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Calculus, MET1 SM-Bank 30

A function is given by  .

  1. Find the coordinates of the stationary points of    and determine their nature.   (3 marks)

  2. Hence, sketch the graph    showing the stationary points.   (2 marks)

  3. For what values of is the function increasing?   (1 mark)

  4. For what values of will    have no solution?   (1 mark)

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  4. 2UA HSC 2012 14ai
     
Show Worked Solution
i.
 
 

 

 

 

 

 
 
 

 

ii.  2UA HSC 2012 14ai

 

♦ Mean mark (HSC) 42%
MARKER’S COMMENT: Be careful to use the correct inequality signs, and not carelessly include  or by mistake.

 

iii.
 

 

iv.   

♦♦♦ Mean mark (HSC) 12%.

Calculus, MET2 2016 VCAA 2

Consider the function  

  1.  i. Given that  ,
  2.      show that  .   (1 mark)

  3. ii. Find the values of for which the graph of    has a stationary point.   (1 mark)

The diagram below shows part of the graph of  , the tangent to the graph at    and a straight line drawn perpendicular to the tangent to the graph at  . The equation of the tangent at the point with coordinates    is  .

The tangent cuts the -axis at . The line perpendicular to the tangent cuts the -axis at .
 


 

  1.   i. Find the coordinates of .   (1 mark)

  2.  ii. Find the equation of the line that passes through and and, hence, find the coordinates of .   (2 marks)

  3. iii. Find the area of triangle .   (2 marks)

  4. The tangent at is parallel to the tangent at . It intersects the line passing through and at .

     


     
     i. Find the coordinates of .   (2 marks)

  5. ii. Find the length of .   (3 marks)

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  1.  i.
  2. ii.
  3.   i.
  4. ii.
  5.      
  6. iii. ²
Show Worked Solution
a.i.   
   
   
 

 

  

 

a.ii.   


 

b.i.   


 

b.ii.   

   

   


 

b.iii.  
   
    ²

 

♦ Mean mark part (c)(i) 43%.
MARKER’S COMMENT: Many students gave an incorrect -value here. Be careful!
c.i.   
 
 
   

  

 

c.ii.  

 

♦♦ Mean mark 32%.

 

 

 

Calculus, MET2 2010 VCAA 4

Consider the function  

  1. Find the -coordinate of each of the stationary points of   and state the nature of each of these stationary points.   (4 marks)

In the following, is the function   where and are real constants.

  1. Write down, in terms of and , the possible values of for which is a stationary point of .   (3 marks)

  2. For what value of does have no stationary points?   (1 mark)

  3. Find in terms of if has one stationary point.   (2 marks)

  4. What is the maximum number of stationary points that can have?  (1 mark)

  5. Assume that there is a stationary point at and another stationary point  where  .
  6. Find the value of .   (3 marks)

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Show Worked Solution

a.  

 


 

 

vcaa-graphs-fur2-2010-4ai


 

b.  

♦ Mean mark (b) 46%.
MARKER’S COMMENT: Issues with use of CAS caused significant difficulties in this question.

 


 

c.  

♦ Mean mark part (c) 47%.


 

d.  

♦♦♦ Mean mark (d) 16%.

 

♦ Mean mark (e) 37%.

e.   

 

f.    


 

♦♦♦ Mean mark (f) 9%.

 

   


 

 

Calculus, MET2 2014 VCAA 5

Let  

  1. Express    in the form  .   (2 marks)

  2. Describe the translation that maps the graph of    onto the graph of  .   (1 mark)

  3. Find the values of such that the graph of   has
    1. one positive -axis intercept.   (1 mark)

    2. two positive -axis intercepts.   (1 mark)

  4. Find the value of for which the equation    has one solution.   (1 mark)

  5. At the point  , the gradient of    is and at the point , the gradient is  , where is a positive real number.
    1. Find the value of  .   (2 marks)

    2. Find and if  .   (1 mark)

    1. Find the equation of the tangent to the graph of    at the point  .   (1 mark)

    2. Find the equations of the tangents to the graph of    that pass through the point with coordinates  .   (3 marks)

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  3.  i.
  4. ii.
  6.  i.
  7. ii.
  8. i.
  9. ii.
Show Worked Solution

a. 


 

 
 
 

 

b.   
   
   

 

♦ Mean mark (b) 37%.


 

c.i. 

♦♦♦ Mean mark 7%.

 
met2-2014-vcaa-sec5-answer
 


 

c.ii.  

♦♦♦ Mean mark part (c)(ii) 19%.


 

d.   

♦♦♦ Mean mark 17%.

 

met2-2014-vcaa-sec5-answer1
 

 
 

 

e.i.  

♦♦ Mean mark (e.i.) 28%.

 

e.ii.  

♦♦♦ Mean mark (e.ii.) 10%.


 

f.i.   

 

♦♦ Mean mark (f.i.) 22%.

 

f.ii.  

♦♦♦ Mean mark (f.ii.) 14%.