Project 3. Predicting the Slope of a Tangent Line

[Pages:1]MATH 163, Spring 2000 Due Date:

Name(s):

Project 3. Predicting the Slope of a Tangent Line

Objective

To predict the slope of a tangent line to a curve at a point. This will not only provide an application of zooming-in, but also lay the foundation for concepts to be discussed later in this course.

Narrative

If you have not already done so, read A Preview of Calculus (see p. 2) in the text. In this project we investigate the tangent problem.

Task

a) Type the command lines in the left-hand column below into Maple in the order in which they are listed. These commands will allow you to predict the slope m of the tangent line to the graph of f (x) = 0.5 sin(x) at the origin by computing the slope m = (f (h) - f (0))/h of the secant line that passes through the origin and the point whose coordinates are (h, f (h)).

> # Project 3. Predicting the Slope of a Tangent Line

> restart;

Clear Maple's memory.

> f := x -> 0.5*sin(Pi*x);

Let f (x) = 0.5 sin(x).

> h := 0.8;

Let h be 0.8.

> m := evalf((f(h)-f(0))/h);

Compute the slope m of the secant line through

(0, 0) and (h, f (h)).

> plot({f(x),m*x},x=-1..1,y=-1..1); Plot the graph of f and the secant line y = mx.

b) Repeat the last three steps of (a) for h values of 0.4, 0.2, and 0.1. (The easiest way to do this is by copying, pasting, editing, and re-entering the last 3 lines of (a).) Observe how the secant line gets closer and closer to the tangent line as h decreases. Also observe that if h = 0.1 then near the origin, the graph of f is indistinguishable from the secant / tangent line.

c) To 2 decimal places of accuracy, guess the slope of the tangent line to the graph of f (x) = 0.5 sin(x) at the origin. Justify your answer. (You may need to look at values for h smaller than 0.1.)

d) What happens if you try to compute m when h = 0?

Your lab report will be a hard copy of your typed input and Maple's responses (including both text and graphics) to the various steps in this task, together with your written responses.

Comments

1. Since f (0) = 0:

(a) we didn't need to include f (0) in our fifth command line (we did so only for pedagogical reasons), and

(b) the tangent line to the graph of f at the origin must pass through the origin, simplifying the slope-intercept equation of the tangent line from y = mx + b to y = mx.

2. If you zoom in over and over again, the graph of f and the graph of the tangent line will eventually look the same, and this brings up an interesting question: "What happens if you zoom in any further: do they continue to look the same?" After all, even though you may think you know what the graph of f looks like, remember that in Project 2 -- in graphing f (x) = |x2 - 4| - 2x -- we thought we knew what the graph looked like only to find out that we didn't!

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