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Question

Compute Δy/Δx for the interval [3,4], where y=3x−3.

(Use decimal notation.

Give your answer to three decimal places.)

Δy/Δx= (decimals)

What is the instantaneous rate of change of y with respect to x at x=7?

Rate =

Answer 1

Δy/Δx is the (change in y) / (change in x) = the slope

So the slope is ( y(4) - y(3) )/ (4 -3)

y(4) = 3(4) - 3 = 9

y(3) = 3(3) - 3 = 6

So the slope is (9 - 6) / (4 -3) = 3

The instantaneous rate in change is the derivative. y'(x) = 3

The instantaneous rate of change of y with respect to x at x = 7 = y'(7) = 3

Answer 2

This is a linear function which has a constant rate of change.

When x = 3, y = 6.

When x = 4, y = 9.

Think of the line passing through the points (3,6) and (4,9). The Δy/Δx notation is equivalent to rise/run, or slope of the line passing through those two points. Using the slope formula gives: (9-6) / (4-3) = 3/1 = 3.

This is no surprise because the slope calculated for any two points on the line will be the same, i.e. m = 3.

We could have recognized this immediately by seeing the y = 3x-3 is already in slope-intercept form, i.e. the slope is 3.

That means the instantaneous rate of change at any point (even where x = 7) will be 3.