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The Derivative and the Tangent Line Problem

Learn the definition of tangent, slope of tangent lines, and how to find derivatives. Discover tangent line approximations and use GeogebraDemo for interactive learning. Understand derivative notation and application in finding slopes and functions.

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The Derivative and the Tangent Line Problem

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  1. The Derivative and theTangent Line Problem Lesson 3.1

  2. Definition of Tan-gent

  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)

  4. • • 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

  5. Animated Tangent

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

  7. Definition ofa Tangent • Let Δx shrinkfrom the left

  8. Definition ofa Tangent • Let Δx shrinkfrom the right

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

  10. Animated Secant Line

  11. 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

  12. 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

  13. 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 !

  14. 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)

  15. 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

  16. 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

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

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

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