egr 1101 unit 8 lecture #1
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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: Slope of a curve Rates of change (such as velocity and acceleration) - PowerPoint PPT PresentationTRANSCRIPT
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.