What is the differentiation. Essential Mathematics for Economics and Business, 4th Edition What is the differentiation. © John Wiley and Sons 2013 www.wiley.com/college/Bradley © John Wiley and Sons 2013 www.wiley.com/college/Bradley

Slopes/rates of change Recall linear functions For linear functions slope is the change in y per unit increase in x Slope is the rate of change The slope or rate of change is the same over any interval of any size See the following example

Example: calculate the rate of change for a linear function Suppose the distance, (x m) that a ball after y (seconds) is given by the equation y = x Slope is the rate of change The average speed of the ball is 1 meter per second, between the 1st and 4th sec Δy = 4 -1 = 3 y (distance in m) Δx = 4-1=3 x (time in seconds)

Example: the rate of change for a linear function.. continued Now take a different interval, 1 to 3 sec Slope is the rate of change = The slope of this linear function (speed) is 1 m/s when measured over any interval of any length! Average speed is 1 meter per second y (distance in m) Δy = 3-1=2 Δx =3-1=2 x (time in seconds)

The slope of a curve varies along the curve Suppose the distance y that a ball travels after x seconds is given by the equation y = x2 The slope varies along the curve. y (distance in m) x (time in seconds)

How do find the rate of change (speed) at x = 1? Start by getting the average slope over progressively smaller intervals where one end is fixed at x = 1. The average slope over the smallest possible interval at x = 1approximates the slope at x = 1. y (distance in m) x (time in seconds)

What is the rate of change (speed) at x = 1 Start by getting the average slope over progressively smaller intervals, where one end is fixed at x = 1. The average slope over the smallest possible interval at x = 1 approximates the slope at x = 1. y (distance in m) x (time in seconds)

Average slope of the curve between x = 1 and x = 3 or the average speed between 1st and 3rd second is 4 m/sec Slope of chord BC C Δy = 9-1=8 y (distance in m) B Δx = 3-1=2 x (time in seconds)

Average slope of the curve between x = 1 and x = 2.5 or the average speed between 1 and 2.5 seconds is 3.5 m/sec Slope of chord BC C y (distance in m) Δy = 6.25-1=5.25 B Δx = 2.5-1=1.5 x (time in seconds)

Average slope of the curve between x = 1 and x = 2 or the average speed between the 1st and 2nd seconds is 3 m/sec Slope of chord BC C y (distance in m) Δy = 4-1=3 B Δx = 2 - 1=1 x (time in seconds)

Average slope of the curve between x = 1 and x = 1.5 or the average speed between 1 and 1.5 seconds is 2.5 m/sec Slope of chord BC C B Δy = 2.25-1=1.25 Δx = 1.5 - 1= 0.5

Average slope of the curve between x = 1 and x = 1.1 or the average speed between 1 and 1.1 seconds is 2.1 m/sec Slope of chord BC C B Δy = 1.21-1=0.21 Δx = 1.1 - 1= 0.1

The slope of a curve at x = 1 Zero divided zero is not defined! So…. we need another method to find the slope at a point Slope of chord at B B

Summary, so far y = x2 is the distance travelled, y after x seconds Time interval: between Average speed 1st and 3rd second 4 m/s 1 and 2.5 seconds 3.5 m/s 1st and 2nd second 3 m/s 1 and 1. 5 second 2.5 m/s 1 and 1.1 second 2.1 m/s at 1 second Not defined ??? Slope (average speed) is approaching the value 2 as the interval Δx is becoming progressively smaller towards x = 1

Find slope of a curve at a point, graphically Slope of the curve at B = slope of the tangent at B But…the accuracy of graphical methods is limited! Measure the slope (m) of the tangent line B h = 4.0 d = 2.0

The slope of a curve at a point is determined analytically by differentiation. Method: given the equation of the curve, y = x2 First, find the derivative of y = x2 (see Worked Example 6.1 for proof ) Then evaluate the derivative at the given point At x = 1,

The slope of a curve at a point, graphically and analytically curve y = x2 At x = 1, the slope, m = Slope of the curve at B = slope of the tangent at B Graphically, measure the slope (m) of the tangent line B h = 4.0 d = 2.0

Summary: Slope of chord = 4, the average slope of the curve between x = 1 and x = 3 Slope of chord =3.5, the average slope of the curve between x = 1 and x = 2.5 Slope of chord =3, the average slope of the curve between x = 1 and x = 2 Slope of chord =2.5, the average slope of the curve between x = 1 and x = 1.5 Slope of curve at x = 1 is the same as the slope of the tangent at x =1 By differentiation slope of curve at x = 1 is

Procedure The slope of a curve at a point is determined analytically by differentiation Method consists of two simple steps. Given the curve, y = f(x), find its derivative , Evaluate the derivative at the given point First Next The slope of a curve at a point is also described as the instantaneous rate of change

At x = -1, dy/dx = 2(-1) = - 2 At x = 1.5, y = 2.25, dy/dx = 2(1.5) =3 Worked Example 6.1 Figure 6.3: Equation of the curve y = x2 Equation for slope of curve, dy/dx = 2x At x = -1, dy/dx = 2(-1) = - 2 At x = 1.5, y = 2.25, dy/dx = 2(1.5) =3 -2 -1.5 -1 -0.5 0.5 1 1.5 2 2.5 3 3.5 4 -3 -2.5 x y (i) y = 3x -2.25 (ii) y = -2x - 1 Figure 6.3: Slope of curve y = x2 at x = -1 and x = 1.5

Equation of the curve y = x2 Equation for slope of curve, dy/dx = 2x Worked Example 6.1 Figure 6.3: Equation of the curve y = x2 Equation for slope of curve, dy/dx = 2x At x = -1, dy/dx = 2(-1) = - 2 At x = 1.5, y = 2.25, dy/dx = 2(1.5) 1 2 3 4 5 6 7 8 9 -3 -2 -1 Figure 6.3: Slope of curve y = x2 at x = -1 and x = 1.5