Question

find the percentage rate of change of f(x) at the indicated value of x. f(x)=290 + 55x; x = 8. the percentage rate of change of f(x) at x = 8 is %. (type an integer or decimal rounded to the nearest tenth as needed.)

Explanation:

Step1: Find the derivative of f(x)

The derivative of $f(x)=290 + 55x$ with respect to $x$ is $f^\prime(x)=55$ (since the derivative of a constant 290 is 0 and the derivative of $55x$ using the power - rule $\frac{d}{dx}(ax)=a$ is 55).

Step2: Calculate f(8)

Substitute $x = 8$ into $f(x)$: $f(8)=290+55\times8=290 + 440=730$.

Step3: Calculate the percentage rate of change

The percentage rate of change of a function $y = f(x)$ at a point $x$ is given by $\frac{f^\prime(x)}{f(x)}\times100\%$. Substitute $f^\prime(8) = 55$ and $f(8)=730$ into the formula: $\frac{55}{730}\times100\%=\frac{5500}{730}\% \approx 7.5\%$.

Answer:

7.5