The Tangent Line Problem In 1637, mathematician Rene Descartes said this about the tangent line problem. I dare say that this is not only the most useful and general problem in geometry that I know, but even that I ever desire to know. Finding the tangent line at point P is the same as finding the slope of the tangent line. We can approximate the slope of the tangent line by using a line through the point of tangency and a second point on the curve. This creates a secant line. Q Secant Line P

Definition of The Tangent Line with Slope m If f is defined on an open interval containing c, and if the limit exists, then the line passing through (c, f (c)) with slope m is the tangent line to the graph of f at the point (c, f (c)). The slope of the tangent line to the graph off at point (c, f (c)) is also called the slope of the graph of f at x = c.

The Slope of a Linear Function Find the slope of the graph of f(x) = 2x – 3 at the point (2, 1). I know just by looking at the equation that the slope is 2, but let’s see if we get the same answer when we use the definition of the slope of a tangent line. To find the slope of f when c = 2, we can apply the definition of the slope of a tangent line as follows. Let’s find the slope of the same graph at the point (5, 4) Hey, it’s Sam Ting!

Tangent Lines of Nonlinear Functions Find the slopes of the tangent lines to the graph of at the points (0, 1) and (-1, 2). Let (x, f (x)) represent any point on the graph. Then the slope of the tangent line at (x, f (x)) is given by Therefore, the slope at any point (x, f (x)) on the graph of f is m = 2x. At the point (0, 1), the slope is At the point (-1, 2), the slope is That was easy

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Finding the Equation of a Tangent Line Find the equation of the tangent lines to the graph of f at the given point. At the point (4, 11), the slope is That was easy

Finding the Equation of a Tangent Line Find the equation of the tangent lines to the graph of f at the given point. That was easy At the point (3, 8), the slope is

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The Derivative of a Function The limit used to define the slope of a tangent line is also used to define the Derivative of a Function. Definition of the Derivative of a Function. The Derivative of f at x is given by provided the limit exists. The process of finding the derivative of a function is called Differentiation. A function is differentiable at x if its derivative exists at x and is differentiable on an open interval (a, b) if it is differentiable at every point in the interval.

Derivative Notations f prime of x. y prime. The most common notations for the derivative of a function and the proper way of reading them are as follows: f prime of x. The derivative of y with respect to x. y prime. The derivative of f(x) with respect to x. The derivative of y with respect to x.

Finding the Derivative of a Function Find the derivative of This takes a while, but it’s pretty easy.

Using the Derivative to Find the Slope at a Point Find the derivative of Then find the slope of the graph at the points (1, 1) and (4, 2). Discuss the behavior of the function at (0, 0). At (1, 1), the slope is: At (4, 2), the slope is: At (0, 0), the slope is: The graph has a vertical tangent line at (0, 0)

More Derivatives Find the derivative with respect to t of

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Differentiability and Continuity You can find the derivative of a function at a specific point. The derivative of f at c can be found by using In order for the derivative to exist at x = c, the limit and the derivative from the left must be equal to the limit and the derivative from the right.

Graph with a Sharp Curve The function is continuous at x = 2, however, let’s see if it is differentiable at x = 2. They are not Sam Ting. X f is not differentiable at x = 2 and the graph does not have a tangent line at the point (2, 0)

Graph with a Vertical Tangent Line The function is continuous at x = 0, however, let’s see if it is differentiable at x = 0. Since the limit is infinite, there is a vertical tangent line at x = 0, therefore, f is not differentiable at x = 0.

Summarizing Differentiability and Continuity If a function is differentiable at x = c, then it is continuous at x = c. Differentiability implies Continuity. It is possible for a function to be continuous at x = c and not differentiable at x = c. Continuity does not imply Differentiability.