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The Derivative and the Tangent Line Problem

The Derivative and the Tangent Line Problem. Section 2.1. After this lesson, you should be able to:. find the slope of the tangent line to a curve at a point use the limit definition of a derivative to find the derivative of a function

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The Derivative and the Tangent Line Problem

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  1. The Derivative and the Tangent Line Problem Section 2.1

  2. After this lesson, you should be able to: • find the slope of the tangent line to a curve at a point • use the limit definition of a derivative to find the derivative of a function • understand the relationship between differentiability and continuity

  3. Tangent Line A line is tangent to a curve at a point P if the line is perpendicular to the radial line at point P. Note: Although tangent lines do not intersect a circle, they may cross through point P on a curve, depending on the curve. P

  4. The Tangent Line Problem Find a tangent line to the graph of f at P. Why would we want a tangent line??? f Remember, the closer you zoom in on point P, the more the graph of the function and the tangent line at P resemble each other. Since finding the slope of a line is easier than a curve, we like to use the slope of the tangent line to describe the slope of a curve at a point since they are the same at a particular point. A tangent line at P shares the same point and slope as point P. To write an equation of any line, you just need a point and a slope. Since you already have the point P, you only need to find the slope. P

  5. Definition ofa Tangent • Let Δx shrinkfrom the left

  6. Definition of a Tangent Line with Slope m

  7. The Derivative of a Function Differentiation- the limit process is used to define the slope of a tangent line. Really a fancy slope formula… change in y divided by the change in x. Definition of Derivative: (provided the limit exists,) This is a major part of calculus and we will differentiate until the cows come home! Also, = slope of the line tangent to the graph of f at (x, f(x)). = instantaneous rate of change of f(x) with respect to x.

  8. Definition of the Derivative of a Function

  9. Notations For Derivative Let If the limit exists at x, then we say that f is differentiable at x.

  10. Note: dx does not mean d times x ! dy does not mean d times y !

  11. does not mean ! does not mean ! Note: (except when it is convenient to think of it as division.) (except when it is convenient to think of it as division.)

  12. does not mean times ! Note: (except when it is convenient to treat it that way.)

  13. The derivative is the slope of the original function. The derivative is defined at the end points of a function on a closed interval.

  14. A function is differentiable if it has a derivative everywhere in its domain. It must be continuous and smooth. Functions on closed intervals must have one-sided derivatives defined at the end points. p

  15. Theorem 2.1 Differentiability Implies Continiuty

  16. The Slope of the Graph of a Line Example: Find the slope of the graph of at the point (2, 5).

  17. The Slope of the Graph of a Line Example: Find the slope of the graph of at the point (2, 5).

  18. The Slope of the Graph of a Line Example: Find the slope of the graph of at the point (2, 5).

  19. The Slope of the Graph of a Line Example: Find the slope of the graph of at the point (2, 5).

  20. The Slope of the Graph of a Line Example: Find the slope of the graph of at the point (2, 5).

  21. The Slope of the Graph of a Line Example: Find the slope of the graph of at the point (2, 5).

  22. The Slope of the Graph of a Non-Linear Function Example: Given , find f ’(x) and the equation of the tangent lines at: a) x = 1 b) x = -2 a) x = 1:

  23. The Slope of the Graph of a Non-Linear Function Example: Given , find f ’(x) and the equation of the tangent lines at: a) x = 1 b) x = -2 a) x = 1:

  24. The Slope of the Graph of a Non-Linear Function Example: Given , find f ’(x) and the equation of the tangent lines at: a) x = 1 b) x = -2 a) x = 1:

  25. The Slope of the Graph of a Non-Linear Function Example: Given , find f ’(x) and the equation of the tangent lines at: a) x = 1 b) x = -2 a) x = 1:

  26. The Slope of the Graph of a Non-Linear Function Example: Given , find f ’(x) and the equation of the tangent lines at: a) x = 1 b) x = -2 a) x = 1:

  27. The Slope of the Graph of a Non-Linear Function Example: Given , find f ’(x) and the equation of the tangent lines at: a) x = 1 a) x = 1: b) x = -2

  28. The Slope of the Graph of a Non-Linear Function Example: Given , find f ’(x) and the equation of the tangent line at: b) x = -2

  29. The Slope of the Graph of a Non-Linear Function Example: Find f ’(x) and the equation of the tangent line at x = 2 if

  30. The Slope of the Graph of a Non-Linear Function Example: Find f ’(x) and the equation of the tangent line at x = 2 if

  31. The Slope of the Graph of a Non-Linear Function Example: Find f ’(x) and the equation of the tangent line at x = 2 if

  32. The Slope of the Graph of a Non-Linear Function Example: Find f ’(x) and the equation of the tangent line at x = 2 if

  33. Example-Continued If x = 2, the slope is, -¼. So, y = 1/4x + b. Going back to the original equation of y = 1/x, we see if x = 2, y = 1/2. So:

  34. Derivative Example: Find the derivative of f(x) = 2x3 – 3x.

  35. Derivative Example: Find the derivative of f(x) = 2x3 – 3x.

  36. Derivative Example: Find the derivative of f(x) = 2x3 – 3x.

  37. Derivative Example: Find the derivative of f(x) = 2x3 – 3x.

  38. Derivative Example: Find the derivative of f(x) = 2x3 – 3x.

  39. Derivative Example: Find for

  40. Derivative Example: Find for

  41. Derivative Example: Find for

  42. Derivative Example: Find for THIS IS A HUGE RULE!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

  43. Example-Continued Let’s work a little more with this example… Find the slope of the graph of f at the points (1, 1) and (4, 2). What happens at (0, 0)?

  44. Example-Continued Let’s graph tangent lines with our calculator…we’ll draw the tangent line at x = 1. Graph the function on your calculator. 1 3 Select 5: Tangent( Type the x value, which in this case is 1, and then hit  4 (I changed my window) 2 Now, hit  DRAW Here’s the equation of the tangent line…notice the slope…it’s approximately what we found

  45. Differentiability Implies Continuity If f is differentiable at x, then f is continuous at x. • Some things which destroy differentiability: • A discontinuity (a hole or break or asymptote) • A sharp corner (ex. f(x)= |x| when x = 0) • A vertical tangent line (ex: when x = 0)

  46. 2.1 Differentiation Using Limits of Difference Quotients • Where a Function is Not Differentiable: • 1) A function f(x) is not differentiable at a point x = a, if there is a “corner” at a.

  47. 2.1 Differentiation Using Limits of Difference Quotients • Where a Function is Not Differentiable: • 2) A function f (x) is not differentiable at a point • x = a, if there is a vertical tangent at a.

  48. 3. Find the slope of the tangent line to at x = 2. This function has a sharp turn at x = 2. Therefore the slope of the tangent line at x = 2 does not exist. • Functions are not differentiable at • Discontinuities • Sharp turns • Vertical tangents

  49. 2.1 Differentiation Using Limits of Difference Quotients Where a Function is Not Differentiable: 3) A function f(x) is not differentiable at a point x = a, if it is not continuous at a. Example: g(x) is not continuous at –2, so g(x) is not differentiable at x = –2.

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