1. A Little History  Seventeenth-century mathematicians faced at least four big problems that required new techniques: 1. Slope of a curve 2. Rates of change (such as velocity and acceleration) 3. Maxima and minima of functions 4. Area under a curve

2. Maxima & Minima  Mona Kapoor

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  Differentiation is just the process of finding a function’s derivative.  The following sentences are equivalent: “Find the derivative of y(t) = 3t 2 + 12t + 7” “Differentiate y(t) = 3t 2 + 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. Maxima and Minima  Given a function y(t), the function’s local maxima and local minima occur at values of t where

13. 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

14. 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

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

16. 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

17. 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.