Step 1 :Find the derivative of the function \( s(x) \) with respect to \( x \): \( s'(x) = -3600(60+x)^{-2} \)
Step 2 :Substitute \( x=0 \) into the derivative to find \( s'(0) \): \( s'(0) = -\frac{1}{3600} \)
Step 3 :Use the linear approximation formula: \( f(a+h) \approx f(a) + f'(a)h \)
Step 4 :Substitute \( a=0 \) and \( h=6 \) into the linear approximation formula: \( f(6) \approx f(0) + f'(0) \cdot 6 \)
Step 5 :Calculate \( f(6) \) using the values: \( f(6) \approx \frac{3599}{60} \)
Step 6 :The person's approximate average speed if they travel one mile in 66 seconds is \( \frac{3599}{60} \) miles per hour
Step 7 :Substitute \( x=66 \) into the original function \( s(x) \) to find the person's exact speed: \( s(66) = \frac{200}{7} \)
Step 8 :The person's exact speed is \( \frac{200}{7} \) miles per hour