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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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Definition of the Derivative Using Average Rate () a a+h f(a) Slope of the line = h f(a+h) Average Rate of Change = f(a+h) – f(a) h f(a+h) – f(a) h.

Definition of the Derivative Using Average Rate () a a+h f(a) Slope of the line = h f(a+h) Average Rate of Change = f(a+h) – f(a) h f(a+h) – f(a) h.

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