Calculus I Chapter Three1. 2 Calculus Timeline: Descartes 1596-1650 Cavalieri 1598-1647 Fermat 1601-1665 Wallis 1616-1703 Barrow 1630-1677 Gregory 1638-1675.

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Presentation transcript:

Calculus I Chapter Three1

2 Calculus Timeline: Descartes Cavalieri Fermat Wallis Barrow Gregory Newton Leibniz

Calculus I Chapter Three3 GeoGebra

Calculus I Chapter Three4

5 Explain how slopes of secant lines approach the slopes of the tangent line at a point.(Definition(1)) Definition(1) Derivative at a point GeoGebra fileDerivative at a point

Calculus I Chapter Three6 Explain how slopes of secant lines approach the slopes of the tangent line at a point. (Definition(2)) Definition(2) Derivative at a point GeoGebra file

Calculus I Chapter Three7 The Derivative Function

Calculus I Chapter Three8 Average Rates The pattern of the average rates looks quadratic! xAverage Rate

Calculus I Chapter Three9 The derivative can be interpreted in two ways: the slope of the tangent line to the graph of the relation at the given x value. the rate of change of the function with respect to x at the given x value.

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Calculus I Chapter Three14 a-D b-C c-B d-A

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Calculus I Chapter Three16 Differentiable Implies Continuous

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Calculus I Chapter Three49 Alternative proof of the product rule using linearization

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Derivative of lnx:

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Derivative of :

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Derivative of sine inverse function:

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Derivative of cosine inverse function:

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Derivative of tangent inverse function:

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Derivative of cotangent inverse function:

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Derivative of secant inverse function:

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Derivative of cosecant inverse function:

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Calculus I Chapter Three105 Higher-order Derivatives Geogebra’s answer

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Calculus I Chapter Three107 Tangent lines and Normal lines The normal line to a curve C at a point A is the line through A that is perpendicular to the tangent line at A. The concept of the normal line to a curve has applications in the study of optics where one needs to consider the angle between a light ray and the normal line to a lens.

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