The Derivative and the Tangent Line Problem Lesson 3.1

Definition of Tan-gent

Tangent Definition From geometry –a line in the plane of a circle –intersects in exactly one point We wish to enlarge on the idea to include tangency to any function, f(x)

Slope of Line Tangent to a Curve Approximated by secants – two points of intersection Let second point get closer and closer to desired point of tangency View spreadsheet simulation View spreadsheet simulation Geogebra Demo Geogebra Demo

Animated Tangent

Slope of Line Tangent to a Curve Recall the concept of a limit from previous chapter Use the limit in this context

Definition of a Tangent Let Δx shrink from the left

Definition of a Tangent Let Δx shrink from the right

The Slope Is a Limit Consider f(x) = x 3 Find the tangent at x 0 = 2 Now finish …

Animated Secant Line

Calculator Capabilities Able to draw tangent line Steps Specify function on Y= screen F5-math, A-tangent Specify an x (where to place tangent line) Note results

Difference Function Creating a difference function on your calculator –store the desired function in f(x) x^3 -> f(x) – Then specify the difference function (f(x + dx) – f(x))/dx -> difq(x,dx) –Call the function difq(2,.001) Use some small value for dx Result is close to actual slope Use some small value for dx Result is close to actual slope

Definition of Derivative The derivative is the formula which gives the slope of the tangent line at any point x for f(x) Note: the limit must exist –no hole –no jump –no pole –no sharp corner A derivative is a limit !

Finding the Derivative We will (for now) manipulate the difference quotient algebraically View end result of the limit Note possible use of calculator limit ((f(x + dx) – f(x)) /dx, dx, 0)

Related Line – the Normal The line perpendicular to the function at a point –called the “normal” Find the slope of the function Normal will have slope of negative reciprocal to tangent Use y = m(x – h) + k

Using the Derivative Consider that you are given the graph of the derivative … What might the slope of the original function look like? Consider … –what do x-intercepts show? –what do max and mins show? –f `(x) 0 means what? To actually find f(x), we need a specific point it contains f '(x)

Derivative Notation For the function y = f(x) Derivative may be expressed as …

Assignment Lesson 3.1 Page 123 Exercises: 1 – 41 EOO, 63, 65