EGR 1101 Unit 8 Lecture #1

The Derivative

(Sections 8.1, 8.2 of Rattan/Klingbeil text)

A Little History

Seventeenth-century mathematicians faced at least four big problems that required new techniques:

1. Slope of a curve2. Rates of change (such as velocity and

acceleration)3. Maxima and minima of functions4. Area under a curve

Slope

We know that the slope of a line is defined as

(using t for the independent variable). Slope is a very useful concept for lines.

Can we extend this idea to curves in general?

tym

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

5

10

15

20

25

30

35

40

t

yy1(t) = 3*t + 4

(3,13)

(1,7)

(2,10)

(4,16)

Derivative

We define the derivative of y with respect to t at a point P to be the limit of y/t for points closer and closer to P.

In symbols:

ty

dtdy

t

0lim

Alternate Notations There are other common notations for the

derivative of y with respect to t. One notation uses a prime symbol ():

Another notation uses a dot:

ty

dtdyty

t

0lim)(

ty

dtdyty

t

0lim)(

Tables of Derivative Rules

In most cases, rather than applying the definition to find a function’s derivative, we’ll consult tables of derivative rules.

Two commonly used rules (c and n are constants):

0)( cdtd

1)( nn nttdtd

Differentiation

Differentiation is just the process of finding a function’s derivative.

The following sentences are equivalent: “Find the derivative of y(t) = 3t2 + 12t + 7” “Differentiate y(t) = 3t2 + 12t + 7”

Differential calculus is the branch of calculus that deals with derivatives.

Second Derivatives

When you take the derivative of a derivative, you get what’s called a second derivative.

Notation:

Alternate notations: dtd

dtyd dt

dy)(2

2

)()(tyty

Forget Your Physics

For today’s examples, assume that we haven’t studied equations of motion in a physics class.

But we do know this much: Average velocity:

Average acceleration:

tyvavg

tvaavg

From Average to Instantaneous

From the equations for average velocity and acceleration, we get instantaneous velocity and acceleration by taking the limit as t goes to 0.

Instantaneous velocity:

Instantaneous acceleration:dtdytv )(

dtdvta )(

Today’s Examples

1. Velocity & acceleration of a dropped ball2. Velocity of a ball thrown upward

Maxima and Minima

Given a function y(t), the function’s local maxima and local minima occur at values of t where

0dtdy

Maxima and Minima (Continued)

Given a function y(t), the function’s local maxima occur at values of t where

and

Its local minima occur at values of t where

and

0dtdy

02

2

dtyd

0dtdy

02

2

dtyd

EGR 1101 Unit 8 Lecture #2

Applications of Derivatives: Position, Velocity, and Acceleration

(Section 8.3 of Rattan/Klingbeil text)

Review

Recall that if an object’s position is given by x(t), then its velocity is given by

And its acceleration is given by

dtdx

txtv

t

0lim)(

2

2

0lim)(

dtxd

dtdv

tvta

t

Review: Two Derivative Rules

Two commonly used rules (c and n are constants):

0)( cdtd

1)( nn nttdtd

Three New Derivative Rules

Three more commonly used rules ( and a are constants):

)cos())(sin( ttdtd

)sin())(cos( ttdtd

atat aeedtd

)(

Today’s Examples

1. Velocity & acceleration from position 2. Velocity & acceleration from position3. Velocity & acceleration from position

(graphical)4. Position & velocity from acceleration

(graphical)5. Velocity & acceleration from position

Review from Previous Lecture

Given a function x(t), the function’s local maxima occur at values of t where

and

Its local minima occur at values of t where

and

0dtdx

02

2

dtxd

0dtdx

02

2

dtxd

Graphical derivatives

The derivative of a parabola is a slant line. The derivative of a slant line is a horizontal

line (constant). The derivative of a horizontal line

(constant) is zero.