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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

EGR 1101 Unit 8 Lecture #1

The Derivative

(Sections 8.1, 8.2 of Rattan/Klingbeil text)

a little history
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
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
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
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
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
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
Second Derivatives
  • When you take the derivative of a derivative, you get what’s called a second derivative.
  • Notation:
  • Alternate notations:
forget your physics
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:
from average to instantaneous
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
Today’s Examples
  • Velocity & acceleration of a dropped ball
  • Velocity of a ball thrown upward
maxima and minima
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
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

EGR 1101 Unit 8 Lecture #2

Applications of Derivatives: Position, Velocity, and Acceleration

(Section 8.3 of Rattan/Klingbeil text)

review
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
Review: Two Derivative Rules
  • Two commonly used rules (c and n are constants):
three new derivative rules
Three New Derivative Rules
  • Three more commonly used rules ( and a are constants):
today s examples1
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
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
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.