A bicyclist is riding on a path modeled by the function f(x) = 0.03x, where x and f(x) are measured in miles. Find the rate of change of elevation when x = 3.
Rate of change of elevation with horizontal distance or with time?
For the latter, you need to know the speed.
For the former, f' = df/dx = 0.03 at all times.
0.09
.09
To find the rate of change of elevation when x = 3, we need to find the derivative of the function f(x) = 0.03x with respect to x.
Let's start by finding the derivative of the function.
f(x) = 0.03x
To find the derivative, we need to use the power rule, which states that if we have a function of the form f(x) = ax^n, where a and n are constants, the derivative is given by f'(x) = anx^(n-1).
In this case, a = 0.03 and n = 1. So, applying the power rule, we get:
f'(x) = d/dx (0.03x)
= 0.03 * d/dx (x)
= 0.03 * 1
= 0.03
The derivative of f(x) = 0.03x is f'(x) = 0.03.
Now, we can find the rate of change of elevation when x = 3 by evaluating the derivative at that point.
f'(3) = 0.03
Therefore, the rate of change of elevation when x = 3 is 0.03 miles per mile.
Note: The unit "miles per mile" means that for every 1 mile traveled in the x-direction, there is a change in altitude of 0.03 miles.