Derivative as a Rate of Change

Chapter 3 Section 4

Usually omit instantaneous

Interpretation: The rate of change at which f is changing at the point x

Interpretation: Instantaneous rate are limits of average rates.

Example• The area A of a circle is related to its diameter by

the equation• How fast does the area change with respect to

the diameter when the diameter is 10 meters?• The rate of change of the area with respect to

the diameter

• Thus, when D = 10 meters the area is changing with respect to the diameter at the rate of

2

4DA

2

2444

22 DDDdDdAD

dDdA

mmD /71.152

)10(2

2

Motion Along a Line

• Displacement of object over time Δs = f(t + Δt) – f(t)

• Average velocity of object over time interval

ttfttf

tsvaverage

)()(

timetravelnt displaceme

Velocity

• Find the body’s velocity at the exact instant t– How fast an object is moving along a horizontal line– Direction of motion (increasing >0 decreasing <0)

Speed

• Rate of progress regardless of direction

Graph of velocity f ’(t)

Acceleration

• The rate at which a body’s velocity changes– How quickly the body picks up or loses speed– A sudden change in acceleration is called jerk• Abrupt changes in acceleration

Example 1: Galileo Free Fall

• Galileo’s Free Fall Equations distance falleng is acceleration due to Earth’s gravity

(appx: 32 ft/sec2 or 9.8 m/sec2)– Same constant acceleration– No jerk

2

21 gts

0121 2

dtdgt

dtd

dtdgt

dtds

dtd

dtdj

Example 2: Free Fall Example

• How many meters does the ball fall in the first 2 seconds?

• Free Fall equation s = 4.9t2 in meters

s(2) – s(0) = 4.9(2)2 - 4.9(0)2 = 19.6 m

22 9.4)8.9(21 tts

Example 2: Free Fall Example

• What is its velocity, speed and acceleration when t = 2?– Velocity = derivative of position at any time t

– So at time t = 2, the velocity is

ttdtdtstv 8.99.4)(')( 2

sec/6.19)2(8.98.99.4)(')( 2 mttdtdtstv

Example 2: Free Fall Example

• What is its velocity, speed and acceleration when t = 2?– Velocity = derivative of position at any time t

– So at time t = 2, the speed is

ttdtdtstv 8.99.4)(')( 2

sec/6.19sec/6.19 mmvelocityspeed

Example 2: Free Fall Example

• What is its velocity, speed and acceleration when t = 2?– Velocity = derivative of position at any time t

– The acceleration at any time t

– So at t = 2, acceleration is (no air resistance)

ttdtdtstv 8.99.4)(')( 2

2sec/8.98.9)('')(')( mtdtdtstvta

2sec/8.9)( mta

Derivatives of Trigonometric Functions

Chapter 3 Section 5

Derivatives

Application: Simple Harmonic Motion• Motion of an object/weight bobbing

freely up and down with no resistance on an end of a spring

• Periodic, repeats motion• A weight hanging from a spring is

stretched down 5 units beyond its rest position and released at time t = 0 to bob up and down. Its position at any later time t is

s = 5 cos(t) • What are its velocity and acceleration at

time t?

Application: Simple Harmonic Motion

• Its position at any later time t is s = 5 cos(t)– Amp = 5– Period = 2

• What are its velocity and acceleration at time t?– Position: s = 5cos(t)– Velocity: s’ = -5sin(t)

• Speed of weight is 0, when t = 0– Acceleration: s’’ = -5 cos(t)

• Opposite of position value, gravity pulls down, spring pulls up

Chain Rule

Chapter 6 Section 6

Implicit Differentiation

Chapter 3 Section 7

Implicit Differentiation

• So far our functions have been y = f(x) in one variable such as y = x2 + 3– This is explicit differentiation

• Other types of functionsx2 + y2 = 25 or y2 – x = 0

• Implicit relation between the variables x and y

• Implicit Differentiation– Differentiate both sides of the equation with respect to x, treating y

as a differentiable function of x (always put dy/dx after derive y term)

– Collect the terms with dy/dx on one side of the equation and solve for dy/dx

Circle Example

Folium of Descartes• The curve was first proposed

by Descartes in 1638. Its claim to fame lies in an incident in the development of calculus.

• Descartes challenged Fermat to find the tangent line to the curve at an arbitrary point since Fermat had recently discovered a method for finding tangent lines.

• Fermat solved the problem easily, something Descartes was unable to do.

• Since the invention of calculus, the slope of the tangent line can be found easily using implicit differentiation.

Folium of Descartes• Find the slope of the folium

of Descartes

• Show that the points (2,4) and (4,2) lie on the curve and find their slopes and tangent line to curve

0933 xyyx

Folium of Descartes

• Show that the points (2,4) and (4,2) lie on the curve and find their slopes and tangent line to curve

0933 xyyx

0)4)(2(942 33 072648

07272 00

0)2)(4(924 33

072864 07272

00

Folium of Descartes

• Show that the points (2,4) and (4,2) lie on the curve and find their slopes and tangent line to curve– Find slope of curve by implicit differentiation by

finding dy/dx

0933 xyyx

0933

dxdxy

dxdy

dxdx

dxd

0)(933 22

x

dxdy

dxdyx

dxdyyx

PRODUCT RULE

09933 22 ydxdyx

dxdyyx

09933 22 ydxdyx

dxdyyx

Factor out dy/dx

09393 22 yxxydxdy

22 3993 xyxydxdy

xyxy

dxdy

9339

2

2

xyxy

dxdy

33

2

2

Divide out 3

xyxy

dxdy

33

2

2

Evaluate at (2,4) and (4,2)

54

108

616412

)2(342)4(3

33

2

2

)4,2(2

2

)4,2(

xyxy

dxdy

Slope at the point (2,4)

45

810

124166

)4(324)2(3

33

2

2

)2,4(2

2

)2,4(

xyxy

dxdy

Slope at the point (4,2)

0933 xyyx

Find Tangents54

)4,2( dxdy

11 xxmyy

2544 xy

58

544 xy

458

54

xy

520

58

54

xy

512

54

xy

45

)2,4( dxdy

4452 xy

5452 xy

345

xy

512

54

xy

345

xy

Folium of Descartes• Can you find the slope of

the folium of Descartes

• At what point other than the origin does the folium have a horizontal tangent?– Can you find this?

0933 xyyx

Derivatives of Inverse Functions and Logarithms

Chapter 3 Section 8

Examples 1 & 2