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MAT 213 Brief Calculus Section 2.2 Instantaneous Rates of Change

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“At the instant the horse crossed the finish line, it was traveling at 42 miles per hour.”

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There is a paradox in trying to study motion at a particular instant in time. By focusing on a single instant, we stop motion!!! How can we know the speed of a horse at the instant it crosses the finish line?

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Instantaneous Rate of Change The instantaneous rate of change at a point on a curve is the slope of the curve at that point. Slope is the measure of tilt of a LINE! In the last section we discussed average rate of change as the slope of the secant line between two points Now instantaneous rate of change gives us the slope of the tangent line at a single point

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Local Linearity If we look close enough near any point on a smooth curve, the curve will look like a straight line! Example Grapefruit

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Local Linearity If we look close enough near any point on a smooth curve, the curve will look like a straight line! This is called the tangent line at that point The slope of a graph at a point is the slope of the tangent line at that point

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Example On this graph, indicate where the slope of the tangent line is: positive greatest negative zero Find two points on the curve where the slopes are about the same.

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Average Rate of Change Graphically, the average rate of change over the interval a ≤ x ≤ b is the slope of the secant line connecting (a,f(a)) with (b,f(b)) Instantaneous Rate of Change Instantaneous rate of change at a is the slope of the line TANGENT to the curve at a single point, (a,f(a))

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Secant line becoming tangent line

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The Tangent Line The tangent line at a point Q on a smooth, continuous graph is the limiting position of the secant lines between point Q and point P as P approaches Q along the graph (if the limiting position exists)

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The Tangent Line General Rule Lines tangent to a smooth, nonlinear curve do not “cut through” the graph of the curve at the point of tangency and lie completely on one side of the graph near the point of tangency Except at an inflection point. Examples Concave up Concave down

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Where instantaneous rate of change DOES NOT exist… -Point of discontinuity -Sharp point -Vertical tangent (no “run”) Let’s look at some examples

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Consider the graph of f(x) = |x| Is it continuous at x = 0? Is it differentiable at x = 0? –Let’s zoom in at 0

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No matter how close we zoom in, the graph never looks linear at x = 0 –Therefore there is no tangent line there so it is not differentiable at x = 0

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Example- Piecewise Defined Function

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Note: This is a graph of It has a vertical tangent at x = 0 Example

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In groups let’s try the following from the book 7, 25

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