At the end of this session you will be able to
So far in this course, you have learned the following concepts:
You learned these concepts because they were the foundations for the study of calculus and will be what you built the next concept on. Therefore, if you need to review them, then review the Session 2 materials for a function, Session 4 materials for a limit, and Session 5 materials for continuity.
THE DEFINITION OF A DERIVATIVE
In your studying for this lesson, you saw the derivative defined on page 139 and on pages 147 and 148. Although these definitions might appear different to you, they are actually the same definition. On page 147, the definition is for a derivative at a value x. On page 139, the definition is for a derivative at a value x0.
Because derivative is the fourth major concept of this semester, it is imperative that you can write the definition of the derivative without looking at the book or looking at your notes. As you learn the definition of the derivative, notice the following about it:
The derivative is a limit of a fraction.
The numerator of the fraction is the subtraction of functional dependent-variable values, that is second coordinates of points on the graph of a function. The numerator is f(x + h) - f(x) for y = f(x).
The denominator of the fraction is the subtraction of the independent variable values. The denominator is h = delta x for y = f(x).
Let us see what we have learned.
The derivative of a function f is a
The numerator of a derivative of a function f is the
The denominator of a derivative of a function f is the
FINDING THE DERIVATIVE USING THE DEFINITION OF THE DERIVATIVE
Examples 1 and 2 on pages 148 and 149 are excellent illustrations of applying the definition of a derivative.
In the third line of Example 1 on page 149, there is an error. There should be a limit symbol in between the equal sign and the fraction.
In Example 2 on page 149 your textbook authors factored the denominator. You could also do this using rationalizing the numerator by multiplying the numerator and denominator by √z + √x. Try it.
If you want the derivative at a particular value of x, then you substitute in your particular value AFTER you find the derivative. If you look at Example 2 on page 149, then you will see that the textbook substituted in x = 4 AFTER finding the limit for part 2(a). In Example 2(b), f'(4) is the derivative of f(x) at x = 4.
SYMBOLS FOR THE DERIVATIVE
Your textbook gives various symbols for the derivative on the top of page of page 150. The most common symbols used in your book are f'(x) and d/dx (f(x)). Notice that the symbols that appear as fractions are not fractions at all. For example, dy/dx is not a fraction. It is the "derivative of y with respect to x".
In the middle of page 150, the textbook gives the "evaluation symbols" and explains them. Also, f'(a) means "evaluate the derivative f'(x) at x = a. Thus, in example 2 (b) on page 149, f'(4) is also the evaluation of the derivative at x = 4.
The process of calculating the derivative is called "differentiation". If a function has a derivative, then we say that the function is differentiable.
Now you can go to the Session 6 assignments to continue your assignment.
At the end of this session you will be able to
GEOMETRIC INTERPRETATION OF THE DERIVATIVE
When we reviewed functions in the Session 2, we studied the geometric interpretation of a function. The geometric interpretation of the function was important because it permitted us to visualize the abstract notion of a function. If you have forgotten the geometric interpretation of a function, then now review the geometric interpretation on pages 21 and 22 of your textbook. What is the geometric interpretation of a function?
As we saw with the geometric interpretation of a function, the geometric interpretation of a derivative gives us a way to visualize the abstract notion of a derivative. By looking at the definition of the slope of the curve on page 137 of your textbook and looking at the definition of the derivative on page 147, you see that the definitions are identical. Therefore, the geometric interpretation of a derivative is
Therefore, as you leaf through your textbook and every time you see a derivative, you can think of it geometrically as
In other words, the derivative is always the slope of the tangent line to the graph of the function being differentiated.
EQUATION OF TANGENT LINE
In the Session 2, I strongly recommend that you use the slope-intercept form of the equation to write the equation of a straight line. There are only two pieces of information that we need: the slope and a point on the line.
In the same way, if we want to write the equation of the tangent line to a curve at a point, there are only two pieces of information that you need: the slope of the tangent line and a point on the line. Because the geometric interpretation of the derivative is the slope, we will use the derivative to find the slope of the line. Thus, how will we write the equation of the tangent line? The answer is:
- Find the derivative of the function.
- Evaluate the derivative at the point of tangency. This will give you the slope of the tangent line.
- Now write the equation using this slope and using the point of tangency as the given point.
HORIZONTAL TANGENT LINE
Because geometrically the derivative is the slope of the tangent and because a horizontal line has the slope of zero, the horizontal tangent line must have slope zero. That will be the same place where the derivative is zero. That will be where you will have a horizontal tangent line.
Therefore, if we desire a horizontal tangent line, we first find the derivative. Then we set the derivative equal to zero and solve for the independent variable. That is exactly what your textbook is doing in Example 7 on page 152 of Section 2.1. To find the second coordinates of the points, you substitute the x-values into the original equation and calculate y. For example, if you substitute x = 0 into the original equation, then you have y = 0.
Now you can go to the Session 6 assignments to continue your assignment.
At the end of this session you will be able to
DIFFERENTIABLE FUNCTION
You should have noticed on page 152 of Section 2.1 of your textbook that differentiable function is one that can be differentiated at every point in an open interval.
Also at the endpoints, we can only have one-sided derivatives, analogous to one-sided limits. Thus, h must approach zero so that you remain within the domain as described at the middle of page 152. In example 5, make sure that you understand why the absolute value of h is -h when he approaches zero from the left. That is because h is less than zero when you approach zero from the left.
DIFFERENTIABILITY FAILS
Differentiability can fail at four different situations:
- at a corner
- at a cusp
- at a vertical tangent line
- at a discontinuity
The bottom of page 153 and the top of page 154 shows why you have discontinuity in all four of these cases. Differentiability failing means that the derivative does not exist at each of these four situations.
Now you can go to the Session 6 assignments to continue your assignment.
At the end of this session you will be able to
CONTINUITY AT A POINT
On page 125 of your textbook, you saw that for the "test" for continuity at a point required meeting three conditions. They are the following:
DERIVATIVE TO DETERMINE CONTINUITY FOR A FUNCTION
Theorem 1 on page 154 of your textbook is telling you that, if you have a derivative at a point where x = c, then your function f is continuous at x = c. In other words, differentiability guarantees continuity.
Do not assume that the converse of Theorem 1 is true, because it is not. As Example 8 shows you, continuity does not imply you have differentiability. The function is continuous at the corner in Example 5 on page 152, but the derivative does not exist.
Now you can go to the Session 6 assignments to continue your assignment.