AP Calculus AB Fall 2009

AP Calculus AB Fall 2011 Name ______Derivative Applications Test Form B

1. Sketch the graph of f(x) with the following conditions. 6 points The graph of f( x ) is continuous. Use the information below to sketch the graph.

f( -2)=1, f(3)=3

f( x ) = 0 x = -2 and x = 1

f( x ) undefined x = -1

f( x ) > 0 (-�, � 2) -( 去 2, 1) ( 1, )

f( x ) < 0 (-1,1)

fⅱ( x ) = 0 x = - 2 and x = 3

fⅱ( x ) undefined x = -1

fⅱ( x ) > 0 (-2, - 1) �( 1,3)

fⅱ( x ) < 0 (-�, 去 2) ( 3, )

2. Given the graph of f '(x), find the following intervals or x values where: (estimate to the nearest ¼ unit)

This is the graph of f ' (x) a) f(x) in increasing. Justify. 2 pt b) f(x) has horizontal tangents. Justify. 2 pt c) f(x) has a local extrema. Justify. Be sure and identify the local extrema as a local max or local min. 4 pt d) f(x) is concave down. Justify. 2 pt e) f(x) has a point of inflection. Justify. 2 pt 3. Hermione is looking to climb a 17 foot ladder that is leaning against a rack of books. She needs to grab “A History of Hogwarts” book from the top shelf, but the ladder starts sliding away at a rate of 4 feet/sec. How fast is the top of the ladder sliding down the bookcase when the foot of the ladder is 8 feet from the bookcase. 5 pt

4. It’s just another day sailing through the ocean when Captain Barbosa exiles Captain Jack from the Black Pearl again. Jack is stranded 60 feet away from the Black Pearl. The Pearl’s flag is rising at a rate of 3 ft/sec. At what rate does Jack’s eyes travel to follow the flag, when the flag is 45 feet in the air? 5 pt

5. The top cone of the Wicked Witches’ Hourglass, which is counting down to Dorothy’s death, has a diameter of 10 inches and a height of 12 inches. If the sand is 8 inches deep and sinking at a rate of 5 inches per hour, at what rate is the volume changing? 5 pt Wow, I wish I could solve this.

If I only had a brain. 6. Harry Potter is walking through Hogsmeade Village one fine evening, heading back to Hogwarts castle. Harry walks past a lamppost that is 20 feet tall at a constant rate of 4 feet per second. Recently enduring a growth spurt, Harry is now 6 feet tall. a) Determine the rate at which Harry’s shadow is lengthening when three seconds have passed since he walked by the light. 4 pt

b) Determine the rate at which the tip of Harry’s shadow is lengthening. 2 pt

7. Find the absolute maximum and absolute minimum values guaranteed by the Extreme Value Theorem of f( x )= 2 x3 - 15 x 2 + 24 x + 7 on the interval [- 3, 5]. 5 pt

x 8. Let f( x ) = . Find the value of c that satisfies the conclusion of the Mean Value Theorem x +1 on the interval [2, 3]. 5 pt 9. A farmer has 800 ft of fencing and wants to fence off a rectangular field that borders a relatively straight river. He needs no fence along the river. What are the dimensions of the field that has the largest area? 5 pt

10. Find the dimensions of the rectangle with maximum area that has its base on the x-axis and its other two vertices above the x axis and lying on the parabola y = 48 - x 2 . 5 pt

11. Find the following limits. ( 3 points each = 9 pt)

2 sin x x 2x+ sin(5 x ) lim a) lim (b) lim c) 3p x 5x x 0 x p - x e sin(3x ) 2

Multiple Choice: You must show reasoning and all work for the multiple choice questions. 3 points each

______12. A function f is continuous for all x and has a local maximum at (3,5).Which statement must be true?

a. f ' (3) = 0 b. the graph of f is concave up at x = 3 c. f ' (x) exists at x = 3 d f ' (x) is positive if x < 3 and f ' (x) is negative if x > 3 e. f ' (x) is negative if x < 3 and f ' (x) is positive if x > 3

______15. f(x0 is graphed to the right. f( x ) How many critical numbers does f( x ) have? a. 3 b. 4 c. 5 d. 6 e. infinitely many

f( x )= 4 x3 - 27 x 2 + 54 x + 2 f'( x ) = 1 pt

critical numbers 2 pt

y ' line 1 pt

Find all relative extrema JUSTIFY 4 pt

f''( x ) = 1 pt

y '' line

1 pt points of inflection JUSTIFY 2 pt