AP Calculus

Class of December 15, 2008

1)    We handed in the Bonus

2)    We started applications of derivatives (see below)

3)    We finished and handed in the worksheet we started last class

Derivative Applications

• Maximum and minimum points of a graph are where the derivative is zero
• Pg. 371 #2
• y= 4-x² from x=0 to x=2; the parabola is rotated, and there is a cone inscribed; we need to find the maximum volume of the cone
• VOCAB: paraboloid: 3D parabola rotated

• In order to find the maximum volume of the cone, we need to determine the height and radius of the cone.
• V=1/3πr²h; this is the equation of volume of the cone. In order to find the maximum volume, we need to find the maximum height, or where the derivative is equal to zero. However, this equation is in terms of two variables, and we need it to have only one.
• The radius of the cone is equal to the x value at the outer-most edge of the cone; the height of the cone is equal to the y value at the top of the cone.
• Therefore, V (r)= 1/3πr²(4-r²)= 4/3πr²-1/3πr⁴
• V’=8/3πr-4/3r³ then becomes the derivative of the above equation. In order to find the maximum point, we need to find where the derivative is equal to zero, so set this equation equal to zero. However, the answers this way aren’t necessary maximum points. In order to be maximum points, the derivative to the left has to be positive, and the derivative on the right has to be negative. REMEMBER TO WRITE THIS OUT, ESPECIALLY ON THE AP TEST!!
• V’= 4/3πr(2-r²)=0
  • r=0 or 2-r²= 0 or r= +/- √2, only +√2 (-√2 is not within the given domain)
• While r=√2 is the only possible maximum point, because maximum points cannot be at zero in this problem, it cannot be assumed that it will be the maximum. It is only the maximum if:
  • x<√2, V’>0
  • x> √2, V’<0
• In order to check to see if these two are right, we must plug a number less than √2 and greater than √2 into the derivative equation and see if they give us a number larger than zero and less than zero (respectively).
• REMEMBER: Always check the endpoints! The endpoints could be a maximum or minimum point. Plug the endpoints into the equation and see which volume is the highest.
• NOTE: The technique for analysis of maximum and minimum point problems is on page 372!

HOMEWORK: pg. 367 #39 and max./min. problems on pg. 372 # 1, 7, 11, 15; quiz corrections due next class, and the MC packet is due on Friday. No quiz on Friday!!!

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