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# The Derivative and the Tangent Line Problem

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). •. •.

## The Derivative and the Tangent Line Problem

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1. Definition of Tan-gent

2. 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)

3. • Slope of Line Tangent to a Curve • Approximated by secants • two points of intersection • Let second point get closerand closer to desiredpoint of tangency • View spreadsheet simulation GeogebraDemo

4. Animated Tangent

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

6. Definition ofa Tangent • Let Δx shrinkfrom the left

7. Definition ofa Tangent • Let Δx shrinkfrom the right

8. The Slope Is a Limit • Consider f(x) = x3 Find the tangent at x0= 2 • Now finish …

9. Animated Secant Line

10. 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

11. 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 functiondifq(2, .001) • Use some small value for dx • Result is close to actual slope

12. 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 !

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

14. 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

15. Using the Derivative • Consider that you are given the graph of the derivative … • What might theslope of the originalfunction look like? • Consider … • what do x-intercepts show? • what do max and mins show? • f `(x) <0 or f `(x) > 0 means what? f '(x) To actually find f(x), we need a specific point it contains

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

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

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