Question

suppose that the cost (in dollars) for a company to produce x pairs of a new line of jeans is c(x)=3000 + 3x+.01x^{2}+.0002x^{3}. (a) find the rate of change of the cost function. (b) find c(100). (a) c(x)= (b) c(100)= dollars/pair

Explanation:

Step1: Apply power - rule for differentiation

The power - rule states that if $y = ax^n$, then $y^\prime=anx^{n - 1}$, and the derivative of a constant $k$ is 0. Given $C(x)=3000 + 3x+0.01x^{2}+0.0002x^{3}$.
The derivative of 3000 is 0, the derivative of $3x$ is $3\times1\times x^{1 - 1}=3$, the derivative of $0.01x^{2}$ is $0.01\times2x^{2 - 1}=0.02x$, and the derivative of $0.0002x^{3}$ is $0.0002\times3x^{3 - 1}=0.0006x^{2}$. So $C^\prime(x)=3 + 0.02x+0.0006x^{2}$.

Step2: Evaluate $C^\prime(x)$ at $x = 100$

Substitute $x = 100$ into $C^\prime(x)$.
$C^\prime(100)=3+0.02\times100 + 0.0006\times100^{2}$.
First, $0.02\times100 = 2$, and $0.0006\times100^{2}=0.0006\times10000 = 6$.
Then $C^\prime(100)=3 + 2+6=11$.

Answer:

(a) $C^\prime(x)=3 + 0.02x+0.0006x^{2}$
(b) $C^\prime(100)=11$