Kelly Swain

Write-up #2

Explore the relationship between the graph of a function and the graph of the derivative of the function. Given a function f(x), what is the equation of the tangent line at any given point. Using Graphing Calculator 2.2, graph f(x) and the tangent line.

First, let's start by looking at the graph of a function and its derivative on the same coordinate plane!

I will use f(x)=x^2

f'(x)=2x

The derivative is a slope-predictor of a tangent line to a particular function. A lot of times we use it to get the equation of the tangent line to a particular function. Here we see that the two intersect at exactly two points, the origin and the point (2,4).

To find the equation of the tangent line to a function--we'll use y=x^2:

1- Take the derivative.

2- Plug in the x-value of any point on the graph of the function in for x in the derivative. (This gives you the slope of the tangent line.)

3- Now, plug in an x and y-values and the slope into the point-slope equation.

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f(x)=x^2

f'(x)=2x

We will call the point at which the line is tangent (a,f(a)).

The slope of the tangent line is 2a. (m=2a)

So, using the Point-Slope Equation of a Line, we get y-f(a)=(2a)(x-a).

Therefore, y=(2a)(x-a)+f(a).

To show the graph of the tangent line at any point, in graphing calculator we can type in the equation y+x^2 and for the equation of the tangent line y=(2a)(x-a)+a^2.

General form of the tangent line for any function f(a) in terms of the derivative: y-f(a)=f'(a)(x-a) !!!!!!

Isn't this exciting!----yea, yippi!

To view my movie of the tangent line at any point click here

Return

To see a neat combination, graph f(x)=sin x and the tangent lines to it at any point. The equation for the tangent lines would be f(x)=(cos a)(x)-a(a-(cos a)).