1. [22 marks] Using the method outlined in Section 5.4 of the textbook, and showing the working,...

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1. [22 marks] Using the method outlined in Section 5.4 of the textbook, and showing the working, sketch the graph of each of the following functions. Note: you do not need to consider f ''(x), concavity and points of inflection (step 6). (a) f(x) = (1/4)(x 2 - 4)2 (b) f(x) = 8 x 2 v x + 5 2. [8 marks] Find the absolute maximum and minimum of the function f(x) = x 2/3 (x 2 - 6) for x ? [-1, 3] Express your answers in simple exact form. 3. [8 marks] A company wants to manufacture an open cylindrical bucket of volume 10 litres (10000 cm3 ). The plastic used for the base of the bucket costs 0.05 cents per cm2 while the plastic used for the curved side of the bucket costs 0.02 cents per cm2 . Find the radius and height of the bucket for which the bucket has minimum cost. What is the minimum cost? Show all the reasoning and evaluate your answers to 2 decimal places. 4. [7 marks] (a) Find dy/dx for xy3 + v x 2 + 5y = 5 (b) Show that the point (x, y) = (2, 1) lies on the curve defined by

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MAS182: Applied Mathematics Semester 1, 2013 Assignment 2 Due by: 4:00pm Thursday, 28 March 2013 (This is the correct due date, not the one printed in the Unit Information.) 1. [22 marks] Using the method outlined in Section 5.4 of the textbook, and showing the working, sketch the graph of each of the following functions. Note: you do not need to '' consider f (x), concavity and points of in?ection (step 6). 2 2 (a) f(x) = (1=4)(x 4) 8 (b) f(x) = p 2 x x+5 2. [8 marks] Find the absolute maximum and minimum of the function 2=3 2 f(x) = x (x 6) for x2 [1;3] Express your answers in simple exact form. 3. [8 marks]Acompanywantstomanufactureanopencylindricalbucketofvolume10litres 3 2 (10000 cm ). The plastic used for the base of the bucket costs 0:05 cents per cm while 2 the plastic used for the curved side of the bucket costs 0:02 cents per cm . Find the radius and height of the bucket for which the bucket has minimum cost. What is the minimum cost? Show all the reasoning and evaluate your answers to 2 decimal places. 4. [7 marks] v 3 2 (a) Find dy=dx for xy + x +5y = 5 (b) Show that the point (x;y) = (2;1) lies on the curve de?ned by the equation in part (a), and ?nd the slope of the tangent line at this point. Notes: 10% of the marks for this assignment are reserved for presentation. There are penalties for late assignments. You must contact your tutor before the due date if you have di?culties making the deadline. 1

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David · Accepted Answer

Sol: 1(a)   
2
21( ) 4
4
f x x 
 
        
Domain: All real numbers.
 
        
intercept:  2x x    
            intercept:  0, 1y x y    
          
2
2
vertical asymptotes:
1
                      4
4
y x 
 
Symmetry:
    
    
   
2
2
2
2
1
4
4
1
4
4
f x x
f x x
f x f x
   
  
 
 
On taking a differentiation,
    
   
2
2
2
' 4 2
4
' 4
f x x x
f x x x
 
 
 
Again taking a differentiation  
                             
     
 
2
2
'' 4 2
'' 3 4
f x x x x
f x x
  
   
To find a critical point, 
                     ' 0f x 
    
 
2
2
2
' 4 2
4
4 0
2,0,2
f x x x
x x
x
 
 
 
 
See the critical point in below graph,
Sol: 1(b)    
2
8
( )
5
f x
x x

  
                   
 Domain: x R: 5 0 or 0x x    
 
                   
intercept:  2x x    
            intercept:  0, 1y x y    
         
2
2
vertical asymptotes:
8
     as 5
5
8
      as 0
5
x
x x
x
x x
 

 


Symmetry:
 
 
 
 
   
   
2
2
8
5
8
5
f x
x x
f x
x x
f x f x
or
f x f x
 
  
 
 
 
  
 
On taking a differentiation,

1. [22 marks] Using the method outlined in Section 5.4 of the textbook, and showing the working, sketch the graph of each of the following functions. Note: you do not need to consider f ''(x),...

Answer To: 1. [22 marks] Using the method outlined in Section 5.4 of the textbook, and showing the working,...

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