Q. 1TF - Calculus

Question

Approximating the length of a curve: Suppose you want to

calculate the driving distance between New York City and

Dallas, Texas.

Print out a highway map of the United States, and

highlight a route, snaking along with the paths of the

major highways.

How could we use the method of “subdivide, approximate, and add up” to approximate this driving distance?

Make an actual approximation of the driving distance

from New York City to Dallas, Texas, using the route

and method you just described.

How accurate do you think your approximation is? Is

it an over-approximation or an under-approximation?

Step-by-Step Solution

Answer

A highway map is shown below .

Step 1 . To find

$\mathrm{A} \mathrm{print} \mathrm{out} \mathrm{of} \mathrm{a} \mathrm{high} \mathrm{way} \mathrm{map} \mathrm{of} \mathrm{the} \mathrm{united} \mathrm{states} \mathrm{and} \mathrm{high} \mathrm{light} \mathrm{a} \mathrm{route} \mathrm{taking} \mathrm{along} \mathrm{with} \mathrm{the} \mathrm{paths} \mathrm{of} \mathrm{the} \mathrm{major} \mathrm{highways}.$

Step 2. Explanation

$$\begin{gathered} \mathrm{Consider} \mathrm{the} \mathrm{driving} \mathrm{distance} \mathrm{between} \mathrm{New} \mathrm{York} \mathrm{City} \mathrm{and} \mathrm{Dallas}, \mathrm{Texas}. \\ \mathrm{A} \mathrm{high} \mathrm{way} \mathrm{map} \mathrm{of} \mathrm{the} \mathrm{United} \mathrm{States} \mathrm{is} \mathrm{given} \mathrm{below}. \\ \mathrm{The} \mathrm{below} \mathrm{route} \mathrm{taking} \mathrm{along} \mathrm{with} \mathrm{the} \mathrm{paths} \mathrm{of} \mathrm{the} \mathrm{major} \mathrm{highways} \mathrm{such} \mathrm{as} \mathrm{the} \mathrm{route} \mathrm{between} \mathrm{the} \mathrm{cities} \\ \mathrm{of} \mathrm{New} \mathrm{York} \mathrm{City} \mathrm{and} \mathrm{Dallas}, \mathrm{Texas} \mathrm{has} \mathrm{been} \mathrm{high} \mathrm{lighted}. \end{gathered}$$