The function $s(x)=\frac{3600}{60+x}=3600(60+x)^{-1}$ gives a person's average speed in miles per hour if he or she travels one mile in $60+x$ seconds. Use a linear approximation to $s$ at 0 to find a person's approximate average speed if he or she travels one mile in 66 seconds. What is his or her exact speed?

The person's approximate average speed is $\square \frac{\mathrm{mi}}{\mathrm{hr}}$.

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