OBJECTIVES: To introduce the ideas of average and instantaneous rates of change, and show that they are closely related to the slope of a curve at a point.

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Presentation transcript:

OBJECTIVES: To introduce the ideas of average and instantaneous rates of change, and show that they are closely related to the slope of a curve at a point on the curve. Rates of Change and Tangents to Curves

Rates of change of functions You have learned how the slope of a line indicates the rate at which a line rises or falls. For a line, this rate (or slope) is the same at every point on the line. For graphs other than lines, the rate at which the graph rises or falls changes from point to point. For instance, points on a parabola rise more quickly away from the vertex, level off at the vertex and then fall or rise after the vertex. The average rate of change is identical with the slope of secant. Average Rate of Change of with respect to over the interval is

Difference Quotient If is the point of tangency and is a second point on the graph of, the slope of the secant line through the two points is The right side of this equation is called the difference quotient. The beauty of this procedure is that you obtain better and better approximations of the slope of the tangent line by choosing the second point closer and closer to the point of tangency.

Slope of a Graph The slope of the graph of at the point is equal to the slope of its tangent (or secant) line at and is given by 1.Find the slope of the graph of

Slope of a Graph 2.Find the slope of 3.Find the formula for the slope of the graph Then find the slope at the points

Slope of a Graph 4.Find the formula for the slope of the graph 5.Find the formula for the slope of the graph

Find the equation of the tangent line at point P 6. 7.