1151 Exam 3 Review - MSLCMath1151 Exam3Review 1...

MSLC – Math 1151
Exam 3 Review
1. A particle is traveling along a one ‐ dimensional path (such as a number line). The position of the particle is governed by the
time function
4 3 2 ( ) 3 16 18 2 x t t t t    
, where
t
is in minutes and 5 0   t . Answer the following questions.
a) At what times is the particle stationary?
b) For which time intervals is the particle moving in a positive direction? A negative direction?
c) What is the particle’s most positive position? Most negative position?
d) What is the particle’s displacement? What is the total distance the particle has traveled?
e) When does the particle’s acceleration undergo a sign change? What is the particle’s acceleration at the times
when the particle is stationary?
f) Sketch a graph (on a t ‐ x coordinate plane) of the particle’s position using the information above.
2. Sketch the graph
2 3 2
1
x x
y
x
 


using your knowledge of limits, functions, and derivatives (first and second derivatives).
Find all the points of interest (x and y intercepts, critical values, max and mins, concavities, etc).
3. Sketch a graph with the following properties:

lim ( ) 0
x
f x


and
7
lim ( )
x
f x


 
,
( 2) (2) 1 f f   
,
'(0) '(4) 0 f f  

'( ) 0 f x 
on
 
0,4
, 0 ) ( '  x f on    
,0 4,7 

, 0 ) ( "  x f on  
2,2 
, 0 ) ( "  x f on
   
, 2 2,7  

 Domain
 
,7 
4. Find the area of the largest rectangle that can be inscribed in the ellipse
2 2
2 2
1
x y
a b
 
5. Suppose a rectangular box is to contain 10 cubic feet. The length of the box is to be twice the width. Material for the box
costs $2.50 per square foot for the bottom of the box, $1.50 per square foot for the sides of the box and $2.00 per square
foot for the top of the box. Find the dimensions of the box that would minimize the cost.
6. Find the most general antiderivatives of the following functions.
a)
2
5
2
( ) 20 f x x
x
  
b)
2 ( ) 4sec sin f x x x  
7. Find the particular antiderivative of the following.
a)
( ) x f x e 
,
(0) 2 F 
b)
 
3
2
( )
3
x
g x
x

,
(1) 3 G 
8. A lighthouse is located on an island 5 miles from the nearest point
P
on a straight shoreline and its light makes 6
revolutions per minute. How fast is the beam of light moving along the shoreline when it is 2 miles from
P
?