MTH 251 – Differential Calculus Chapter 2 – Limits and ContinuityCalculus: A tool for Measuring ChangeAverage Rate of ChangeInstantaneous Rate of ChangeIn general … Average Rate of Change over an IntervalIn general … Instantaneous Rate of Change at x0ExampleMTH 251 – Differential CalculusChapter 2 – Limits and ContinuitySection 2.1Rates of Change and Tangents to CurvesCopyright © 2010 by Ron Wallace, all rights reserved.Calculus: A tool for Measuring Change•What does it mean to say that y is a function of x?Values of x uniquely determined by values of yx – independent variabley – dependent variable•Change?Difference in a variable between two pointsx = change in x = end value – start value = x1 – x0y = change in y = end value – start value = y1 – y0•Rate of change?Measurement of how a dependent variable is changing per unit of change of the independent variable. (examples: slope of a line & mph)( )y f x=Average Rate of Change•Compound Interest Formula•ExampleA0 = $100r = 10% per term0( ) (1 )tA t A r= +A(t) = value of the investment after t periodsA0 = original amount of the investmentr = interest rate per periodt = number of periods( ) 100(1.1)tA t =t A(t)0 $100.002 $121.004 $146.41Average Rate of Change:AtDD1st 2 periods: $10.50/period2nd 2 periods: $12.71/period1st 4 periods: $11.60/periodInstantaneous Rate of Change•Average Rate of Change  Difference Quotient (let t = h)•To find the “instantaneous rate of change”, use smaller and smaller values for h.•Example – Find the IRC at t = 2A0 = $100r = 10% per term0( ) (1 )tA t A r= +t A(t)A/ t2 $121.00 ----------2.1 $122.1588 $11.58772.01 $121.1154 $11.53802.001 $121.0115 $11.53310 0( ) ( )A A t h A tt hD + -=D( ) 100(1.1)tA t =Actual Value?$11.5325...In general …Average Rate of Change over an Interval•The average rate of change of y = f(x) over the interval [x0,x1] is …y/x = slope of a secant line1 0 0 01 0( ) ( ) ( ) ( )f x f x f x h f xyx x x h- + -D= =D -f(x)Secant LineP(x0, f(x0))Q(x1, f(x1))hMoving Q closer and closer to P causes the secant line to approach a limiting line called the tangent.Tangent LineIn general …Instantaneous Rate of Change at x0•The instantaneous rate of change of y = f(x) at x0 is the “limiting” value of the average rate of change as a second value of x gets closer and closer to x0.•Tangent line = Limiting line for the secant line.Slope of a curve at a point is the slope of the tangent line at that point.0 0 00( ) ( ) ( ) ( ) w/ near 0kkf x f x f x h f xyhx x x h- + -D= =D -Example•Find the average rate of change of the function over the interval; the slope at the given point; and the instantaneous rate of change at the given point.2( ) 4[1, 4](1, 3)f x x xxP=