1. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 1 of 115 § 1.2 The Slope of a Curve at a Point

2. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 2 of 115  Tangent Lines  Slopes of Curves  Slope of a Curve as a Rate of Change  Interpreting the Slope of a Graph  Finding the Equation and Slope of the Tangent Line of a Curve Section Outline

3. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 3 of 115 Tangent Lines DefinitionExample Tangent Line to a Circle at a Point P: The straight line that touches the circle at just the one point P

4. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 4 of 115 Slope of a Curve & Tangent Lines DefinitionExample The Slope of a Curve at a Point P: The slope of the tangent line to the curve at P (Enlargements)

5. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 5 of 115 Slope of a GraphEXAMPLE SOLUTION 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

6. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 6 of 115 Slope of a Curve: Rate of Change

7. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 7 of 115 Interpreting Slope of a GraphEXAMPLE Refer to the figure below to decide whether the following statements about the debt per capita are correct or not. Justify your answers. (a) The debt per capita rose at a faster rate in 1980 than in 2000. (b) 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.

8. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 8 of 115 Interpreting Slope of a GraphSOLUTION (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. CONTINUED

9. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 9 of 115 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. CONTINUED

10. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 11e – Slide 10 of 115 Equation & Slope of a Tangent LineEXAMPLE Find the slope of the tangent line to the graph of y = x 2 at the point (-0.4, 0.16) and then write the corresponding equation of the tangent line. SOLUTION The slope of the graph of y = x 2 at the point (x, y) is 2x. The x-coordinate of (-0.4, 0.16) is -0.4, so the slope of y = x 2 at this point is 2(-0.4) = -0.8. We shall write the equation of the tangent line in point-slope form. The point is (-0.4, 0.16) and the slope (which we just found) is -0.8. Hence the equation is: