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AP Calculus AB :: 2006-2007 :: Shubleka
Name______________________________ Page 1
Problem 1 Alice Gardiner wants to build a rectangular enclosure with a dividing fence in the middle of the rectangle. On one side, she plans to put some goats; on the other side she wants to raise some vegetables.
The fence along the outside of the rectangle costs $3 per foot, but the dividing fence costs $12 per foot. a) Alice decides to spend $240 on the fencing. What is the maximum area she can enclose? b) If Alice decides that she wants to enclose 300 square feet, what is the minimum cost? Problem 2 Let f ( x ) =
ln( x ) x
a) Find the value of x that maximizes f ( x ) . Justify your answer. b) Let h( x ) =
k ln( x ) . Is there a value of k so that x = 1 gives the maximum value of h ( x) ? If so, x
find this value. c) Let g ( x ) =
C + ln( x ) . Is there a value of C such that x = 1 gives the maximum value of g ( x ) ? If x
so, find this value. Problem 3 If f ( x) = e x , a) Find the local linearization near x = 0. b) Using the linearization from part a), approximate c) Is the approximation of
1 . e0.1
1 1 determined in part b) larger or smaller than the true value of 0.1 ? 0.1 e e
Defend your answer with geometric or graphical argument.
Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate. (Euler, Leonhard 1707-1783)
1
Name______________________________ Page 1
Problem 1 Alice Gardiner wants to build a rectangular enclosure with a dividing fence in the middle of the rectangle. On one side, she plans to put some goats; on the other side she wants to raise some vegetables.
The fence along the outside of the rectangle costs $3 per foot, but the dividing fence costs $12 per foot. a) Alice decides to spend $240 on the fencing. What is the maximum area she can enclose? b) If Alice decides that she wants to enclose 300 square feet, what is the minimum cost? Problem 2 Let f ( x ) =
ln( x ) x
a) Find the value of x that maximizes f ( x ) . Justify your answer. b) Let h( x ) =
k ln( x ) . Is there a value of k so that x = 1 gives the maximum value of h ( x) ? If so, x
find this value. c) Let g ( x ) =
C + ln( x ) . Is there a value of C such that x = 1 gives the maximum value of g ( x ) ? If x
so, find this value. Problem 3 If f ( x) = e x , a) Find the local linearization near x = 0. b) Using the linearization from part a), approximate c) Is the approximation of
1 . e0.1
1 1 determined in part b) larger or smaller than the true value of 0.1 ? 0.1 e e
Defend your answer with geometric or graphical argument.
Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate. (Euler, Leonhard 1707-1783)
1
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