Presentation on theme: "Section 2.2 Instantaneous Rates of Change"— Presentation transcript:
1Section 2.2 Instantaneous Rates of Change MAT 213Brief CalculusSection 2.2 Instantaneous Rates of Change
2“At the instant the horse crossed the finish line, it was traveling at 42 miles per hour.”
3There 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?
4Instantaneous 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 pointsNow instantaneous rate of change gives us the slope of the tangent line at a single point
5Local LinearityIf we look close enough near any point on a smooth curve, the curve will look like a straight line!ExampleGrapefruit
6This is called the tangent line at that point Local LinearityIf 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 pointThe slope of a graph at a point is the slope of the tangent line at that point
7ExampleOn this graph, indicate where the slope of the tangent line is: positive greatestnegative zeroFind two points on the curve where the slopes are about the same.
8Instantaneous Rate of Change Average Rate of ChangeGraphically, 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 ChangeInstantaneous rate of change at a is the slope of the line TANGENT to the curve at a single point, (a,f(a))
10The Tangent LineThe 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)
12Except at an inflection point. Examples Concave up Concave down The Tangent LineGeneral RuleLines 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 tangencyExcept at an inflection point.ExamplesConcave up Concave down
13Where instantaneous rate of change DOES NOT exist… Point of discontinuitySharp pointVertical tangent (no “run”)Let’s look at some examples
14Consider the graph of f(x) = |x| Is it continuous at x = 0?Is it differentiable at x = 0?Let’s zoom in at 0