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**Sec. 2.1: The Derivative and the Tangent Line**

Goal: To calculate the slope of a curve at a point on the curve. This is the same as calculating the slope of the tangent line to the curve at a point.

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Slope of a Secant Line Need: Slope formula between the two points (c, f(c)) and (c + ∆x, f(c + ∆x)). Slope of secant line = (difference quotient)

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Slope of a Tangent Line By using the formula for finding the slope of a secant line and letting ∆x approach zero, the resulting slopes approach the slope of the tangent line.

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**Definition of Tangent Line with Slope m**

If f is defined on an open interval containing c, and if the limit exists, then the line passing through (c, f(c)) with slope m is the tangent line to the graph of f at the point (c, f(c)). The slope of the tangent line is also known as the slope of the curve at x = c.

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**Definition of the Derivative of a Function**

The derivative of f at x is given by provided the limit exists. The derivative of f at x = c is equivalent to the slope of the tangent line to f at x = c.

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2.4 Rates of Change and Tangent Lines Devil’s Tower, Wyoming Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1993.

2.4 Rates of Change and Tangent Lines Devil’s Tower, Wyoming Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1993.

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