|Module 11 - The Relationship between a Function and Its First and Second Derivatives|
|Introduction | Lesson 1 | Lesson 2 | Self Test|
|Lesson 11.1: What the First Derivative Says About a Function|
In Module 10 we saw that the value of the derivative of a function at x is given by the slope of the line tangent to the graph of f at x. In this lesson you will explore what the first derivative says about the graph of the original function by using the Derivative and Tangent features of the TI-83.
As noted earlier, the first derivative of a function f is denoted by f ', which is read "f prime." An alternate notation for the derivative is
, or simply df(x)/dx. The value of the first derivative at
Using the Derivative Feature on the Graph Screen
The TI-83 has several features that can be used to approximate the derivative of a function. The Derivative feature of the Graph screen's CALCULATE menu can be used to estimate a derivative.
The value of the derivative of f(x) = x3 - 2x2 - 5x + 6 at x = -2 appears to be approximately 15. That is, .
Using the Tangent Feature
The Tangent feature in the Draw menu can also be used to approximate the derivative of a function at a given point because it graphs the tangent line at the point and displays the equation for that tangent line.
After you make the selection, you should see the graph of the function.
The TI-83 displays the graph of the tangent line and shows its equation at the bottom of the screen. The equation of the tangent line in the example above is approximately y = 15x + 30.
11.1.1 What part of the equation of the tangent line to the curve at x = -2 represents the value of the derivative at that point? Explain. Click here for the answer.
11.1.2 The table below contains x-values of points on the graph of the function f(x) = x3 - 2x2 - 5x + 6. Determine each corresponding value for f(x) and use the Derivative feature to find an approximate value for the derivative at each point. Record the values.
Click here for the answer.
11.1.3 Use the graph of the function and the table you just completed to answer the following questions.
Click here for the answer.
Finding Turning Points from the Derivative
If a function is differentiable at one of its
The Numerical Derivative, which is found in the Math menu, can be used to graph the derivative of a function defined in the Y= editor. The syntax for the Numerical Derivative is
To graph the derivative, designate the function defined in Y1 as the expression with X as the variable. The point where nDeriv is computed needs to be the graphing variable X, so the expression shown below muse be entered on the Y= screen.
The function is increasing exactly where the derivative is positive and decreasing exactly where the derivative is negative. The zeros of the derivative function identify the endpoints of the intervals of interest.
On the graph of the derivative find the x-value of the zero that is left of the origin.
One of the zeros of the derivative is approximately x = -0.7863. We have therefore found the x-coordinate of a turning point.
11.1.4 Find the other zero of the derivative using the Zero feature of the Graph screen's CALCULATE menu. Click here for the answer.
Summarizing the Relationship Between f and f '
The following characteristics of the function f(x) = x3 - 2x2 - 5x + 6 can be determined from the graph of its first derivative.
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