EGR 1101 Unit 8 Lecture #1

1. EGR 1101 Unit 8 Lecture #1 The Derivative (Sections 8.1, 8.2 of Rattan/Klingbeil text)

2. 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

3. 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?

5. 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:

6. 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:

7. 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):

8. 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.

9. Second Derivatives • When you take the derivative of a derivative, you get what’s called a second derivative. • Notation: • Alternate notations:

10. 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:

11. 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:

12. Today’s Examples • Velocity & acceleration of a dropped ball • Velocity of a ball thrown upward

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

14. 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

15. EGR 1101 Unit 8 Lecture #2 Applications of Derivatives: Position, Velocity, and Acceleration (Section 8.3 of Rattan/Klingbeil text)

16. 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

17. Review: Two Derivative Rules • Two commonly used rules (c and n are constants):

18. Three New Derivative Rules • Three more commonly used rules ( and a are constants):

19. 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

20. 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

21. 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.