Definition of the Derivative Lesson 3.4

Tangent Line Recall from geometry Tangent is a line that touches the circle at only one point Let us generalize the concept to functions A tangent will just "touch" the line but not pass through it Which of the above lines are tangent?

A Secant Line Crosses the curve twice The slope of the secant will be x = a h

Tangent Line Now let h get smaller and smaller x = a Try this with Geogebra h The slope of the tangent line

Tangent Line Try it out …given Evaluate Determine the slope of the tangent line at x = 0 Evaluate

Tangent Line Use the limit command on your calculator to determine the slope of for x = 1 Once you have the slope and the point You can determine the equation of the line

The Derivative We will define the derivative of f(x) as Note The derivative is the rate of change function for f(x) The derivative is also a function of x The limit must exist

Comparison Difference Quotient Slope of secant Average rate of change Average velocity Derivative Slope of tangent Instantaneous rate of change Instantaneous velocity

Finding f'(x) from Definition Strategy Find f(x + h) Find and simplify f(x + h) – f(x) Divide by h to get Let h → 0

Hint: rationalize the numerator Try It Out Use the strategy to find the derivatives of these functions Hint: rationalize the numerator

Equation of the Tangent Line We stated previously that once we determine the slope of the tangent We can "cut to the chase" and state it as

Warning Our definition of derivative included the phrase "if the limit exists" Derivatives do not exist at "corners" or "sharp points" on the graph The slope is different on each side of the point The limit does not exist f(x) = | x – 3 |

Assignment Lesson 3.4 Page 210 Exercises 1 – 51 odd