Lesson 3 1 Derivatives AP Calculus Mrs Mongold

Lesson 3 -1: Derivatives AP Calculus Mrs. Mongold

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

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

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

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

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

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

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

Example • Differentiate f(x)=x 3

Example • Differentiate f(x)=x 3 To differentiate we take the limit

Example • Differentiate f(x)=x 3 To differentiate we take the limit

Example • Differentiate f(x)=x 3 To differentiate we take the limit

Example • Differentiate f(x)=x 3 To differentiate we take the limit

Example • Differentiate f(x)=x 3 To differentiate we take the limit

Example • Differentiate f(x)=x 3 To differentiate we take the limit

Example • Differentiate f(x)=x 3 To differentiate we take the limit

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Alternate Definition • Derivative at a point – The derivative of a function f at the point x=a is the limit Provided the limit exists

Example • Use Alt. Def. to differentiate f(x)= at x=a

Example • Use Alt. Def. to differentiate f(x)= • Alt. Def. Limit is at x=a

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Example • Use Alt. Def. to differentiate f(x)= at a=2

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Lots to Remember with derivatives • The derivative is the slope at a point • When graphing a derivative the x values stay the same, but the y-values for the graph of f’ are the slopes from the points on f – So positive slope means f’ graph is above x axis – So negative slope means f’ graph is below x axis – 0 slope means f’ graph crosses the x axis

Example • Graph the derivative of f

Example • Graph the derivative of f

Example • Graph the derivative of f

Example • Graph the derivative of f

Example • If f(x) = x 3 -x, find a formula for f’(x) and illustrate by comparing f and f’ graphs

Example • Graph f from f’ – Sketch a graph of a function f that has the following properties • f(0)=0, f( • The graph of f’, the derivative of f, is below on left • F is continuous for all x

Example • Graph f from f’ – Sketch a graph of a function f that has the following properties • f(0)=0 • The graph of f’, the derivative of f, is below on left • F is continuous for all x

Example • Sketch the graph of a continuous function f , with f(0) = -1 and

Example • Sketch the graph of a continuous function f , with f(0) = -1 and -1

Homework • Page 101/ 1 -12 in Blue and Red Calc Book