12 Derivative Function and Graph PDF

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Questions ADV: Calculus (Adv), C3 Applications of Calculus (Adv)

The Derivative Function and its Graph (Y12) Teacher: Michael Boulus Exam Equivalent Time: 63 minutes (based on HSC allocation of 1.5 minutes approx. per mark)

1.Calculus, 2ADV C3 2017 HSC 4 MC The function 

is defined for 

On this interval,  Which graph best represents   (A) y

? (B) y

a

b

x

(C) y

a

b

x

a

b

x

(D) y

HISTORICAL CONTRIBUTION C3 Applications of Calculus is the biggest topic in the Advanced course, contributing a massive 18% to past Mathematics exams, on average over the past decade. This topic has been split into five sub-topics for analysis purposes: 1-The Derivative Function and its Graph (3.2%), 2-Curve Sketching (5.5%), 3-Tangents (1.4%), 4-Maxima and Minima (6.1%) and 5-Rates of Change (1.8%). This analysis looks at The Derivative Function and its Graph. HSC ANALYSIS - What toexpect and commonpitfalls The Derivative Function and its Graph is a challenging sub-topic that demands a solid conceptual understanding of the first and second derivative. Students are regularly asked to explore the graphical relationship between f(x) and f ´(x). This area has caused problems in the past and deserves attention. This sub-topic has been tested in seven exams over the past decade (notably absent in 2019) and has produced sub-50% mean marks on a majority of occasions. A revision focus is recommended. 

a

b

x

2.Calculus, 2ADV C3 2012 HSC 4 MC The diagram shows the graph  

.

3.Calculus, 2ADV C3 2013 HSC 8 MC The diagram shows points  ,  ,  

and 

on the graph 

 Which of the following statements is true? (A)  (B) 

 

(C) 



(D) 



 At which point is (A) 



(B) 



(C) 



(D) 



 and

?

.

4.Calculus, 2ADV C3 2017 HSC 9 MC The graph of  

5.Calculus, 2ADV C3 2018 HSC 9 MC

is shown.

The diagram shows the graph of 

,the derivative of a function.

y

4 y = f ′( x )

O

2

x

For what value of

 The curve 

has a maximum value of 12.

What is the equation of the curve 

?

does the graph of the function

 have a point of inflexion?

(A) (B) (C)

(A)

(D)

(B) (C) (D)

6.Calculus, 2ADV C3 2020 10 MC The graph shows two functions

Define

0

B.

1

C.

2

D.

3

.

.

How many stationary points does A.

 and

 have for

?

7.Calculus, 2ADV C3 2013 HSC 12a The cubic  Show that 

13.Calculus, 2ADV C3 2011 HSC 9c

has a point of inflexion at 

The graph  in the diagram has a stationary point when   , and a horizontal asymptote  . 

.

.  (2 marks)

, a point of inflexion when

8.Calculus, 2ADV C3 2011 HSC 4c The gradient of a curve is given by 

. The curve passes through thepoint 

.

What is the equation of the curve?  (2 marks)

9.Calculus, 2ADV C3 2014 HSC 11f The gradient function of a curve  through the point  .

is given by 

. The curve passes

Find the equation of the curve. (2 marks)

  Sketch the graph  shape of the graph as 

, clearly indicating its features at  .  (3 marks)

and at 

, and the

10.Calculus, 2ADV C3 2014 HSC 14a Find the coordinates of the stationary point on the graph 

, anddetermine its nature. 

(3 marks)

14.Calculus, 2ADV C3 2014 HSC 14e The diagram shows the graph of a function 

11.Calculus, 2ADV C4 2008 HSC 5a The gradient of a curve is given by 

.

The graph has a horizontal point of inflexion at  , a point of inflexion at  turning point at  .  . The curve passes throughthe point 

. What is the equation of the curve?  (3 marks)

12.Calculus, 2ADV C3 2010 HSC 8d Let  Find the values of  for which 

, where  is a constant. is an increasing function.  (2 marks)

  Sketch the graph of the derivative 

.  (3 marks)

anda maximum

15.Calculus, 2ADV C3 2018 HSC 14c Let

, where

Find the values of

for which

17.Calculus, 2ADV C3 2009 HSC 8a is a constant.

 has NO stationary points. (3 marks)

16.Calculus, 2ADV C3 2010 HSC 9b Let 

be a function defined for 

, with 

The diagram shows the graph of the derivative of  , 

. .

The diagram shows the graph of a function  i. For which values of  is the derivative,  ii. What happens to  iii. Sketch the graph 

. , negative? (1 mark)

for large values of  ?  (1 mark) . (2 marks)

18.Calculus, 2ADV C4 2008 HSC 9c A beam is supported at   The shaded region 

has area  square units. The shaded region 

i. For which values of  is 

iv. Draw a graph of 

as shown in the diagram.

has area  square units.

increasing?(1 mark)

ii. What is the maximum value of  iii. Find the value of 

and 

? (1 mark)

.  (1 mark) for 

.  (2 marks)

 It is known that the shape formed by the beam has equation   and   

, where 

satisfies

( is a positive constant) .

 i. Show that 

.  (2 marks)

ii. How far is the beam below the  -axis at 

? (2 marks)

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4.Calculus, 2ADV C3 2017 HSC 9 MC

Worked Solutions 1.Calculus, 2ADV C3 2017 HSC 4 MC

 

2.Calculus, 2ADV C3 2012 HSC 4 MC



3.Calculus, 2ADV C3 2013 HSC 8 MC ♦ Mean mark 48%

5.Calculus, 2ADV C3 2018 HSC 9 MC

♦ Mean mark 41%.

6.Calculus, 2ADV C3 2020 10 MC

8.Calculus, 2ADV C3 2011 HSC 4c 



  



7.Calculus, 2ADV C3 2013 HSC 12a

9.Calculus, 2ADV C3 2014 HSC 11f

  



10.Calculus, 2ADV C3 2014 HSC 14a

12.Calculus, 2ADV C3 2010 HSC 8d







♦♦ Mean mark 28%. MARKER'S COMMENT: The arithmetic required to solve  proved the undoing of too many students in this question. TAKE CARE!

 

 

11.Calculus, 2ADV C4 2008 HSC 5a 13.Calculus, 2ADV C3 2011 HSC 9c 

♦ Mean mark 43% IMPORTANT: Examiners regularly ask questions that require the graphing of an given the graph and vice-versa. KNOW



IT!





14.Calculus, 2ADV C3 2014 HSC 14e

15.Calculus, 2ADV C3 2018 HSC 14c

♦ Mean mark 49%.





EXTREMESwhen given a defined domain. In this case, the origin is obvious graphically, and the other extreme at , is CLEARLY LABELLED!

16.Calculus, 2ADV C3 2010 HSC 9b i.     ii.   

17.Calculus, 2ADV C3 2009 HSC 8a i.   





♦♦♦ Parts (ii) and (iii) proved particularly difficult for students with mean marks of 12% and 11% respectively.



♦♦ Exact data not available.

ii. 



 iii.

♦♦Exact data not available. MARKER'S COMMENT: Poorly drawn graphs with axes not labelled and inaccurate scales were common.

iii. 







 



  iv.

♦♦ Mean mark 28% EXAM TIP:Clearly identify THE

18.Calculus, 2ADV C4 2008 HSC 9c i. 



 



 ii.











 



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