113Q, Math Exam 1 Feb 18, 2008
Name Discussion Section
Directions: Please read each question carefully. Show all work clearly in the space provided. For full credit, solution methods must be complete, logical and understandable. Answers must be clearly labeled in the spaces provided after each question. Please mark out or fully erase any work that you do not want graded. The point value of each question is indicated after its statement. No books or other references are permitted. Calculators are allowed, but you must show all your work to receive full credit on a problem.
1) For each part, if the statement is true, circle the printed capital T following the statement. If the statement is false, circle the printed capital F. (6 points) 3 3
a) One may apply Newtons Method to f (x) = T
x 5 to compute F
5.
b) An antiderivative of f (x) = e
x2
ex is F (x) = . 2x T F
2
c) If F (x) is an antiderivative of f (x) and G(x) = F (x) + 4, then G(x) is an antiderivative of f (x). T F
x2 dx as a limit of Riemann sums. Your nal expression 3 ln x needs to have the actual function being integrated in it and not the symbol f (x). It does not 2) a) Find an expression for f (x) = need to be simplied any further than that. DO NOT EVALUATE THE LIMIT! (7 points)
8
b) Let f (x) be a continuous function given by the following Use graph. the graph to evaluate the following integrals. (8 points)
" &
#
!
% % !! !# ! # & " ' ) $ + ( *
2
i)
0
f (x)dx
7
ii)
0
f (x)dx
3) Find the most general antiderivative of the following functions. 1 x1 3
a) f (x) = 4x2 + 6
(5 points)
b) f (x) =
3
x4 +
2 x
(5 points)
4) A rectangular swimming pool is to be built with an area of 1800 square feet. The owner wants 5-ft wide decks along either side and 10-ft wide decks at the two ends (as shown). Find the dimensions of the smallest piece of property on which the pool can be built satisfying these conditions. (15 points)
Please circle a) the equation that you are optimizing and b) your nal solution to the problem
5 y 10 x 5 10
5) a) Use Newtons Method to nd the third approximation (x3 ) to the r...