1. Draw a curve on the graph paper. You can use any curve you like, but it should have a clear point of maximum or minimum.
2. Choose a point on the curve near the point of maximum or minimum. This will be the point where you will find the slope of the tangent line.
3. Draw a tangent line to the curve at the chosen point. You can do this by drawing a straight line that touches the curve at only one point.
4. Use the slope formula (y2-y1)/(x2-x1) to find the slope of the tangent line at the chosen point. You can choose a second point on the tangent line and a third point on the curve near the chosen point to calculate the slope.
5. Repeat steps 2-4 for several different points on the curve near the point of maximum or minimum.
6. Compare the slopes of the tangent lines at different points. What
do you notice? Do the slopes change as you move closer to the
point of maximum or minimum?
7. Use the knowledge you gained from the previous steps to answer
the following questions:
a. What is the relationship between the slope of the tangent
line and the slope of the curve at the point of maximum or
minimum?
b. What is the slope of the tangent line at the point of
maximum or minimum?
c. Can you predict whether the slope of the tangent line at the
point of maximum or minimum will be positive or negative?