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Chapter 1 Functions

Chapter 1 Functions. § 1.1. The Slope of a Straight Line. Nonvertical Lines. Lines – Positive Slope. EXAMPLE. The following are graphs of equations of lines that have positive slopes. Lines – Negative Slope. EXAMPLE.

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Chapter 1 Functions

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  1. Chapter 1Functions

  2. §1.1 The Slope of a Straight Line

  3. Nonvertical Lines

  4. Lines – Positive Slope EXAMPLE The following are graphs of equations of lines that have positive slopes.

  5. Lines – Negative Slope EXAMPLE The following are graphs of equations of lines that have negative slopes.

  6. Interpretation of a Graph EXAMPLE A salesperson’s weekly pay depends on the volume of sales. If she sells x units of goods, then her pay is y = 5x + 60 dollars. Give an interpretation of the slope and the y-intercept of this straight line.

  7. Properties of the Slope of a Nonvertical Line

  8. Properties of the Slope of a Line

  9. Finding Slope and y-intercept of a Line EXAMPLE Find the slope and y-intercept of the line

  10. Making Equations of Lines EXAMPLE Find an equation of the line that passes through the points (-1/2, 0) and (1, 2).

  11. Making Equations of Lines EXAMPLE Find an equation of the line that passes through the point (2, 0) and is perpendicular to the line y = 2x.

  12. Slope as a Rate of Change EXAMPLE Compute the rate of change of the function over the given intervals.

  13. §1.2 The Slope of a Curve at a Point

  14. Tangent Lines

  15. Slope of a Curve & Tangent Lines

  16. Slope of a Graph EXAMPLE Estimate the slope of the curve at the designated point P. The slope of a graph at a point is by definition the slope of the tangent line at that point. The figure above shows that the tangent line at P rises one unit for each unit change in x. Thus the slope of the tangent line at P is

  17. Slope of a Curve: Rate of Change

  18. Interpreting Slope of a Graph EXAMPLE • Refer to the figure below to decide whether the following statements about the debt per capita are correct or not. Justify your answers . • The debt per capita rose at a faster rate in 1980 than in 2000. • The debt per capita was almost constant up until the mid-1970s and then rose at an almost constant rate from the mid-1970s to the mid-1980s.

  19. Interpreting Slope of a Graph (a) The slope of the graph in 1980 is marked in red and the slope of the graph in 2000 is marked in blue, using tangent lines. It appears that the slope of the red line is the steeper of the two. Therefore, it is true that the debt per capita rose at a faster rate in 1980.

  20. Interpreting Slope of a Graph (b) Since the graph is a straight, nearly horizontal line from 1950 until the mid-1970s, marked in red, it is therefore true that the debt per capita was almost constant until the mid-1970s. Further, since the graph is a nearly straight line from the mid-1970s to the mid-1980s, marked in blue, it is therefore true that the debt per capita rose at an almost constant rate during those years.

  21. Equation & Slope of a Tangent Line EXAMPLE Given the slope of the graph of y = x2 at the point (x, y) is 2x.Find the slope of the tangent line to the graph of y = x2 at the point (-0.4, 0.16) and then write the corresponding equation of the tangent line.

  22. §1.3 The Derivative

  23. The Derivative

  24. Differentiation

  25. Differentiation Examples These examples can be summarized by the following rule.

  26. Differentiation Examples EXAMPLE Find the derivative of

  27. Differentiation Examples EXAMPLE Find the slope of the curve y = x5 at x = -2.

  28. Equation of the Tangent Line to the Graph of y = f(x) at the point (a, f(a))

  29. Equation of the Tangent Line EXAMPLE Find the equation of the tangent line to the graph of f(x) = 3x at x = 4.

  30. Leibniz Notation for Derivatives Ultimately, this notation is a better and more effective notation for working with derivatives.

  31. Calculating Derivatives Via the Difference Quotient The Difference Quotient is

  32. Differentiable

  33. Limit Definition of the Derivative

  34. Use TI89 to Graph • Slope and tangent lines • 1) Graph the function. • 2) 2ndDraw5, then type x value • or graph2ndcalc6, then type x value. • Graph y and y’ at the same time • 1) graph the function in y1. • 2) Enter y2 = nDerive(y1, x, x), then graph: • y2 =MathnDerive(VarsYvarsy1 then press , x, x)graph

  35. §1.4 Limits and the Derivative

  36. Definition of the Limit

  37. Finding Limits EXAMPLE Determine whether the limit exists. If it does, compute it. SOLUTION Let us make a table of values of x approaching 4 and the corresponding values of x3 – 7. As x approaches 4, it appears that x3 – 7 approaches 57. In terms of our notation,

  38. Finding Limits EXAMPLE For the following function g(x), determine whether or not exists. If so, give the limit. SOLUTION We can see that as x gets closer and closer to 3, the values of g(x) get closer and closer to 2. This is true for values of x to both the right and the left of 3.

  39. Limit Theorems

  40. Finding Limits EXAMPLE Use the limit theorems to compute the following limit.

  41. Limit Theorems

  42. Finding Limits EXAMPLE Compute the following limit.

  43. Using Limits to Calculate a Derivative

  44. Limit Calculation of the Derivative EXAMPLE Using limits, apply the three-step method to compute the derivative of the following function:

  45. Using Limits to Calculate a Derivative EXAMPLE Use limits to compute the derivative for the function

  46. Limits as x Increases Without Bound EXAMPLE Calculate the following limit.

  47. §1.6 Some Rules for Differentiation

  48. Rules of Differentiation

  49. Differentiation EXAMPLE Differentiate

  50. Differentiation EXAMPLE Differentiate

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