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Business Calculus Rates of Change

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Types of Change Average rate of change: the average rate of change of y with respect to x is a ratio of change in y to change in x. as long as x 2 – x 1 ≠ 0. This is the same formula as the slope of a line. So for any function f, average rate of change can be thought of as the slope of the line connecting two points of f, (x 1, y 1 ) and (x 2, y 2 ). This line is called a secant line of the function f.

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Note: In application problems, the units of an average rate of change are a ratio of output to input. Difference Quotient: a second formula for the average rate of change is where x 2 = x 1 + h. This formula gives the slope of the secant line connecting two points of f.

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Instantaneous Rate of Change Instantaneous rate of change is a way to find the slope of a curve. It means finding a rate of change at a single point instead of between two points. We do this by finding the slope of the tangent line at the point. In other words: the instantaneous rate of change at point P = the slope of the curve at point P = the slope of the line tangent to the curve at point P.

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Drawing tangent lines: a tangent line can be found by drawing successive secant lines moving the second point closer and closer to the point P. Eventually, drawing the secant lines should not be necessary to draw a good tangent line. Existence of tangent lines: a function may have characteristics that will make a tangent line at a particular point impossible to draw, or to find its slope. At those places, we say the slope of the tangent line does not exist, and so the instantaneous rate of change does not exist (in a mathematical sense).

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The slope of the secant line through (c, f(c)) and (c+h, f(c+h)) is: The slope of the tangent line at (c, f(c)) is: To find average rate of change from x = c to x = c + h : To find instantaneous rate of change at a point x = c :

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