The cost (in dollars) of producing x units of a certain commodity is given by C(

Question

The cost (in dollars) of producing x units of a certain commodity is given by C(x) = Squareroot x + 50. What is the average rate of change of C when the production level changes from x = 100 to x = 196? What is the instantaneous rate of x when the production level is x = 144? The power delivered by a battery to an apparatus of resistance R (in ohms) is given by p = 2.25 R/(R + 0.5)^2 watts Find the instantaneous rate of change of power with respect to resistance at R = 4 ohms. The function s(t) = 1/3 t^3 - 3/2t^2 + 2t - 2, for t greaterthanorequalto 0, describes the position of a particle moving along a horizontal axis, where s is measured in feet and t greaterthanorequalto 0 is measured in seconds. Find the position, velocity and acceleration at time t = 0.5. Is the particle moving to the left or to the right at t = 0.5? Explain. Is the particle speeding up or slowing down at t = 0.5? Explain. What is the position of the particle when it is at instantaneous rest? Let S_A = 15t^2 + 10t + 20 and S_B = 5t^2 + 40t, t greaterthanorequalto 0, be the position functions of cars A and B that are moving along parallel straight lines of a highway. S is measured in feet and t is measured in seconds. At what instants of time are the cars next to one another? At what instant of time do they have the same velocity? Which car is ahead at this instant? A toy rocket fired straight up into the air has height sit) = 320t - 16t^2 feet after t seconds. What is the velocity after 2 seconds? What is the highest height achieved by the rocket? At what time will the rocket return to the ground? At what velocity, will the rocket be traveling just as it returns to the ground? The height at time t (in seconds) of a mass, oscillating at the end of a spring, is x(t) = 30e^0.5t cos(2t) cm. Find the velocity of the mass t = pi/6 seconds. Give your answer accurate te four decimal places.

Explanation / Answer

1)given C(x)=x +50

a)average rate of change from x =100 to x =196 is =[C(196)-C(100)]/(196-100)

average rate of change=[(196 +50 )-(100 +50)]/96

average rate of change=[(14 +50 )-(10 +50)]/96

average rate of change=4/96

average rate of change=1/24

b)C(x)=x +50

differentiate with respect to x

C'(x)=(1/2x) +0

C'(x)=(1/2x)

at x =144

C'(144)=(1/2144)

C'(144)=(1/2*12)

C'(144)=1/24

instantanous rate of change =1/24

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2)P=2.25R/(R+0.5)2

differentiate with repect to R

dP/dR=[2.25(R+0.5)2 -2.25R2(R+0.5)]/((R+0.5)2)2

dP/dR=[2.25(R+0.5)-4.5R]/(R+0.5)3

dP/dR=[2.25R+1.125-4.5R]/(R+0.5)3

dP/dR=[1.125-2.25R]/(R+0.5)3

at R=4

dP/dR=[1.125-(2.25*4)]/(4+0.5)3

dP/dR=-7.875/91.125

dP/dR-0.08642

instantaneous rate of change is approximately -0.08642

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