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)

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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)
• Maxima and minima of functions
• 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?
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:
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:
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):
Differentiation
• Differentiationis 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:
• 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:
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:
Today’s Examples
• Velocity & acceleration of a dropped ball
• 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
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

### 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
Review: Two Derivative Rules
• Two commonly used rules (c and n are constants):
Three New Derivative Rules
• Three more commonly used rules ( and a are constants):
Today’s Examples
• Velocity & acceleration from position
• Velocity & acceleration from position
• Velocity & acceleration from position (graphical)
• Position & velocity from acceleration (graphical)
• 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
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.