We have the cost function for x units as:
[tex]C(x)=7700+4x+0.01x^2+0.0002x^3[/tex]
a) We can find the marginal cost as the first derivative of the cost function:
[tex]\begin{gathered} \frac{dC}{dx}=7700(0)+4(1)+0.01(2x)+0.0002(3x^2) \\ \frac{dC}{dx}=4+0.02x+0.0006x^2 \end{gathered}[/tex]
b) We can find the marginal cost when x = 100 by replacing x with 100 in the marginal cost function:
[tex]\begin{gathered} \frac{dC}{dx}(100)=4+0.02(100)+0.0006(100^2) \\ \frac{dC}{dx}(100)=4+2+0.0006\cdot10000 \\ \frac{dC}{dx}(100)=4+2+6 \\ \frac{dC}{dx}(100)=12 \end{gathered}[/tex]
c) To calculate the cost at x = 100, we replace x with 100 in the cost function and calculate:
[tex]\begin{gathered} C(100)=7700+4(100)+0.01(100^2)+0.0002(100^3) \\ C(100)=7700+400+0.01\cdot10000+0.0002\cdot1000000 \\ C(100)=7700+400+100+200 \\ C(100)=8400 \end{gathered}[/tex]
Answer:
a) 4 + 0.02x + 0.0006x²
b) 12
c) 8400
