The Derivative and the Tangent Line Problem 12
The Derivative and the Tangent Line Problem 12. 3
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)
Animated Tangent
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 • • •
Animated Secant Line
Slope of Line Tangent to a Curve • How do you find the slope of a line? • What does this have to do with the • • difference quotient?
The Slope Is a Limit • Consider f(x) = x 3 x= 2 • Now finish … Find the tangent at
Calculator Capabilities • Able to draw tangent line Steps • Specify function on Y= screen • F 5 -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
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, 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 f '(x) slope of the original function look like? • Consider … – what do x-intercepts show? To actually find f(x), we – what do max and mins show? need a specific point it contains – f `(x) <0 or f `(x) > 0 means what?
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
Slope of Line Tangent to a Curve • Approximated by secants – two points of intersection • • Let second point get closer and closer to desired View spreadsheet point of tangency simulation Geogebra Demo
Definition of a Tangent • Let Δx shrink from the left
Definition of a Tangent • Let Δx shrink from the right
- Slides: 19