Problem Set 4 – Instantaneous Rate of Change and the Derivative
Problem Set 4 – Instantaneous Rate of Change and the Derivative
Note that the chart and the graph that we completed in #3 of Problem Set 3 gives an approximation of a function that describes the slope of. Our goal is to find a function that gives the exact slope of . Recall that the slopes in the chart came from average rates of change of our function. In our effort to find the exact slopes, we begin by computing the average rates of change over smaller and smaller intervals.
1)Use function notation to write the coordinates of
points A and B. Find the average rate of change of
f on the interval [x, x+h]. In other words, find the
slope of the line through the points A and B.
(This line is called a secant line.)
2)Let’s think about the average rate of change of the function near one. Complete the table below. For this discussion, hrepresents a small change in x.
Copyright 2007. Concepts of Calculus for Middle Level Students. First developed for the La Meta (Mathematics Educators Targeting Achievement) Summer Institute, University of New Mexico. Adapted for the Math in the Middle Institute Partnership, University of Nebraska, Lincoln. 1
Problem Set 4 – Instantaneous Rate of Change and the Derivative
h / / / Average rate of change1 / (1,1) / (2,4) / = 3
.5 / (1,1) / (1.5, 2.25)
.1 / (1,1) / (1.1, ____ )
.01 / (1,1) / (1.01, ____ )
What do you think happens to the average rates of change as
happroaches 0?
From your data, estimate the slope of the tangent line at (1, 1).
On the graph below, sketch the secant lines corresponding to each row of the table.
Copyright 2007. Concepts of Calculus for Middle Level Students. First developed for the La Meta (Mathematics Educators Targeting Achievement) Summer Institute, University of New Mexico. Adapted for the Math in the Middle Institute Partnership, University of Nebraska, Lincoln. 1
Problem Set 4 – Instantaneous Rate of Change and the Derivative
h / / / Average rate of change1 / (0,0) / (1,1) / = 1
.5 / (0,0) / (0.5, 0.25)
.1 / (0,0) / (0.1, ____ )
.01 / (0,0) / (0.01, ____ )
3)Complete the table below to investigate the average rate of change of the function near zero.
What do you think happens to the average rates of change as h approach 0?
From the data estimate the slope of the tangent line at (0,0).
On the graph below, sketch the secant lines corresponding to each row of the table.
Copyright 2007. Concepts of Calculus for Middle Level Students. First developed for the La Meta (Mathematics Educators Targeting Achievement) Summer Institute, University of New Mexico. Adapted for the Math in the Middle Institute Partnership, University of Nebraska, Lincoln. 1
Problem Set 4 – Instantaneous Rate of Change and the Derivative
4)We’ve examined the average rates of change of near x= 0 and x= 1. We now wish to study the average rate of change near any value of x.
Compute and simplify for .
5)We say the limit of this slope as h goes to zero is the instantaneous rate of change of fat x. We write . Compute this limit by letting h approach 0 in the expression you obtained in part 4).
6)Notice that the answer we got in problem 5 is itself a function. We call this function the derivative of f, and we write, which reads “f prime.” We think of the derivative as the “slope function” because it gives the slope of the line tangent to the graph at any point on the graph. The formal definition of the derivative of f at x is
.
In this example, the function was and we found that.
a)Using this formula, what isWhat is
b)Compare these answers with what your results in problems 2 and 3.
7)Now let .
a)Find and simplify it.
b)Find(here is where we let h go to 0). This is the derivative of (again the answer is a function).
c)Compute . Sketch a graph of , and sketch the tangent line to the curve at . Confirm that the slope of this line is.
d)Choose two additional points on your graph. Sketch the tangent lines to the curve at these points. Verify that the slopes agree with the derivative at these points.
e)Now write the equation of the tangent line we sketched in part c). (Recall that is the point-slope form of the equation of a line.)
f)Graph this line and the functionwith your calculator on the same coordinate axes to confirm that the line in part e) is indeed tangent to the curve.
8)Now let.
a)Compute.
b)Think about the graph of. Explain how your answer makes sense if we think of the derivative as the “slope function.”
9)Now let’s go back and think about the speedometer. Describe how a speedometer and derivative are related.
10) Go back and revisit #1 and #2 in Problem Session 3. Why is the slope not defined at “corners”?
Copyright 2007. Concepts of Calculus for Middle Level Students. First developed for the La Meta (Mathematics Educators Targeting Achievement) Summer Institute, University of New Mexico. Adapted for the Math in the Middle Institute Partnership, University of Nebraska, Lincoln. 1