2.1 Rates of Change and Tangent Lines Devil’s Tower, Wyoming Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1993.

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Presentation transcript:

2.1 Rates of Change and Tangent Lines Devil’s Tower, Wyoming Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1993

AP Practice Find the following limit:

AP Practice Find the following limit:

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:

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.

Recap : Average rate of change between (a, f(a)) and (b, f(b) Is also called the average velocity: Instantaneous rate of change at (a, f(a)) is also called The actual velocity at that point or the derivative:

Alternatively, you can find the instantaneous rate of change at (x, f(x)) and evaluate it at x = a.

Example: Use the limit of the difference quotient to find the slope of the graph of At the point (2,1 )

Tangent lines can be found using the point-slope form of a line:. The slope of a curve at a point is the same as the slope of the tangent line at that point. If you want the normal line, use the opposite reciprocal of the slope. (The normal line is perpendicular.)

Using the limit of the difference quotient, find the slope of the line tangent to the graph of the given function at x= -1, then use the slope to find the equation of the tangent line: Important example:

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

Derivatives

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

“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 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 ! (except when it is convenient to think of it as division.)

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

You must be able to do this : Find the equation of a line tangent to the graph at (2,2)

You must be able to do this : Find the equation of a line tangent to the graph at (-2,-1)

Example: Use the limit of the difference quotient to find the derivative of What is the derivative of the function at x = 2?

Homework P ,3, 7, 17, 21, 23, 25, 29, 33, all