )6 d ) 6-7 d ) ( + ) d ) + + d

2 6) 0 d ) (8-7 + ) d 7) e- d ) - 9 d 8) e-0. d ) + e d 9) ( - 6)e-9 d 0) (ln ) d ) ( + -) d ) d (ln ) ) dr 6r - 7 6) ( + ) d ) 8e d 7) e e + e d ) e z 8 z dz 8) d ) (7 + ) d 9) 7 - e -. d ) ( ) d 6) ln 7 d Find d/d b implicit differentiation. 0) + + = 8 ) e + = 7) (ln )6 d 8) (t + t - ) dt ) + = ) / - / = ) + = 9) (/- /) d ) / + / = 0) 8 d ( - 9) 6) ln + = 7) + + = 8) - =

3 9) = 0) + = Solve the problem. ) S() = , 0 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. ) The price P of a certain computer sstem decreases immediatel after its introduction and then increases. If the price P is estimated b the formula P = 0t - 800t + 600, where t is the time in months from its introduction, find the time until the minimum price is reached. ) A piece of molding 98 cm long is to be cut to form a rectangular picture frame. What dimensions will enclose the largest area? 8) For a simpl supported beam with a load that increases uniforml from left to right, the bending moment M (in ft lb) at a distance of (in ft) from the left end is given b M = 6 (wl - w). Determine the location of the maimum bending moment. In the formula, w is the rate of load increase (in lb ft ) and l is the length (in ft) of the beam. 9) If the price charged for a cand bar is p() cents, then thousand cand bars will be sold in a certain cit, where p() = 8-6. How man cand bars must be sold to maimize revenue? 60) In a certain state, the rate (per 00,000 inhabitants) at which automobiles were stolen each ear during the ears are given in the figure. Consider the closed interval [990, 000]. ) P() = , 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. ) Find the dimensions that produce the maimum floor area for a one-stor house that is rectangular in shape and has a perimeter of ft. Rate per 00K People A B C D E F G 6) The velocit of a particle (in ft ) is given b v = s t - 6t + 8, where t is the time (in seconds) for which it has traveled. Find the time at which the velocit is at a minimum. 7) P() = , 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 Year A (990, 68) D (99, 8) G (996 B (99, 0) E (99, ) H (997 C (99, ) F (99, ) K (998 Give the absolute maimum and minimum on the interval and the ears when the occur. 6) The demand equation for a certain product is p + q = 00, where p is the price per unit in dollars and q is the number of units demanded. Find dp/dq.

4 6) The cost of a computer sstem increases with increased processor speeds. The cost C of a sstem as a function of processor speed is estimated as C = 6S - S + 00, where S is the processor speed in MHz. Find the processor speed for which cost is at a minimum. 6) P() = , 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. 6) An architect needs to design a rectangular room with an area of 87 ft. What dimensions should he use in order to minimize the perimeter? 70) The graph gives the profit P() as a function of production level. Use graphical optimization to estimate the production level that gives the maimum profit per item produced P() ) The demand equation for a certain product is 8p + q = 900, where p is the price per unit in dollars and q is the number of units demanded. Find dq/dp. 66) A compan wishes to manufacture a bo with a volume of cubic feet that is open on top and is twice as long as it is wide. Find the width of the bo that can be produced using the minimum amount of material. 67) A rectangular field is to be enclosed on four sides with a fence. Fencing costs $7 per foot for two opposite sides, and $8 per foot for the other two sides. Find the dimensions of the field of area 670 ft that would be the cheapest to enclose. 68) The correlation between respirator rate and bod mass in the first three ears of life can be epressed b the function log R(w) = log (w), where w is the bod weight (in kg) and R(w) is the respirator rate (in breaths per minute). Find R'(w) using implicit differentiation. 69) S() = , 0 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. 7) S() = , 0 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. 7) An architect needs to design a rectangular room with an area of 9 ft. What dimensions should he use in order to minimize the perimeter? Find the equation of the tangent line at the given point on the curve. 7) = -7; (-, ) 7) + = ; (, ) 7) - = ; (, ) 76) - + = -; (, -) 77) + + = 0; (0, -) 78) = ; (, ) 79) = ; (, -)

5 Use a calculator to find, to the nearest tenth, the location of the indicated etremum. 80) The absolute minimum for the function f() = ( - 9)/ with domain all real numbers 8) The absolute maimum for the function f() = - + with domain [-, ]. + + Find the locations of all absolute etrema if the eist. 8) f() = ) f() = Find the equation of the tangent line at the given value of on the curve. 8) + + = +, = 8) + =, = 86) - =, = 87) + =, =

6 Answer Ke Testname: BLENDED-PT-CHAP ) = ) = - ) = 0 ) = - ) = 6) = - 7) No minimum 8) = 9) = - 0) (z - 7)/ + C ) - e -7t + C ) (ln ) C ) 9 (6-7) / + C ) ( + ) + C ) ln C 6) 0 ln 0 + C 7) - e - + C 8) -0e-0. + C 9) e -9 + C 0) - ) 0(ln )0 + C 6r C ) e + C ) e z + C ) C ) ln + C 6) (ln 7) + C 7) (ln ) C 8) t + t - t + C 9) 7 7/ / + C 0) - ( - 9) + C ) C ) ln + 9 -/ + C ) ln + e + C ) + ln + C ) ln ln + C 6) / + / + C 7) ln(e + e) + C 8) + 6 ln + C 9) 7 ln + 0e-. + C 0) ) d d = - e - e - ) - ) / ) - + ) - / 6) d d = 6 - ln - + 7) ) - 9) - 6 0) - ( + ) ) C ) 6 months ) 9. cm 9. cm ) 8 hundred thousand ) 8.7 ft 8.7 ft 6) s 7) hundred thousand 8) = l 9) 6 thousand cand bars 60) Absolute maimum of 8 in 99 Absolute minimum of in 99 6) dp/dq = -q/p 6) 0. MHz 6) 6 hundred thousand 6) 9. ft 9. ft 6) dq/dp = -8p/q 66).6 ft 67) 7.7 $7 b. $8 68) R'(w) = -6.w-. 69) C 70) units 7) C 7) 9.7 ft 9.7 ft 7) = - 7) = - + 7) = 76) = 6-77) = - 78) = ) = - 80) There is no absolute minimum. 8) = -.8 8) Absolute maimum at = ; no absolute minima 8) Absolute minimum at = -; absolute maimum at = 6 8) = - - 8) = - +

7 Answer Ke Testname: BLENDED-PT-CHAP 86) = 87) = - + 7