Time 2
Module 2 10:30
1. Find the dimensions of the rectangle of maximum area, with sides parallel to the coordinate
axes, that can be inscribed in the ellipse given by:
x2 y2
+
= 1.
16 4
In order to get the full credit for the exercise you must show all
Show all of your work and explain your reasoning. A correct answer without supporting work will
not awarded credit.
1. A cylindrical container with no top has surface area of 3 square feet. What should
be the radius and the height of the cylinder to maxim
Time 1
Module 2 12:30
Solutions
1. Use the Intermediate Value Theorem and Rolles Theorem to prove that the following
equation has exactly one real solution:
10 x 5 + 3 sin x = 0 .
Solution:
Let denote f ( x) = 10 x 5 + 3 sin x. The function f is continuou
Show all of your work and explain your reasoning. A correct answer without supporting work will
not awarded credit.
6
1. Find 5 x 3
+ cos x dx .
x
Exit Exam Module 2 Time 2 12:30 Fall 2007
Show all of your work and explain your reasoning. A correct answe
Show all of your work and explain your reasoning. A correct answer without supporting work will
not awarded credit.
1. Find the relative and absolute extrema of the function f ( x) =
2x
on the interval
2 + x2
[2, 2].
Exit Exam Module 2 Time 2 10:30 Fall 2
Directions:
Please read the cover sheet carefully; you are responsible for the notices on the
sheet.
1.
Show all your work. A correct answer without supporting work will not be
awarded any credit.
Solve the differential equation f ' ( t ) = 2et + 3sin t
CALCULUS 1 MATH 1411
SPRING 2008
Time 3
Module 2 12:30
1. Locate the absolute extrema of the function: f ( x) = sin x + cos x on the closed interval
0, 2 . (12p)
Solution:
f ' ( x) = cos x sin x , so that f ' ( x) = 0 becomes
cos x sin x = 0
tan x = 1
CALCULUS 1 MATH 1411
SPRING 2008
Time 3
Module 2 10:30
1. Locate the absolute extrema of the function f ( x) = x 2 2 cos x on the closed
interval [-1,3].
2. Let p ( x) = Ax 2 + Bx + C . Prove that for any interval [a,b], the value c guaranteed
by the Mean
CALCULUS 1 MATH 1411
SPRING 2008
Time 3
Module 2 8:30
1. Find the absolute and relative extrema of the function f ( x) =
x
on its entire
x2 + 1
domain.
IMPORTANT: In order to get the full credit you need to show all the work.
CALCULUS 1 MATH 1411
SPRING 2