SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

Calculus, MET2 2020 VCAA 4

The graph of the function  , where  , is shown below.
 

  1. Find the slope of the tangent to at  .   (1 mark)

  2. Find the obtuse angle that the tangent to at    makes with the positive direction of the horizontal axis. Give your answer correct to the nearest degree.   (1 mark)

  3. Find the slope of the tangent to at a point  . Give your answer in terms of  .   (1 mark)

  4.  i. Find the value of for which the tangent to at  and the tangent to at    are perpendicular to each other. Give your answer correct to three decimal places.   (2 marks)

  5. ii. Hence, find the coordinates of the point where the tangents to the graph of at    and    intersect when they are perpendicular. Give your answer correct to two decimal places.   (3 marks)

Two line segments connect the points   and    to a single point  , where  , as shown in the graph below.
 
         
 

  1.   i. The first line segment connects the point and the point , where .
  2.      Find the equation of this line segment in terms of  .   (1 mark)

  3.  ii. The second line segment connects the point and the point  , where  .
  4.      Find the equation of this line segment in terms of .   (1 mark)

  5. iii. Find the value of , where  , if there are equal areas between the function and each line segment.
  6.      Give your answer correct to three decimal places.   (3 marks)

Show Answers Only
  4.  i.
  5. ii.
  6.   i.
  7.  ii.
  8. iii.
Show Worked Solution

a.   

♦ Mean mark part (b) 37%.

 

b.   


 

c.  


 

d.i.   


 

d.ii. 

♦ Mean mark part (d)(ii) 41%.

♦ Mean mark part (e)(i) 44%.

 
e.i. 

♦♦ Mean mark part (e)(ii) 28%.
 

e.ii. 

 
 
♦♦ Mean mark part (e)(iii) 28%.

 

e.iii. 

Calculus, MET2 2021 VCAA 3

Let  .

  1. State the maximal domain and the range of .   (2 marks)

  2.  i. Find the equation of the tangent to the graph of when  .   (1 mark)

  3. ii. Find the equation of the line that is perpendicular to the graph of when    and passes through the point  (-2, 0).   (1 mark)

Let 

  1. Explain why is not a one-to one function.   (1 mark)

  2. Find the gradient of the tangent to the graph of at  .   (1 mark)

The diagram below shows parts of the graph of and the line  .
 
 
                     
 
The line    and the tangent to the graph of at    intersect with an acute angle of between them.

  1. Find the value(s) of for which  . Give your answer(s) correct to two decimal places.   (3 marks)

  2. Find the -coordinate of the point of intersection between the line  and the graph of , and hence find the area bounded by  , the graph of and the -axis, both correct to three decimal places.   (3 marks)

Show Answers Only

  2. i. 
    ii.
Show Worked Solution

a.   


 

 


 
 

b.     i.   


 

ii.   


  

c.    


 

d.  


 
 

e. 

 


 


 

f.    


 

 

Calculus, MET1 2007 VCAA 9

The graph of  ,    is shown. The normal to the graph of where it crosses the -axis is also shown.
 

MET1 2007 VCAA Q9
 

  1. Find the equation of the normal to the graph of where it crosses the -axis.   (2 marks)

  2. Find the exact area of the shaded region.   (3 marks)

Show Answers Only
  2. ²
Show Worked Solution

a.  

MARKER’S COMMENT: A common error was made in calculating the point of tangency. Be careful!

  

  

 

♦ Mean mark 44%.
b.   
   
   
   
    ²