The Derivative and the Tangent Line Problem Section 2.1

The Tangent Line 0 You don’t have to copy these first few slides, but it’s background knowledge for understanding where “Calculus” came from!

The Tangent Line 0 A tangent line (in geometry, usually) is thought of as a line in the same plane as a circle that intersects the circle in exactly one point. 0

The Tangent Line Problem 0 You are given a function f and a point P on its graph. Find an equation of the tangent line to the graph at point P.

The Tangent Line Problem 0 This problem was one of four that European mathematicians were working on during the 17 th century. Credit for the solution to this problem was given to Isaac Newton and Gottfried Liebniz (called the “Father of Calculus” in some references).

The Tangent Line Problem 0 To solve the problem, mathematicians instead drew a secant line to the curve that would hit two points on the curve. (A secant line is a line that intersects a circle in two points.)

The Tangent Line Problem 0 The nearer this line gets to the actual point, the more it resembles a tangent line, and you can use the two points to find the slope of the tangent line, so that you can then write an equation of the line.

The Tangent Line Problem 0 In 1637, mathematician Rene Descartes stated this about the tangent line problem, “And 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.”

Slope of the Graph at f(x) = c

Example

The Slope of the Tangent Line 0 Once we know that the slope is 2c, we can take any point on that graph, plug the x value in for c, and we’ll have the slope of the tangent line. 0 0 At point (0, 1), the slope is 0; at (-1, 2), the slope is -2.

Differentiation 0 The limit that is used to define the slope of a tangent line is also used to define differentiation.

The Derivative 0 The derivative of a function is the equation for the slope of the line tangent to the function. 0 The process of finding the derivative is called differentiation, or you are going to “differentiate.”

The Derivative

0 The new function that you derive gives the slope of the tangent line to the graph of f at the point (x, f(x)). (a repeat of what was said two slides ago!)

Notation for Differentiation

Example: Use the definition of a derivative… 0 To find the derivative of f(x) = x 3 + 2x.

Example

Continuity and Differentiability 0 Theorem: If f is differentiable at x = c, then f is continuous at x = c.

In other words…. 0 If a function is not continuous at a point, then it can also not be differentiable at that point.

In other words… 0 If you can’t differentiate from both sides and get the same answer, there will not be a derivative.

In other words… 0 You cannot find a derivative of f(x) = |x – 2| at x = 2 because the derivative from the left and right are two different values. (This is called a graph with a sharp turn!)

Vertical Lines 0 You also cannot find a derivative for a vertical tangent line.

The End!