2.7 and 2.8 Derivatives Great Sand Dunes National Monument, Colorado Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2003

The slope of a line is given by: The slope at (1,1) can be approximated by the slope of the secant through (4,16). We could get a better approximation if we move the point closer to (1,1). ie: (3,9) Even better would be the point (2,4).

The slope of a line is given by: If we got really close to (1,1), say (1.1,1.21), the approximation would get better still How far can we go?

slope slope at The slope of the curve at the point is:

The slope of the curve at the point is: is called the difference quotient of f at a. If you are asked to find the slope using the definition or using the difference quotient, this is the technique you will use.

The slope of a curve at a point is the same as the slope of the tangent line at that point. In the previous example, the tangent line could be found using . If you want the normal line, use the negative reciprocal of the slope. (in this case, ) (The normal line is perpendicular.)

Review: p velocity = slope These are often mixed up by Calculus students! average slope: slope at a point: average velocity: So are these! instantaneous velocity: If is the position function: velocity = slope p

is called the derivative of at . We write: “The derivative of f with respect to x is …” An equivalent way to state the derivative: There are many ways to write the derivative of

“the derivative of f with respect to x” “f prime x” or “y prime” “the derivative of y with respect to x” “dee why dee ecks” or “the derivative of f with respect to x” “dee eff dee ecks” or “the derivative of f of x” “dee dee ecks uv eff uv ecks” or

Note: dx does not mean d times x ! dy does not mean d times y !

Note: does not mean ! does not mean ! (except when it is convenient to think of it as division.) does not mean ! (except when it is convenient to think of it as division.)

Note: does not mean times ! (except when it is convenient to treat it that way.)

In the future, all will become clear.

The derivative is the slope of the original function. The derivative is defined at the end points of a function on a closed interval.

Find at x = 2. Example:

A function is differentiable if it has a derivative everywhere in its domain. It must be continuous and smooth. Functions on closed intervals must have one-sided derivatives defined at the end points. p

To be differentiable, a function must be continuous and smooth. Derivatives will fail to exist at: corner cusp discontinuity vertical tangent

Two theorems: If f has a derivative at x = a, then f is continuous at x = a. Since a function must be continuous to have a derivative, if it has a derivative then it is continuous.

p Intermediate Value Theorem for Derivatives If a and b are any two points in an interval on which f is differentiable, then takes on every value between and . Between a and b, must take on every value between and . p