1. § 1.2
The Slope of a Curve at a Point

2. Section Outline 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

3. Tangent Lines Definition Example
Tangent Line to a Circle at a Point P: The straight line that touches the circle at just the one point P

4. Slope of a Curve & Tangent Lines
Definition
Example
The Slope of a Curve at a Point P: The slope of the tangent line to the curve at P
(Enlargements)

5. Slope of a Graph EXAMPLE SOLUTION
Estimate the slope of the curve at the designated point P.
SOLUTION
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. Slope of a Curve: Rate of Change

7. 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.

8. Interpreting Slope of a Graph
CONTINUED
SOLUTION
(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.

9. Interpreting Slope of a Graph
CONTINUED
(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.

10. Equation & Slope of a Tangent Line
EXAMPLE
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.
SOLUTION
The slope of the graph of y = x2 at the point (x, y) is 2x. The x-coordinate of (-0.4, 0.16) is -0.4, so the slope of y = x2 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 Hence the equation is: