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Calculus, MET2 2020 VCAA 5

Let  .

Let    be the function representing the tangent to the graph of at  , where  .

Let be the -intercept of the graph of .

  1. Show that  .   (3 marks)

  2. State the values of for which does not exist.    (1 mark)

  3. State the nature of the graph of when does not exist.   (1 mark)

  4. i.  State all values of for which  . Give your answer correct to four decimal places.   (1 mark)

  5. ii. The graph of has an -intercept at (1, 0).
  6.      State the values of    for which  .
  7.      Give your answers correct to three decimal places.   (1 mark)

The coordinate is the horizontal axis intercept of .

Let be the function representing the tangent to the graph of at  , as shown in the graph below.
 
 
     
 

  1. Find the values of for which the graphs of and , where exists, are parallel and where  .   (3 marks)

Let  , where  .

  1. Show that    for all  .   (1 mark)

A property of the graphs of is that two distinct parallel tangents will always occur at and for all  .

  1. Find all values of such that a tangent to the graph of at , for some  , will have an -intercept at .   (1 mark)

  2. Let  , where    and  .
     
    State any restrictions on the values of , , , and , given that the image of under the transformation always has the property that parallel tangents occur at    and    for all  .   (1 mark)

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  4.  i.
  5. ii.
Show Worked Solution

a.   

 
 

  

 
 
 
 

 
b.   

♦ Mean mark part (b) 46%.

♦♦ Mean mark part (c) 23%.
 

c.   


 

d.i. 


  

d.ii. 

♦♦♦ Mean mark part (d)(ii) 13%.


 

e.   

 
 

 

♦♦♦ Mean mark part (e) 13%.
 
   

 


 

f.   
   
   
   

 
g. 

♦♦♦ Mean mark part (g) 3%.

 
 

 


 

h.   

♦♦♦ Mean mark part (h) 2%.

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)

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  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. 

Algebra, MET2 2017 VCAA 4

Let  . Part of the graph of  is shown below.
 

  1. The transformation    maps the graph of    onto the graph of  .

     

    State the values of and .   (2 marks)

  2. Find the rule and domain for  , the inverse function of  .   (2 marks)

  3. Find the area bounded by the graphs of  and  .   (3 marks)

  4. Part of the graphs of  and  are shown below.
     

         
     
    Find the gradient of  and the gradient of    at  .   (2 marks)

The functions of  , where  , are defined with domain such that  .

  1. Find the value of such that  .  (1 mark)

  2. Find the rule for the inverse functions  of  , where  .   (1 mark)

  3. i. Describe the transformation that maps the graph of  onto the graph of  .   (1 mark)

    ii. Describe the transformation that maps the graph of  onto the graph of  .   (1 mark)

  4. The lines and are the tangents at the origin to the graphs of    and    respectively.
  5. Find the value(s) of for which the angle between and is 30°.   (2 marks)

  6. Let be the value of for which    has only one solution.
  7.  i. Find .   (2 marks)

  8. ii. Let    be the area bounded by the graphs of    and    for all  .
  9.     State the smallest value of such that  .   (1 mark)

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a. 

b. 

c.  ²

d.

e. 

f. 

g.i. 

g.ii. 

h. 

i.i. 

i.ii. 

Show Worked Solution

a.   

   

 

 

b.   

 

 

c.   

MARKER’S COMMENT: Specifically recommends using    in this type of integral to avoid errors in stating the integral.

 

  ²

 

d.   
 

 

e.   

 

f.   

 

♦♦ Mean mark part (g)(i) 31%.

g.i.   
 

 

 

♦♦ Mean mark part (g)(ii) 30%.

g.ii.   
 

 

 

h.   

♦♦♦ Mean mark part (h) 13%.

 

 

 

i.i   

♦♦♦ Mean mark part (i)(i) 4%.

 

 

i.ii 

♦♦♦ Mean mark part (i)(ii) 2%.

 

Calculus, MET1 2017 VCAA 9

The graph of    is shown below.
 

  1. Calculate the area between the graph of and the -axis.   (2 marks)

  2. For in the interval , show that the gradient of the tangent to the graph of is  .   (1 mark)

The edges of the right-angled triangle are the line segments and , which are tangent to the graph of , and the line segment , which is part of the horizontal axis, as shown below.

Let be the angle that makes with the positive direction of the horizontal axis, where  .

  1. Find the equation of the line through and in the form  , for  .   (2 marks)

  2. Find the coordinates of when  .   (4 marks)

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Show Worked Solution
a.  
   
   
   
   
   

 

♦♦ Mean mark part (b) 35%.
MARKER’S COMMENT: Establishing the common denominator in the working was required!
b.  
 
   
   

 

c. 

♦♦♦ Mean mark part (c) 20%.
MARKER’S COMMENT: Most successful answers introduced a pronumeral such as to solve.

 

 

   
 
   

 

 

 

d. 

♦♦♦ Mean mark part (d) 17%.

   
 
   

 

 

 

 

Calculus, MET1 2016 VCAA 2

Let  , where  .

  1. Find  .   (1 mark)

  2. Find the angle from the positive direction of the -axis to the tangent to the graph of   at  , measured in the anticlockwise direction.   (2 marks)

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Show Worked Solution
a.   
 
   

 

♦ Mean mark 40%.
b.