A Preview of Calculus (1.1) September 18th, 2012.

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A Preview of Calculus (1.1) September 18th, 2012

I. The Tangent Line Problem The slope of the tangent line at a point P on a curve is significant in Calculus. To approximate this slope, we use the slope of the secant lines that connect point P to other points on the curve that are increasingly closer to P.

The slope of the secant line is given by The slope of the tangent line is given by the limit of the slope of the secant line. This is also the slope of the curve at point P.

Ex. 1: The following points lie on the graph of f(x)=x 2. Q 1 =(1.5, f(1.5)), Q 2 =(1.1, f(1.1)), Q 3 =(1.01, f(1.01)), Q 4 =(1.001, f(1.001)), Q 5 =(1.0001, f(1.0001)). Each successive point gets closer to the point P(1,1). Find the slope of the secant line through Q 1 and P, Q 2 and P, and so on. Graph these secant lines on a graphing utility. Then use your results to estimate the slope of the tangent line to the graph of f at the point P.

II. The Area Problem To approximate the area under a curve between the vertical lines x=a and x=b and the x-axis, use increasing numbers of rectangles.

Ex. 2: Use the rectangles in each graph to approximate the area of the region bounded by y=sin x, y=0, x=0, and x=