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Slope

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Slope is a measure of the change in the dependent variable for a given change of the independent variable of a function.

The average slope, or slope of the secant line between x1 and x2, can be computed using the formula:

\frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}.

When a linear equation (with no powers other than one and zero) is written in slope-intercept form (y = mx + b), the slope is m, the coefficient of x.

To find the instantaneous slope of a function, or the slope of the tangent line at x0, the limit as x approaches x0 can be taken.

\lim_{ \Delta x \to 0} \frac{\Delta y}{\Delta x} = \lim_{ x \to x_0} \frac{f(x) - f(x_0)}{x - x_0}

This can be extended to every point on a curve by the formula

\lim_{ h \to 0} \frac{f(x + h) - f(x)}{h}

which is the definition of the derivative of f(x).

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