Question

decide whether the following problem can be solved using precalculus or whether calculus is required.
a bicyclist is riding on a path modeled by the function ( f(x) = 0.05(10x - x^2) ), where ( x ) and ( f(x) ) are measured in miles. find the rate of change of elevation at ( x = 1 ).
the problem can be solved using precalculus.
the problem requires calculus to be solved.
when the problem can be solved using precalculus, solve it. when the problem seems to require calculus, use a graphical or numerical approach to estimate the solution.

Explanation:

Step1: Recognize rate of change need

The rate of change of a function at a point is the derivative, which requires calculus. First, rewrite the function in expanded form.
$f(x)=0.05(10x - x^2)=0.5x - 0.05x^2$

Step2: Apply power rule for derivatives

Differentiate term-by-term using $\frac{d}{dx}(x^n)=nx^{n-1}$.
$f'(x)=\frac{d}{dx}(0.5x) - \frac{d}{dx}(0.05x^2)=0.5 - 0.1x$

Step3: Evaluate at $x=1$

Substitute $x=1$ into the derivative.
$f'(1)=0.5 - 0.1(1)$

Answer:

0.4 miles per mile (the rate of change of elevation at $x=1$ is 0.4)