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Calculus, MET2 EQ-Bank 2

Jac and Jill have built a ramp for their toy car. They will release the car at the top of the ramp and the car will jump off the end of the ramp.

The cross-section of the ramp is modelled by the function , where

is both smooth and continuous at .

The graph of    is shown below, where is the horizontal distance from the start of the ramp and is the height of the ramp. All lengths are in centimetres.

  1. Find for .   (2 marks)

  2.   i. Find the coordinates of the point of inflection of .   (1 mark)

  3.  ii. Find the interval of for which the gradient function of the ramp is strictly increasing.   (1 mark)

  4. iii. Find the interval of for which the gradient function of the ramp is strictly decreasing.  (1 mark)

Jac and Jill decide to use two trapezoidal supports, each of width . The first support has its left edge placed at and the second support has its left edge placed at . Their cross-sections are shown in the graph below.

  1. Determine the value of the ratio of the area of the trapezoidal cross-sections to the exact area contained between and the -axis between and . Give your answer as a percentage, correct to one decimal place.   (3 marks)

  2. Referring to the gradient of the curve, explain why a trapezium rule approximation would be greater than the actual cross-sectional area for any interval , where .   (1 mark)

  3. Jac and Jill roll the toy car down the ramp and the car jumps off the end of the ramp. The path of the car is modelled by the function , where

  4. is continuous and differentiable at , and is where the car lands on the ground after the jump, such that .
    1. Find the values of and .   (2 marks)

    2. Determine the horizontal distance from the end of the ramp to where the car lands. Give your answer in centimetres, correct to two decimal places.   (1 mark)

Show Answers Only

a.   

b.i   

b.ii 

b.iii

c.   

d.   

e.i   

e.ii 

Show Worked Solution

a.   

b.i   


  

b.ii 

b.iii

c.   

 

  

 
 

    
d.   


 

e.i   

  


 

e.ii 

Calculus, MET1 2022 VCAA 7

A tilemaker wants to make square tiles of size 20 cm × 20 cm.

The front surface of the tiles is to be painted with two different colours that meet the following conditions:

  • Condition 1 - Each colour covers half the front surface of a tile.
  • Condition 2 - The tiles can be lined up in a single horizontal row so that the colours form a continuous pattern.

An example is shown below.
 

There are two types of tiles: Type A and Type B.

For Type A, the colours on the tiles are divided using the rule , where .

The corners of each tile have the coordinates (0,0), (20,0), (20,20) and (0,20), as shown below.
 

  1.  i. Find the area of the front surface of each tile.   (1 mark)

    ii. Find the value of so that a Type A tile meets Condition 1.   (1 mark)


Type B tiles, an example of which is shown below, are divided using the rule .
 

  1. Show that a Type B tile meets Condition 1.   (3 marks)

  2. Determine the endpoints of and on each tile. Hence, use these values to confirm that Type A and Type B tiles can be placed in any order to produce a continuous pattern in order to meet Condition 2.   (2 marks)

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a.i.    cm²

a.ii.  

b.     cm²

c.     See worked solution

Show Worked Solution
a.i   Area  
  cm²  

  
a.ii 
 

b.   Area  
   
   
   
  cm²  

 
The area of the coloured section of the Type B tile is 200 cm² which is half the 400 cm² area of the tile.
 

c.   
 
   
   
   

 
→
 

   The endpoints for are and and for are also and .

So the tiles can be placed in any order to make the continuous pattern.


♦♦ Mean mark (c) 30%.
MARKER’S COMMENT: Students often only found and , however, and also needed to be found to verify the pattern match.