Tangent Lines and Derivatives 2.7

The First of the TWO GRANDE Sized Questions What are the two primary questions that Calculus seeks to answer?

Tangent Lines and Derivatives In this section, we see how limits arise when we attempt to find: The tangent line to a curve. The instantaneous rate of change of a function.

The Tangent Problem

What is a Tangent Line, Anyways? A tangent line is a line that just touches a curve. The figure shows the parabola y = x 2 and the tangent line t that touches the parabola at P(1, 1).

Equation of Tangent Line We will be able to find an equation of t as soon as we know its slope m. The difficulty is that we know only one point, P, on t. We need two points to compute the slope.

A Road Trip- Average Rate of Change Suppose we went on a road trip to Las Vegas. Suppose further that we drove 210 miles in a 3 hour period of time. What was our average speed? If you left at 7PM and arrived at 10PM, how fast were you driving at 7:35?

Average Rate of Change Earlier in the course, we defined the average rate of change of a function f between the numbers a and x as:

A Road Trip Continued Consider the following diagram:

Derivatives – Tangent Lines We have seen that the slope of the tangent line to the curve y = f(x) at the point (a, f(a)) can be written as: It turns out that this expression arises in many other contexts as well—such as finding velocities and other rates of change. Since this type of limit occurs so widely, it is given a special name and notation.

Derivative—Definition derivative The derivative of a function f at a number x, denoted by f'(x), is: if this limit exists.

Instantaneous Rate of Change—Definition If y = f(x), the instantaneous rate of change of y with respect to x at x = a is the limit of the average rates of change as x approaches a:

Derivative- Instantaneous Rate of Change Suppose we consider the average rate of change over smaller and smaller intervals by letting x approach a. The limit of these average rates of change is called the instantaneous rate of change.

What can we conclude about the slope of the tangent line and the instantaneous rate of change at the same point? That’s right! They’re the same!!!!

Interpreting the Derivative Notice that we now have two ways of interpreting the derivative: f'(a) is the slope of the tangent line to y = f(x) at x = a. f'(a) is the instantaneous rate of change of y with respect to x at x = a.

E.g. 3—Finding a Derivative at a Point Find the derivative of f(x) = 5x + 3 at the number 2.

More Examples

E.g. 4—Finding a Derivative Let. (a) Find f'(a). (b) Find f'(1), f'(4), and f'(9).

E.g. 4—Finding a Derivative We use the definition of the derivative at a: Example (a)

E.g. 4—Finding a Derivative Example (a)

E.g. 4—Finding a Derivative Substituting a = 1, a = 4, and a = 9 into the result of (a), we get: Example (b)

E.g. 4—Finding a Derivative These values of the derivative are the slopes of the tangent lines shown here. Example (b)