Final answer:
The change in cost ΔC when x changes from 50 to 51 for the function C(x) is Δ6849, and the rate of change of the cost function C'(x) at x=50 is Ζ706.
Step-by-step explanation:
The student asked to find ΔC and C'(50) when x=50 and Δx=1 for the total-cost function C(x)=x³−8x²+6x+50. To find ΔC, we evaluate the cost function at x=50 and x=51, and find the difference. To find C'(x), we first calculate the derivative of the cost function.
Step-by-Step Calculation
To compute ΔC:
C(50) = (50)³ − 8(50)² + 6(50) + 50 = 125000 − 20000 + 300 + 50 = 105350
C(51) = (51)³ − 8(51)² + 6(51) + 50 = 132651 − 20808 + 306 + 50 = 112199
ΔC = C(51) − C(50) = 112199 − 105350 = Δ6849
To compute C'(50), first find the derivative:
C'(x) = 3x² − 16x + 6
C'(50) = 3(50)² − 16(50) + 6 = 7500 − 800 + 6 = Ζ706
Therefore, the correct answer is B) ΔC=Δ6849; C'(50)=Ζ706