Estimating speed Use the linear approximation given in Example 1 to answer the following questions.


If you travel one mile in 59 seconds, what is your approximate average speed? What is your exact speed?

Verified step by step guidance

1. Identify the relationship between distance, speed, and time. The formula for speed is given by \( v = \frac{d}{t} \), where \( v \) is speed, \( d \) is distance, and \( t \) is time.
2. Convert the given time from seconds to hours, since speed is typically measured in miles per hour (mph). There are 3600 seconds in an hour, so \( t = \frac{59}{3600} \) hours.
3. Use the linear approximation method to estimate the speed. Linear approximation involves using the derivative of a function to estimate the value of the function at a point close to a known value. Here, the function is \( v(t) = \frac{1}{t} \) and the derivative \( v'(t) = -\frac{1}{t^2} \).
4. Calculate the approximate speed using the linear approximation formula: \( v(t) \approx v(a) + v'(a)(t-a) \), where \( a \) is a point close to \( t \) for which \( v(a) \) is known. Choose \( a = 60 \) seconds (1 minute), where the speed is 60 mph.
5. Finally, calculate the exact speed using the formula \( v = \frac{d}{t} \) with \( d = 1 \) mile and \( t = \frac{59}{3600} \) hours to find the exact speed in mph.