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# Lesson3.1 The Derivative And The Tangent Line

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### Lesson3.1 The Derivative And The Tangent Line

1. 1. The Derivative and the Tangent Line Problem Lesson 3.1
2. 2. Definition of Tan-gent
3. 3. Tangent Definition <ul><li>From geometry </li></ul><ul><ul><li>a line in the plane of a circle </li></ul></ul><ul><ul><li>intersects in exactly one point </li></ul></ul><ul><li>We wish to enlarge on the idea to include tangency to any function, f(x) </li></ul>
4. 4. Slope of Line Tangent to a Curve <ul><li>Approximated by secants </li></ul><ul><ul><li>two points of intersection </li></ul></ul><ul><li>Let second point get closer and closer to desired point of tangency </li></ul>• View spreadsheet simulation • •
5. 5. Animated Tangent
6. 6. Slope of Line Tangent to a Curve <ul><li>Recall the concept of a limit from previous chapter </li></ul><ul><li>Use the limit in this context </li></ul>• •
7. 7. Definition of a Tangent <ul><li>Let Δ x shrink from the left </li></ul>
8. 8. Definition of a Tangent <ul><li>Let Δ x shrink from the right </li></ul>
9. 9. The Slope Is a Limit <ul><li>Consider f(x) = x 3 Find the tangent at x 0 = 2 </li></ul><ul><li>Now finish … </li></ul>
10. 10. Animated Secant Line
11. 11. Calculator Capabilities <ul><li>Able to draw tangent line </li></ul><ul><li>Steps </li></ul><ul><li>Specify function on Y= screen </li></ul><ul><li>F5-math, A-tangent </li></ul><ul><li>Specify an x (where to place tangent line) </li></ul><ul><li>Note results </li></ul>
12. 12. Difference Function <ul><li>Creating a difference function on your calculator </li></ul><ul><ul><li>store the desired function in f(x) x^3 -> f(x) </li></ul></ul><ul><ul><li>Then specify the difference function (f(x + dx) – f(x))/dx -> difq(x,dx) </li></ul></ul><ul><ul><li>Call the function difq(2, .001) </li></ul></ul><ul><li>Use some small value for dx </li></ul><ul><li>Result is close to actual slope </li></ul>
13. 13. Definition of Derivative <ul><li>The derivative is the formula which gives the slope of the tangent line at any point x for f(x) </li></ul><ul><li>Note: the limit must exist </li></ul><ul><ul><li>no hole </li></ul></ul><ul><ul><li>no jump </li></ul></ul><ul><ul><li>no pole </li></ul></ul><ul><ul><li>no sharp corner </li></ul></ul>A derivative is a limit !
14. 14. Finding the Derivative <ul><li>We will (for now) manipulate the difference quotient algebraically </li></ul><ul><li>View end result of the limit </li></ul><ul><li>Note possible use of calculator limit ((f(x + dx) – f(x)) /dx, dx, 0) </li></ul>
15. 15. Related Line – the Normal <ul><li>The line perpendicular to the function at a point </li></ul><ul><ul><li>called the “normal” </li></ul></ul><ul><li>Find the slope of the function </li></ul><ul><li>Normal will have slope of negative reciprocal to tangent </li></ul><ul><li>Use y = m(x – h) + k </li></ul>
16. 16. Using the Derivative <ul><li>Consider that you are given the graph of the derivative … </li></ul><ul><li>What might the slope of the original function look like? </li></ul><ul><li>Consider … </li></ul><ul><ul><li>what do x-intercepts show? </li></ul></ul><ul><ul><li>what do max and mins show? </li></ul></ul><ul><ul><li>f `(x) <0 or f `(x) > 0 means what? </li></ul></ul>To actually find f(x), we need a specific point it contains f `(x)
17. 17. Derivative Notation <ul><li>For the function y = f(x) </li></ul><ul><li>Derivative may be expressed as … </li></ul>
18. 18. Assignment <ul><li>Lesson 3.1 </li></ul><ul><li>Page 123 </li></ul><ul><li>Exercises: 1 – 41, 63 – 65 odd </li></ul>