1007 : The Shortcut

1. 1007 : The Shortcut AP CALCULUS

2. Notation: The Derivative is notated by: Newton L’Hopital Leibniz Derivative of the function With respect to x Notation used in Cal 3

3. Notation: Find the rate of change of the Circumference of a Circle with respect to its Radius. Find the rate of change of the Area of a Square with respect to the length of a Side. Find the rate of change of the Volume of a Cylinder with respect to its Height. L’Hopital Leibniz C A´(s) = V You treat r as a constant

4. Algebraic Rules A). A Constant Function REM: Derivative is the slope of a tangent 0 y = 3 m = 0 0

5. Algebraic Rules B). A Power Function Rewrite in exponent form! y () =-2 y’ (x) =

6. Rewrite in exponent form!

7. c Algebraic Rules C). A Constant Multiplier y =15 y’ =

8. Algebraic Rules D). A Polynomial REM: How do you eat an elephant? One bite at a time

9. Example: Positive Integer Powers, Multiples, Sums, and Differences Calculator: [F3] 1: d( differentiate or [2nd ] [ 8 ] d( d(expression,variable) d( x^4 + 2x^2 - (3/4)x - 19 , x )

10. Do it all ! Step 1: Rewrite using exponents Must rewrite using exponents!

11. A conical tank with height of 4 ft is being filled with water. Write the equation for the volume of the conical tank. Find the instantaneous rate of change equation of the volume with respect to the radius. Find the instantaneous rate of change in Volume when the radius is 9 ft. When the radius is 9 the volume increases 24cu.ft.per minute

12. Second and Higher Order Derivatives function y’ y’’ y’’’ yiv could be y(4)

13. Second and Higher Order Derivatives

14. Example: Find all the derivatives. velocity acceleration jerk } The rest have mathematical uses

15. At a Joint Point Piece Wise Defined Functions: The function must be CONTINUOUS Derivative from the LEFT and RIGHT must be equal. The existence of a derivative indicates a smooth curve; therefore, … At 2=2 the derivative exists and it is a smooth transition Therefore the derivative DNE

16. Last Update • 08/12/10