See the attached file for questioins
Thehonest
2.docMath 331 VB
Spring 2016
DIRECTIONS: Show as much work as possible for each question. This is the only way I can give partial credit if you happen to make a mistake. TI-89’s are wonderful calculators, but they don’t show me if you know anything about calculus! Therefore, don’t expect to receive full credit for a question if little or no work is shown. You may have 2
hours maximum to complete the exam. Show all work on the exam itself, you should NOT use any outside paper, notes.
1. (8 points each) Determine the interval where the functions are increasing and decreasing. Find the critical points (if any).
a.
f x( ) = x x2 +1
fx
()
=
x
x
2
+1
b.
f x( ) = 2x + 8 x
fx
()
=2x+
8
x
2.
(8 points) The 2nd derivative of a function is
′′f x( ) = 9 x + 2( )2 x − 3( )
¢¢
fx
()
=9x+2
()
2
x-3
()
.a. Determine the intervals where the function is concave up & where it is concave down
b. Give the values of any inflection points.
3.
(10 points) A marshy region used for agricultural drainage has become contaminated with selenium. It has been determined that flushing the area with clean water will reduce the selenium for a while, but it will then begin to build up again. A biologist has found that the percent of selenium in the soil x months after the flushing begins is given by
f x( ) = x 2 + 36 2x
fx
()
=
x
2
+36
2x
, 1 ≤ x ≤ 12. When will the selenium bereduced to a minimum?
4. (10 points) You want to build a fence to enclose a rectangular region of 20,000 ft2. You want the side of the rectangle facing the road to look the nicest, so you choose to buy $6 per foot fence for this side, and will buy $3 per foot fence for the other 3 sides. Determine the length and width which will produce the least expensive fence.
5. (10 points) Sketch a graph which meets all of the following conditions:
a) y -intercept at 3
b) when x 0 it is increasing
c) when x 0 it is decreasing
d) when 2 x 2, it is concave up
e) concave down when x 2 or x 2
f) horizontal asymptote at y 7
6.
(8 points) Use differentials to approximate
99
99
7.
(8 points) Calculate
dy dt
dy
dt
for
x2y + y3 =10
x
2
y+y
3
=10
when x = 1, y = 2, and
dx dt
= −4
dx
dt
=-4
8. (10 points) Use implicit differentiation to find the equation of the tangent line to the curve
9.
(10 points) For the function
f x( ) = e4x
fx
()
=e
4x
, calculate the 10th derivative. (Calculate and simplify the first few derivatives, then look for a pattern to predict the 10th derivative.)10.
(10 points) From second derivative test, determine the absolute maximum and absolute minimum values (if exists) for the equation
f x( ) = x8/3 −16x2/3
fx
()
=x
8/3
-16x
2/3
on the interval [1,8].
