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This lecture delves into the foundational concepts of derivatives, including the historical context from the 17th century, key problems that led to their development, and their applications in determining slopes, rates of change, and local extrema. Learn about the definition of derivatives using limits, common notations, differentiation processes, and second derivatives. The lecture also explores the relationships among position, velocity, and acceleration, including graphical interpretations. Master these essential calculus concepts to enhance your understanding of motion and change.
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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:
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 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.
