MAT237Y1 Study Guide - Final Guide: Compact Space, Limit Point, Convex Set

Conjecture

(a) Find formulas for the distances between and , , along the line between these points and along the parabola .

(b) Use the formulas from part (a) to find the distances for .

(c) Make a conjecture about the difference between the two distance as a increases.

Distance Two insects are crawling along different lines in three-space. At time (in minutes), the first insect is at the point on the line Also, at time , the second insect is at the point on the line Assume that distances are given in inches.

(a) Find the distance between the two insects at time .

(b) Use a graphing utility to graph the distance between the insects from to .

(c) Using the graph from part (b), what can you conclude about the distance between the insects?

(d) How close to each other do the insects get?

Approximation

In Exercises 31 and 32, approximate the arc length of the graph of the function over the interval [0, 4] in three ways:

(a) Use the Distance Formula to find the distance between the endpoints of the arc.

(b) Use the Distance Formula to find the lengths of the four line segments connecting the points on the arc when , and . Find the sum of the four lengths.

(c) Use the integration capabilities of a graphing utility to approximate the integral yielding the indicated arc length.