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Calculus, MET2 2020 VCAA 2

An area of parkland has a river running through it, as shown below. The river is shown shaded.

The north bank of the river is modelled by the function  .

The south bank of the river is modelled by the function  .

The horizontal axis points east and the vertical axis points north.

All distances are measured in metres.
 

A swimmer always starts at point , which has coordinates  (50, 30).

Assume that no movement of water in the river affects the motion or path of the swimmer, which is always a straight line.

  1. The swimmer swims north from point .
  2. Find the distance, in metres, that the swimmer needs to swim to get to get to the north bank of the river.   (1 mark)

  3. The swimmer swims east from point .
  4. Find the distance, in metres, that the swimmer needs to swim to get to the north bank of the river.   (2 marks)

  5. On another occasion, the swimmer swims the minimum distance from point to the north bank of the river.
  6. Find this minimum distance. Give your answer in metres, correct to one decimal place.   (2 marks)

  7. Calculate the surface area of the section of the river shown on the graph in square metres.   (1 mark)

  8. A horizontal line is drawn through point . The section of the river that is south of the line is declared a no "no swimming" zone.
  9. Find the area of the "no swimming" zone, correct to the nearest square metre.   (3 marks)

  10. Scientists observe that the north bank of the river is changing over time. It is moving further north from its current position. They model its predicted new location using the function with rule  , where .
  11. Find the values of  for which the distance north across the river, for all parts of the river, is strictly less than 20 m.   (2 marks)

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  5. ²
Show Worked Solution

a.   

 

b. 


 

c.  

♦ Mean mark part (c) 39%.

 

d.  

 

e.  

♦♦ Mean mark part (e) 35%.

 
  ²²  

♦♦♦Mean mark part (f) 15%.

 

f.   

 
     
   

  

Calculus, MET2 2021 VCAA 5

Part of the graph of    is shown below.
 

  1. State the period of .   (1 mark)

  2. State the minimum value of , correct to three decimal places.   (1 mark)

  3. Find the smallest positive value of for which  .   (1 mark)
Consider the set of functions of the form  , where is a positive integer.
  1. State the value of such that    for all .   (1 mark)

  2. i.  Find an antiderivative of in terms of .   (1 mark)

  3. ii. Use a definite integral to show that the area bounded by and the -axis over the interval    is equal above and below the -axis for all values of .  (3 marks)

  4. Explain why the maximum value of cannot be greater than 2 for all values of and why the minimum value of cannot be less than –2 for all values of .   (1 mark)

  5. Find the greatest possible minimum value of .   (1 mark)

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  5. i. 
    ii.
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a.   


 

b.   

♦ Mean mark part (c) 21%.
 

c.     


 

d.   


 

e.i. 

♦ Mean mark part (e)(i) 50%.


 

e.ii.   

♦ Mean mark part (e)(ii) 29%.


 


 

f.   

♦ Mean mark part (f) 13%.


 

g.   

♦ Mean mark part (g) 2%.

Calculus, MET1-NHT 2018 VCAA 7

Let    and  .

  1. Sketch the graph of and the graph of on the axes provided below.   (2 marks)

     
     

     

  2. Let be such that  ,  where 

     
    Find the value of    and the value of  .   (3 marks)

     

  3.  Let be the region enclosed by the horizontal axis, the graph of and the graph of .
    1. Shade the region on the axes provided in part a. and also label the position of on the horizontal axis.   (1 mark)

    2. Calculate the area of the region .   (3 marks)

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  1.  

     

  2.  

  3. i.

     
    ii.
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a.   

 

b.   

 

c. i.

 

   ii.       
 
 
 
 
 
  ²

Calculus, MET1-NHT 2019 VCAA 7

The shaded region in the diagram below is bounded by the vertical axis, the graph of the function with rule    and the horizontal line segment that meets the graph at  , where  .
 


 

Let    be the area of the shaded region.

  1. Show that  .   (3 marks)

  2. Determine the range of values of .   (2 marks)

    1. Express in terms of  , for a specific value of , the area bounded by the vertical axis, the graph of    and the horizontal axis.   (2 marks)

    2. Hence, or otherwise, find the area described in part c.i.   (1 mark)

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a.   

 
 
 

 

b.   


 

 

c.i. 

   
   
   
   

 

 

c.ii. 

 
 
 
 

Calculus, MET2 2019 VCAA 3

During a telephone call, a phone uses a dual-tone frequency electrical signal to communicate with the telephone exchange.

The strength, , of a simple dual-tone frequency signal is given by the function  , where    is a measure of time and  .

Part of the graph of   is shown below

  1. State the period of the function.   (1 mark)

  2. Find the values of    where    for the interval  .   (1 mark)

  3. Find the maximum strength of the dual-tone frequency signal, correct to two decimal places.   (1 mark)

  4. Find the area between the graph of    and the horizontal axis for  .   (2 marks)

Let    be the function obtained by applying the transformation    to the function  , where
 


 

and and are real numbers.

  1. Find the values of and given that    has the same area calculated in part d.   (2 marks)

  2. The rectangle bounded by the line  , the horizontal axis, and the lines    and    has the same area as the area between the graph of    and the horizontal axis for one period of the dual-tone frequency signal.

     

    Find the value of  .   (2 marks)

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  4. ²
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a.   
  

b.   
  

c.   

 

d.   
    ²

 

e.   


  

f.   
 
 

Calculus, MET1 2018 VCAA 9

Consider a part of the graph of  , as shown below.

  1.  i. Given that  , evaluate    when is a positive even integer or 0.
      
    Give your answer in simplest form.   (2 marks)

    ii. Given that  , evaluate    when is a positive odd integer.
    Give your answer in simplest form.   (1 mark)

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

  4. The translation maps the graph of    onto the graph of  , where
  6. and is a real constant.
  7. State the value of .   (1 mark)

  8. Let    and  .
    The line is the tangent to the graph of at the point and the line is the tangent to the graph of at , as shown in the diagram below.
     
         
    Find the total area of the shaded regions shown in the diagram above.   (2 marks)

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  1.  i.
    ii.
Show Worked Solution

a.i. 

♦♦ Mean mark 31%.

 

a.ii.

♦♦♦ Mean mark 19%.

 

b.   
 

 

♦ Mean mark part (b) 49%.

 
 

 

 

 

♦♦ Mean mark part (c) 34%.

c.   
   

 

 

d.   

♦♦♦ Mean mark 7%.

 

π

 


 

Calculus, MET2 2018 VCAA 19 MC

The graphs    and  are shown in the diagram below.
 


 

An integral expression that gives the total area of the shaded regions is

A.  

B.  

C.  

               

D.  

E.  

               

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Show Worked Solution

♦ Mean mark 41%.
COMMENT: In each shaded area, it is critical to note which graph is “higher” and which side of the x-axis the shading is.
 

         

 

Calculus, MET1 SM-Bank 28

The function    has the rule  .

  1. Show that the graph of    cuts the -axis at  .   (1 mark)

  2. Sketch the graph    for    showing where the graph cuts each of the axes.   (2 marks)

  3. Find the area under the curve    between    and  .   (3 marks)

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  1.  
  2.  
       
  3. ²
Show Worked Solution

a.   

 

 

 

b.   2UA HSC 2006 7b

 

c. 
 
  ­­
 
 
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Calculus, MET2 2015 VCAA 4

An electronics company is designing a new logo, based initially on the graphs of the functions

These graphs are shown in the diagram below, in which the measurements in the and directions are in metres.

VCAA 2015 4a

The logo is to be painted onto a large sign, with the area enclosed by the graphs of the two functions (shaded in the diagram) to be painted red.

  1. The total area of the shaded regions, in square metres, can be calculated as  .
  2. What is the value of ?   (1 mark)

The electronics company considers changing the circular functions used in the design of the logo.

Its next attempt uses the graphs of the functions  .

  1. On the axes below, the graph of    has been drawn.
  2. On the same axes, draw the graph of  .   (2 marks)

     

          VCAA 2015 4b

  3. State a sequence of two transformations that maps the graph of    to the graph of  .   (2 marks)

The electronics company now considers using the graphs of the functions  , where and are positive integers with   and .

    1. Find the area enclosed by the graphs of    and    in terms of and if is even.
    2. Give your answer in the form  , where and are integers.   (2 marks)

    3. Find the area enclosed by the graphs of    and    in terms of and  if is odd.
    4. Give your answer in the form  , where and are integers.   (2 marks)

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  2.  

    vcaa-2015-4b-answer   

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a.  
   

♦ Mean mark part (a) 36%.

 

b.   vcaa-2015-4b-answer

 

c.  

 

d.i.  

♦♦♦ Mean mark 20%.
MARKER’S COMMENT: Note that for even.

 

d.ii.  

♦♦♦ Mean mark 14%.
MARKER’S COMMENT: Note that for odd.