|Module 8 - Derivative of a Function|
|Introduction | Lesson 1 | Lesson 2 | Lesson 3 | Self-Test|
|Lesson 8.1: Derivative at a Point|
In Module 7 you saw that velocities correspond to slopes. Average velocity corresponds to the slope of a
Formal Definition of Derivative
The derivative of a function f at x = a is
provided the limit exists. The secant lines converge to a tangent line if one point on the curve is fixed and the second point approaches that fixed point.
Illustrating Secant Line Convergence
The following procedure will find the value of the derivative of the function f(x) = 2x x2 at the point (0.5, 0.75) by using a method similar to the one you used to find instantaneous velocities. First, you will find the slopes of several secant lines and use them to estimate the slope of the tangent line at
The graph below illustrates f(x) = 2x x2 in the window [-1, 3] x [-1, 2], with the three secant lines that approximate to the tangent line at (0.5, 0.75).
Finding Slopes of Secant Lines
The slope of the secant line through the points (0.5, f(0.5)) and (0.5 + h, f(0.5 + h)) can be found by evaluating the difference quotient
We're interested in values of h which are small so that the two points are close together and the resulting secant line will aproximate the tangent line.
The slope of the secant line containing (0.5, f(0.5)) and (0.6, f(0.6)) is 0.9.
Using Smaller Values of h
As the point (0.5 + h, f(0.5 + h)) approaches the point (0.5, f(0.5)), h approaches 0 and the secant lines converge to the tangent line.
Evaluate the difference quotient for smaller values of h.
The slope of the corresponding secant line is 0.99.
The slopes of the secant lines are 0.999 and 0.9999, respectively.
8.1.1 Predict the slope of the tangent line at (0.5, f(0.5)). Click here for the answer.
The slopes of the corresponding secant lines are 1.01 and 1.001. One secant line passes through (0.499, f(0.499)) and (0.5, f(0.5)), the other through (0.4999, f(0.4999)) and
Notice that we get a formula for the difference quotient when no value is specified for h.
Finding a Derivative at a Point
The derivative, which is the slope of the tangent line, is defined to be the limit
The derivative of f(x) = 2x x2 at x = 0.5 is 1.
Using the Derivative Command
You can also evaluate the derivative of the function f at x = 0.5 by using the derivative command d(, which is found above the key and also in the Calc menu. The syntax for finding a derivative at a point is d(expression,variable) | variable=value.
Drawing the Tangent Line
Now that the point on the curve and the derivative at that point are both known, the
Graph the function f and its tangent line at (0.5, 0.75).
The line appears to be tangent to the curve at x = 0.5.
Using a Square Window
It sometimes helps to visualize the value of the slope of the tangent line by drawing the graph in a square viewing window, where a unit on the x-axis is the same width as a unit on the y-axis.
In this window, the slope of the tangent line appears to be 1.
8.1.2 What are the coordinates of the viewing window above? Click here for the answer.
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