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Calculus, MET2 2022 VCAA 5

Consider the composite function , where the function is an unknown but differentiable function for all values of .

Use the following table of values for and .

 

  1. Find the value of .   (1 mark)

The derivative of with respect to is given by .

  1. Show that .   (1 mark)

  2. Find the equation of the tangent to at .   (2 marks)

  3. Find the average value of the derivative function between and .   (2 marks)

  4. Find four solutions to the equation for the interval .   (3 marks)

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a.   

b.   

c.   

d.   

e.   

Show Worked Solution
 

a. 
 
   
   

 

b.   
 
   

 

c.     and 

 
 
 

♦♦ Mean mark (c) 45%.
MARKER’S COMMENT: Some students did not produce an equation as required.

  
d.   The average value of between and

Average  
   
   
   

♦♦ Mean mark (d) 30%.
MARKER’S COMMENT: Those who used the Average Value formula were generally successful.
Some students substituted , not .

e.  

  or 

(1):         
    
 
 
     
(2):  
 
 

  


♦♦ Mean mark (e) 30%.
MARKER’S COMMENT: Some students were able to find . Some solved or but not both.