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Calculus, MET2 2022 VCAA 1

The diagram below shows part of the graph of , where .
 

  1. State the equation of the axis of symmetry of the graph of .   (1 mark)

  2. State the derivative of with respect to .   (1 mark)

The tangent to at point has gradient .

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

The diagram below shows part of the graph of , the tangent to at point and the line perpendicular to the tangent at point .
 

 

  1.  i. Find the equation of the line perpendicular to the tangent passing through point .   (1 mark)

  2. ii. The line perpendicular to the tangent at point also cuts at point , as shown in the diagram above.
  3.     Find the area enclosed by this line and the curve .   (2 marks)

  4. Another parabola is defined by the rule , where .
  5. A tangent to and the line perpendicular to the tangent at , where , are shown below.

  1. Find the value of , in terms of , such that the shaded area is a minimum.   (4 marks)

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a.   

b.   

c.    

di.   

dii.   Area units²

e.   

Show Worked Solution

a.   Axis of symmetry: 
  

b.       
   

  
c.   At gradient

 
 

 
When 
 


 
Equation of tangent at :

 
 
 


d.i   Gradient of tangent

  gradient of normal  

Equation at

 
 
 


d.ii  Points of intersection of and normal are at  and .

So equate and  to find

 
 
 

   
or

Area  
   
   
  units²  

 

e.     

At   

Gradient of tangent

Gradient of normal

Equation of normal at

 
 
 
 

 
Points of intersection of normal and parabola (Using CAS)

solve

  or 

 
Calculate area using CAS


 
Using CAS Solve derivative of   with respect to to find

solve

 
   and 

Given


♦♦ Mean mark (e) 33%.
MARKER’S COMMENT: Common errors: Students found and failed to subtract or had incorrect terminals.

Calculus, MET1 2023 VCAA 9

The shapes of two walking tracks are shown below.
 

 

Track 1 is described by the function  .

Track 2 is defined by the function  .

The unit of length is kilometres.

  1. Given that   and  , verify that   and  .   (1 mark)
  2. Verify that and both have a turning point at .
  3. Give the co-ordinates of .  (2 marks)
  4. A theme park is planned whose boundaries will form the triangle where is the origin, is at and is at , as shown below, where .
  5. Find the maximum possible area of the theme park, in km².   (3 marks)
     

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a.   

b.   

c.   

Show Worked Solution
a.   
 

 

 

b.   
   
 

 

 

 

 

 

 

 


♦ Mean mark (b) 48%.

c.  

 
 

 

 

 
 
 
 

♦♦♦ Mean mark (c) 27%.
MARKER’S COMMENT: Many students made arithmetic errors substituting the fractional values of into to find max area.

Calculus, MET1-NHT 2018 VCAA 9

Let diagram below shows a trapezium with vertices at    and  , where    is a real number and 

On the same axes as the trapezium, part of the graph of a cubic polynomial function is drawn. It has the rule  , where    is a non-zero real number and  .

  1. At the local maximum of the graph, .
  2. Find in terms of .   (3 marks)

The area between the graph of the function and the -axis is removed from the trapezium, as shown in the diagram.
 

  1. Show that the expression for the area of the shaded region is    square units.   (3 marks)

  2. Find the value of    for which the area of the shaded region is a maximum and find this maximum area.   (2 marks)

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a.     
 
 

 


 

 

 

 

b.   

 
 
 
 
 
 

 


 

 

 

c.   


 

 
 

Calculus, MET2-NHT 2019 VCAA 4

A mining company has found deposits of gold between two points, and , that are located on a straight fence line that separates Ms Pot's property and Mr Neg's property. The distance between and is 4 units.

The mining company believes that the gold could be found on both Ms Pot's property and Mr Neg's property.

The mining company initially models he boundary of its proposed mining area using the fence line and the graph of 

where is the number of units from point in the direction of point and is the number of units perpendicular to the fence line, with the positive direction towards Ms Pot's property. The mining company will only mine from the boundary curve to the fence line, as indicated by the shaded area below.
 

  1. Determine the total number of square units that will be mined according to this model.   (2 marks)

The mining company offers to pay Mr Neg $100 000 per square unit of his land mined and Ms Pot $120 000 per square unit of her land mined.

  1. Determine the total amount of money that the mining company offers to pay.   (1 mark)

The mining company reviews its model to use the fence line and the graph of

       where .

  1. Find the value of    for which    for all .   (1 mark)

  2. Solve    for in terms of .   (2 marks)

Mr Neg does not want his property to be mined further than 4 units measured perpendicular from the fence line.

  1. Find the smallest value of , correct to three decimal places, for this condition to be met.   (2 marks)

  2. Find the value of for which the total area of land mined is a minimum.   (3 marks)

  3. The mining company offers to pay Ms Pot $120 000 per square unit of her land mined and Mr Neg $100 000 per square unit of his land mined.
  4. Determine the value of that will minimize the total cost of the land purchase for the mining company. Give your answer correct to three decimal places.   (2 marks)

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a.   
 
 

 

b.   
 

 
c.   

 
 
 

 
d.   

 
 

e.   


 


 

 

f.   

 

 

 

g.   

Calculus, MET2 2019 VCAA 5

Let  . The tangent to the graph of at  , where  , intersects the graph of again at and intersects the horizontal axis at . The shaded regions shown in the diagram below are bounded by the graph of , its tangent at    and the horizontal axis.
 

  1. Find the equation of the tangent to the graph of at  , in terms of .   (1 mark)

  2. Find the -coordinate of , in terms of .   (1 mark)

  3. Find the -coordinate of , in terms of .   (2 marks)

Let    be the function that determines the total area of the shaded regions.

  1. Find the rule of , in terms of .   (3 marks)

  2. Find the value of for which is a minimum.   (2 marks)

Consider the regions bounded by the graph of , the tangent to the graph of at  , where  , and the vertical axis.

  1. Find the value of for which the total area of these regions is a minimum.   (2 marks)

  2. Find the value of the acute angle between the tangent to the graph of and the tangent to the graph of at  .   (1 mark)

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a.   


 

b.   


 

c.   


 

d.   

 
 

 

e.   


 

f.   


 

g.   

 

Calculus, MET1 2019 VCAA 7

The graph of the relation    is shown on the axes below. is a point on the graph of this relation, is the point and is the point .
 

  1. Find an expression for the length in terms of only.   (1 mark) 

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

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a.   

 

b.   
   
 
   
   

 

 
 

Calculus, MET1 SM-Bank 17

The diagram shows a point on the unit circle    at an angle from the positive -axis, where  .

The tangent to the circle at    is perpendicular to  , and intersects the  -axis at  ,  and the line    intersects the  -axis at  

 

  1. Show that the equation of the line is  .   (2 marks)

  2. Find the length of in terms of .   (1 mark)

  3. Show that the area, ,  of the trapezium is given by 
  4.       (2 marks)

  5. Find the angle that gives the minimum area of the trapezium.   (3 marks)

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i.

♦♦ Mean mark 20%

 

ii.  

 

 

 

iii. 

 

               

♦♦ Mean mark (iii) 24%
 
  ²

 

iv.  

 
 
 
 
Mean mark (iv) 19%
IMPORTANT: Look for any opportunity to use the identity as it is an examiner’s favourite and can often be the key to simplifying difficult trig equations.
 

 

 

IMPORTANT: Is the 1st or 2nd derivative test easier here? Students must evaluate and choose. Examiners often make one significantly easier than the other.

.

Calculus, MET1 SM-Bank 35

 

The diagram shows two parallel brick walls and joined by a fence from to .  The wall is metres long and  .  The fence is metres long.

A new fence is to be built from to a point somewhere on .  The new fence will cross the original fence at .

Let    metres, where  .

  1. Show that the total area,  square metres, enclosed by and is given by
  2.    .   (3 marks)

  3. Find the value of that makes as small as possible. Justify the fact that this value of gives the minimum value for .   (3 marks)

  4. Hence, find the length of when is as small as possible.   (1 mark)

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i.

♦♦♦ Students found this question extremely challenging (exact results not available).

 

 
 
 
 
 
 
 

 

ii.   

MARKER’S COMMENT: Students who could not complete part (i) are reminded that they can still proceed to part (ii) and attempt to differentiate the result given.
Note that  and are constants when differentiating. 

 

 


 

iii.   

 
 

 

Calculus, MET2 2016 VCAA 14 MC

A rectangle is formed by using part of the coordinate axes and a point  , where    on the parabola  
 

Which one of the following is the maximum area of the rectangle?

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♦♦ Mean mark 37%.

 

 
 

 

Calculus, MET1 2006 VCAA 9

A rectangle has two vertices, and , on the -axis and the other two vertices, and , on the graph of  , as shown in the diagram below. The coordinates of are where and are positive real numbers.

vcaa-2006-meth-9a

  1. Find the area, , of rectangle in terms of .   (1 mark)

  2. Find the maximum value of and the value of for which this occurs.   (3 marks)

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  2. ²
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♦ Mean mark 37%.
a.   
   
   

 

b.  

♦♦ Mean mark 29%.

 


 

²

Calculus, MET1 2013 VCAA 10

Let  

A right-angled triangle has vertex at the origin, vertex on the -axis and vertex on the graph of , as shown. The coordinates of are
 

 vcaa-2013-meth-10

  1. Find the area, , of the triangle in terms of .   (1 mark)

  2. Find the maximum area of triangle and the value of for which the maximum occurs.   (3 marks)

  3. Let be the point on the graph of on the -axis and let be the point on the graph of with the -coordinate .

     

    Find the area of the region bounded by the graph of and the line segment .   (3 marks)

     

    vcaa-2013-meth-10i

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  2. ²
  3. ²
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a.  
   
 

 

b.  

♦ Mean mark 35%.
 

²

 

c.  

♦♦ Mean mark 32%.


 

 
 
 

 

vcaa-2013-meth-10ii

 
 
 
  ²

Calculus, MET1 2014 VCAA 10

A line intersects the coordinate axes at the points and with coordinates and , respectively, where and are positive real numbers and  .

  1. When , the line is a tangent to the graph of    at the point with coordinates , as shown.

     


    met1-2014-vcaa-q10
     

     

    If and are non-zero real numbers, find the values of and .   (3 marks)

  2. The rectangle has a vertex at on the line. The coordinates of are , as shown.

     
    met1-2014-vcaa-q10_1
     

    1. Find an expression for in terms of .   (1 mark)

    2. Find the minimum total shaded area and the value of for which the area is a minimum.   (2 marks)

    3. Find the maximum total shaded area and the value of for which the area is a maximum.   (1 mark)

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    2. ²
    3. ²
Show Worked Solution

a.  

♦ Mean mark 48%.
MARKER’S COMMENT: The most common error incorrectly stated that (6,0) lay on the parabola.

 

 

 

 

b.i.  

met1-2014-vcaa-q10-answer

♦♦ Mean mark (b.i.) 32%.

 

 
 

 

♦♦♦ Mean mark (b.ii.) 18%.
b.ii.   
   
   

 

 

  ²

 

²
 

b.iii.  

♦♦♦ Mean mark (b.iii.) 8%.

²

Calculus, MET1 2015 VCAA 10

The diagram below shows a point, , on a circle. The circle has radius 2 and centre at the point  with coordinates . The angle is , where  .
  

met1-2015-vcaa-q10
  

The diagram also shows the tangent to the circle at . This tangent is perpendicular to and intersects the -axis at point and the -axis at point .

  1. Find the coordinates of in terms of .   (1 mark)

  2. Find the gradient of the tangent to the circle at in terms of .   (1 mark)

  3. The equation of the tangent to the circle at can be expressed as
  5.  i. Point , with coordinates , is on the line segment .
  6.     Find in terms of .   (1 mark)

  7. ii. Point , with coordinates , is on the line segment .
  8.     Find in terms of .   (1 mark)

  9. Consider the trapezium with parallel sides of length and .
  10. Find the value of for which the area of the trapezium is a minimum. Also find the minimum value of the area.   (3 marks)

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  3.  i.
  4. ii.
  6. ²
Show Worked Solution
a.   
   
 
♦♦♦ Part (a) mean mark 20%.
MARKER’S COMMENT: Many students did not include the “+2” in the -coordinate.

 

 

♦♦♦ Part (b) mean mark 16%.
b.   
   
   
     
 

 

c.i.  

 

c.ii.   

♦ Part (c)(ii) mean mark 47%.
 

 

♦♦♦ Part (d) mean mark 19%.
d.   
   
   

 

 

 
 

 

²

Calculus, MET2 2013 VCAA 4

Part of the graph of a function is shown below.

VCAA 2013 4a

  1. Points and are the positive -intercept and -intercept of the graph , respectively, as shown in the diagram above. The tangent to the graph of at the point is parallel to the line segment
    1. Find the equation of the tangent to the graph of at the point    (2 marks)

    2. The shaded region shown in the diagram above is bounded by the graph of , the tangent at the point , and the -axis and -axis.
    3. Evaluate the area of this shaded region.   (3 marks)
  2. Let be a point on the graph of  .
  3. Find the positive value of the -coordinate of , for which the distance is a minimum and find the minimum distance.   (3 marks)

The tangent to the graph of at a point has a negative gradient and intersects the -axis at point  , where  
 

VCAA 2013 4c
 

  1. Find the gradient of the tangent in terms of    (2 marks)

  2.   i. Find the rule for the function of that gives the area of the shaded region.   (2 marks)

  3.  ii. Find the maximum area of the shaded region and the value of for which this occurs.   (2 marks)

  4. iii. Find the minimum area of the shaded region and the value of for which this occurs.  (2 marks)

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Show Worked Solution

a.i.  

 

 


 

a.ii.   

♦ Mean mark 37%.
MARKER’S COMMENT: A common incorrect answer:
.  Know why it’s wrong!

 

 
  ²

 

  ²

 

b.  

♦♦♦ Mean mark 20%.
 

 


 

c.  

♦♦♦ Mean mark 8%.


 


 

 


 

d.i.  

♦♦♦ Mean mark 8%.


 

 

 

d.ii.   

♦♦♦ Mean mark 5%.

 

 

 met2-2013-vcaa-sec4-answer


 

d.iii.   

♦♦♦ Mean mark 3%.