Intro – average speed is the average rate of change of ...



Lab 3 : Average and Instantaneous Rates of Change

Math 131 C & D 9/8/05

Objectives:

1) To explore the relationship between the tangent line to a function and secant lines as well as their respective slopes.

2) To measure average and instantaneous rates of change of a function graphically, algebraically, & numerically.

The slope of a secant line through two points of a graph measures the average rate of change of a function. The slope of a tangent line through a point on the graph measures the instantaneous rate of change of a function.

Part #1. Secant lines and the difference quotient.

a. What is the slope of a line, L, which contains the points (x1, y1) and (x2, y2)?

m = ___________

b. If the two points also line on the graph of a function, f(x), then the line is a secant line to f(x). We could now denote the points as (x1, f(x1)) and (x2, f(x2)). Express the slope of the secant line with functional notation.

m = ____________

c. We can express the distance between x1 and x2 as h, i.e. x2 = x1 + h. Substitute this expression in the functional notation form of slope and simplify to get a familiar form for the difference quotient.

[pic]

Part #2. Slopes with geometry.

You should have a sheet of graph paper that contains the graph of the parabola, f(x) = 9 – x2.

a. In quadrant 2, you will illustrate secant lines approaching a tangent line at a point. Let

(-1, f(-1) ) be the fixed point. Sketch five secant lines on the graph with h = -2, -1, -0.5, 0.5, 1.

b. Note that as h approaches zero from the right and from the left the lines “move into” a single line. Sketch that line in a different color from your secant lines. Approximate its slope.

c. In quadrant 1 & 4, you will measure the slope of secant lines through successive points on the parabola graphically. In particular, you will construct a table of data.

x 0 1 2 3 3.5

y

change in y ______ ______ _______ _______

change in x ______ ______ _______ _______

secant slope

d. Do the data of the completed table exhibit a trend? Explain your answer.

e. Use your above table to generate an estimate of the slope of the tangent line to the parabola at x = 1, 2, 3. Explain how you calculated the estimate.

f. Use your above table to generate an estimate of the slope of the tangent line to the parabola at x = 0.5, 1.5. Explain how you calculated the estimate.

x 1 2 3 0.5 1.5

Estimate of the

tangent slope

Part #3. Slopes with algebra.

a. Algebraically compute the secant slope or [pic] for the parabola with equation f(x) = 9-x2.

The value of the secant slope when h goes to zero, if it exists, is the slope of the tangent line to a function f (or the derivative of f)

b. Evaluate your algebraic formula for the slope of the tangent line at the table values:

x 1 2 3 0.5 1.5

tangent slope (by algebra)

Part #4. Slopes with numerical calculations.

Now you will compute secant slopes and tangent slopes of the parabola numerically with the TI-89 calculator. The slope of the secant line through two points on the parabola whose x-coordinates differ by h is: [pic]. (This is the difference quotient)

Compute secant slopes for small values of h. Begin with h = 1.0.

Note: You can use SEQ to obtain the values of the slope for multiple values of x. With h=1 and x = 0.5,1, 1.5, 2, 2.5, 3, use SEQ([pic], x, 0.5, 3, 0.5)|h=1.

x 0.5 1 1.5 2 2.5 3

secant slope(h=1.0)

secant slope(h=0.1)

secant slope(h=0.01)

secant slope(h=0.001)

tangent slope

secant slope(h= -0.001)

secant slope(h= -0.01)

secant slope(h= -0.1)

secant slope(h= -1.0)

Use the chart above to estimate the tangent slopes for each x.

Part five. Slope of a curve

a. With the TI-89 calculator, sketch f(x) = 9-x2 in the standard window.

b. Trace to x=2. Note the curve of the graph near this point.

c. Zoom in with x=2 as the new center. Note how the curve “straightens” somewhat in this view.

d. Zoom in again with x=2 as the new center. Note how the curve appears to be linear in this window. Trace to determine points at either end of the “line segment” on your screen. Use these to determine the slope of the “line”. Compare this to the slope of the tangent at x=2.

e. Explain the statement: Smooth, continuous curves are linear locally.

Lab report, one per group.

Estimate the change of the function on the interval [-2, 2]

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