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Calculus, 2ADV C3 2020 HSC 29

The diagram shows the graph of  .
 

  1. Show that the equation of the tangent to    at  , where  , is
     
    .  (2 marks)

  2. Find the value of such that the tangent from part (a) has a gradient of 1 and passes through the origin.  (2 marks)

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a.   

 

 

b.   

♦ Mean mark part (b) 40%.

 

 

Calculus, 2ADV C3 SM-Bank 7

The graph of    for    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.
 


 

  1. Find the equation of the line through and in the form  , for  .  (3 marks)

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i.  
   
   
   
   
   

 

ii.  
 
   
   

 

iii. 

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