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An Introduction to Calculus

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Calculus Study of how things change Allows us to model real-life situations very accurately

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Topics (not in order) Limits Slope of a curve Tangent Lines Extreme values of a curve Area under a curve/volume of a region Lengths Riemann Sums (Finite & Infinite) Rates of Change

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Before Calculus After Calculus Find slope of a line Calculate average speed of a car Calculate area/volume of rigid geometric figures Find slope of a curve at a point Determine exact speed of a car at a particular time Determine area/volume of any figure under any curve/region

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Two Main Problems to Solve: I. Tangent Line Problem: How do you find the equation of the tangent line to the graph of a curve at a particular point? II. Area Under Curve Problem: How can you find the area underneath a curve and above the x-axis?

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f(c) f(c+ x c c+ x Secant line Tangent line Secant line c+ x f(c+ x

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As x decreases ( x0), secant line approaches the tangent line.

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abx1x1 x2x2 x3x3 x x x x

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abx1x1 x2x2 x3x3 x x x x x x x x x4x4 x5x5 x6x6 x7x7

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As the number of rectangles increases, the sum of their areas approaches the exact area of the curve.

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Limits and Their Properties 1 Copyright © Cengage Learning. All rights reserved.

Limits and Their Properties 1 Copyright © Cengage Learning. All rights reserved.

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