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Questions WORKSHEET - Mathematics (Advanced) 1. Geometry and Calculus, 2UA 2017 HSC 4 MC 10. Geometrical Applications of Differentiation Curve Sketching and The Primitive Function The function is defined for Teacher: James Butterworth On this interval, Exam Equivalent Time: 94.5 minutes (based on HSC allocation of 1.5 Which graph best represents ? minutes approx. per mark) (A) y (B) y

x a b x a b x

HISTORICAL CONTRIBUTION x x T10 Geometry and Calculus is the single largest topic within the Mathematics a b a b course, contributing 15.0% to each paper, on average, in the last 10 years. x This topic has been split into three sub-categories: 1-Maxima and Minima (5.3%), 2-Curve Sketching and The Primitive Function (7.6%), and 3-Tangents and Normals (2.1%).

2017 HSC ANALYSIS - What to expect and common pitfalls Curve Sketching and The Primitive Function (7.6%). The most popular curve is easily the cubic, asked 7 times since 2003, including 2015. Note that we have to go back 10 years before cubics have been left off the HSC two years in a row in this topic area. Cubics were left off in the 2016 exam... enough said. Sketches of polynomials of degree 4 are the second most popular, asked in 2007, 2008, 2012 and 2016. Focus Area: An examiner favourite in this topic area requires students to switch between f(x) and f´(x) for a given function, either graphically or by using an equation. This has been tested every year in recent times, except not in 2015 or 2016, in questions worth 1-5 marks. Markers' Comments of note: Many students do not have a clear understanding of concavity and the second derivative. When drawing a graph within a given domain, clearly identify the extremes! 2. Geometry and Calculus, 2UA 2012 HSC 4 MC 3. Geometry and Calculus, 2UA 2013 HSC 8 MC

The diagram shows the graph . The diagram shows points , , and on the graph .

Which of the following statements is true?

(A) (B) (C) At which point is and ? (D) (A) (B) (C) (D) 4. Geometry and Calculus, 2UA 2017 HSC 9 MC 7. Geometry and Calculus, 2UA 2005 HSC 4b

The graph of is shown. A function is defined by .

y (i) Find all solutions of (2 marks) (ii) Find the coordinates of the turning points of the graph of , and determine their nature. (3 marks) 4 (iii) Hence sketch the graph of , showing the turning points and the points where the curve meets the -axis. (2 marks) (iv) For what values of is the graph of concave down? (1 mark) y = f ′(x)

8. Geometry and Calculus, 2UA 2006 HSC 5a

A function is defined by . O 2 x (i) Find the coordinates of the turning points of and determine their nature. ( 3 marks) (ii) Find the coordinates of the point of inflexion. (1 mark) The curve has a maximum value of 12. (iii) Hence sketch the graph of , showing the turning points, the point of inflexion What is the equation of the curve ? and the points where the curve meets the -axis. (3 marks) (A) (iv) What is the minimum value of for ? (1 mark)

(B) 9. Geometry and Calculus, 2UA 2007 HSC 6b (C) Let . (D) (i) Find the coordinates of the points where the curve crosses the axes. (2 marks) 5. Geometry and Calculus, 2UA 2013 HSC 12a (ii) Find the coordinates of the stationary points and determine their nature. (4 marks) (iii) Find the coordinates of the points of inflexion. (1 mark) The cubic has a point of inflexion at . (iv) Sketch the graph of , indicating clearly the intercepts, stationary points and Show that . (2 marks) points of inflexion. (3 marks)

6. Geometry and Calculus, 2UA 2011 HSC 7a 10. Geometry and Calculus, 2UA 2015 HSC 13c

Let . Consider the curve .

(i) Find the coordinates of the stationary points of , and determine their nature. (i) Find the stationary points and determine their nature. (4 marks) (3 marks) (ii) (ii) Hence, sketch the graph showing all stationary points and the -intercept. Given that the point lies on the curve, prove that there is a point of (2 marks) inflexion at . (2 marks) (iii) Sketch the curve, labelling the stationary points, point of inflexion and -intercept. (2 marks) 11. Geometry and Calculus, 2UA 2016 HSC 13a Worked Solutions

Consider the function 1. Geometry and Calculus, 2UA 2017 HSC 4 MC i. Find the two stationary points and determine their nature. (4 marks) ii. Sketch the graph of the function, clearly showing the stationary points and the and intercepts. (2 marks)

12. Geometry and Calculus, 2UA 2014 HSC 11f

The gradient function of a curve is given by . The curve passes through the point . 2. Geometry and Calculus, 2UA 2012 HSC 4 MC Find the equation of the curve. (2 marks)

13. Geometry and Calculus, 2UA 2014 HSC 14a

Find the coordinates of the stationary point on the graph , and determine its nature. (3 marks)

14. Geometry and Calculus, 2UA 2017 HSC 13b 3. Geometry and Calculus, 2UA 2013 HSC 8 MC Consider the curve . ♦ Mean mark 48% (i) Find the stationary points of the curve and determine their nature. (4 marks)

(ii) Sketch the curve, labelling the stationary points. (2 marks) (iii) Hence, or otherwise, find the values of for which

is positive. (1 mark)

(i)

5. Geometry and Calculus, 2UA 2013 HSC 12a

MARKER'S COMMENT: Graphs should be (ii) large (around ½ page), axes drawn with a ruler, with intercepts and turning points clearly shown. The scale can be different on each axe for clarity, as shown in the Worked Solution.

(iii)

7. Geometry and Calculus, 2UA 2005 HSC 4b

(i)

(iv) (ii)

8. Geometry and Calculus, 2UA 2006 HSC 5a

(i)

(iv) (ii)

9. Geometry and Calculus, 2UA 2007 HSC 6b

(i)

(iii)

(ii)

(iv)

10. Geometry and Calculus, 2UA 2015 HSC 13c

(i) (iii)

(ii)

(iii) 11. Geometry and Calculus, 2UA 2016 HSC 13a 12. Geometry and Calculus, 2UA 2014 HSC 11f i.

13. Geometry and Calculus, 2UA 2014 HSC 14a

ii.

14. Geometry and Calculus, 2UA 2017 HSC 13b

(i)

x -2 1

(ii) y

(-2, 27) 27

x -2 -1 (1, 0)

(iii) Questions WORKSHEET - Mathematics (Advanced) 1. Geometry and Calculus, 2UA 2004 HSC 4b 10. Geometrical Applications of Differentiation Curve Sketching and The Primitive Function Consider the function . Teacher: James Butterworth (i) Find the coordinates of the stationary points of the curve and determine Exam Equivalent Time: 100.5 minutes (based on HSC allocation of 1.5 their nature. (3 marks) minutes approx. per mark) (ii) Sketch the curve showing where it meets the axes. (2 marks)

(iii) Find the values of for which the curve is concave up. (2 marks)

2. Geometry and Calculus, 2UA 2010 HSC 6a

Let .

(i) Show that the graph has no stationary points. (2 marks) (ii) Find the values of for which the graph is concave down, and the values for which it is concave up. (2 marks) (iii) Sketch the graph , indicating the values of the and intercepts. (2 marks)

3. Geometry and Calculus, 2UA 2014 HSC 15c

The line is a tangent to the curve at a point .

(i) Sketch the line and the curve on one diagram. (1 mark) HISTORICAL CONTRIBUTION (ii) Find the coordinates of . (3 marks) T10 Geometry and Calculus is the single largest topic within the Mathematics (iii) Find the value of . (1 mark) course, contributing 15.0% to each paper, on average, in the last 10 years. This topic has been split into three sub-categories: 1-Maxima and Minima (5.3%), 2-Curve Sketching and The Primitive Function (7.6%), and 3-Tangents and Normals (2.1%). 4. Geometry and Calculus, 2UA 2008 HSC 8a

Let . 2017 HSC ANALYSIS - What to expect and common pitfalls Curve Sketching and The Primitive Function (7.6%). The most popular curve is easily the (i) Find the coordinates of the points where the graph of crosses the axes. (2 cubic, asked 7 times since 2003, including 2015. Note that we have to go back 10 years marks) (ii) Show that is an even function. (1 mark) before cubics have been left off the HSC two years in a row in this topic area. Cubics were left off in the 2016 exam... enough said. (iii) Find the coordinates of the stationary points of and determine their nature. (4 marks) Sketches of polynomials of degree 4 are the second most popular, asked in 2007, 2008, (iv) Sketch the graph of . (1 mark) 2012 and 2016. Focus Area: An examiner favourite in this topic area requires students to switch between f(x) and f´(x) for a given function, either graphically or by using an equation. This has been 5. Geometry and Calculus, 2UA 2010 HSC 8d tested every year in recent times, except not in 2015 or 2016, in questions worth 1-5 marks. Let , where is a constant. Markers' Comments of note: Many students do not have a clear understanding of concavity and the second derivative. When drawing a graph within a given domain, clearly identify the Find the values of for which is an increasing function. (2 marks) extremes! 6. Geometry and Calculus, 2UA 2011 HSC 9c 8. Geometry and Calculus, 2UA 2004 HSC 9c

The graph in the diagram has a stationary point when , a point of inflexion Consider the function , for . when , and a horizontal asymptote .

(i) Show that the graph of has a stationary point at . (2 marks) (ii) By considering the gradient on either side of , or otherwise, show that the stationary point is a maximum. (1 mark) (iii) Use the fact that the maximum value of occurs at to deduce that for all . (2 marks)

9. Geometry and Calculus, 2UA 2009 HSC 10

Show that the graph of has no turning points. (2 marks) (a) Sketch the graph , clearly indicating its features at and at , and (b) Find the point of inflexion of . (1 mark) the shape of the graph as . (3 marks) (c) (i) Show that for . (1 mark)

7. Geometry and Calculus, 2UA 2014 HSC 14e (ii) Let .

The diagram shows the graph of a function . Use the result in part (c)(i) to show that for all . (2 marks) The graph has a horizontal point of inflexion at , a point of inflexion at and a maximum (d) On the same set of axes, sketch the graphs of and for . turning point at . (2 marks)

(e) Show that . (2 marks)

(f) Find the area enclosed by the graphs of and , and the straight line . (2 marks)

10. Geometry and Calculus, 2UA 2012 HSC 14a

A function is given by .

(i) Find the coordinates of the stationary points of and determine their nature. (3 marks) (ii) Hence, sketch the graph showing the stationary points. (2 marks)

(iii) For what values of is the function increasing? (1 mark)

(iv) For what values of will have no solution? Sketch the graph of the derivative . (3 marks) (1 mark) 11. Geometry and Calculus, 2UA 2010 HSC 9b 12. Geometry and Calculus, 2UA 2009 HSC 8a

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, , negative? (1 mark)

(ii) What happens to for large values of ? (1 mark)

(iii) Sketch the graph . (2 marks) The shaded region has area square units. The shaded region has area square units. Copyright © 2004-16 The State of New South Wales (Board of Studies, Teaching and Educational Standards NSW) (i) For which values of is increasing? (1 mark)

(ii) What is the maximum value of ? (1 mark)

(iii) Find the value of . (1 mark)

(iv) Draw a graph of for . (2 marks) Worked Solutions

1. Geometry and Calculus, 2UA 2004 HSC 4b

(i)

(iii)

2. Geometry and Calculus, 2UA 2010 HSC 6a

(i)

(ii)

3. Geometry and Calculus, 2UA 2014 HSC 15c

(i) (ii)

MARKER'S COMMENT: The significance of the sign of the second derivative was not well understood by most students.

(ii)

♦ Mean mark 40% COMMENT: Given , it follows

. Make sure you understand (iii) the arithmetic behind this (NB. ♦♦ Mean mark 33%. Simply take the of both MARKER'S sides). COMMENT: Students are reminded to bring a ruler to the exam and use it to draw the axes for graphing and to help with an appropriate scale.

(iii)

♦♦ Mean mark 30%.

(iii)

4. Geometry and Calculus, 2UA 2008 HSC 8a

(i)

(iv)

(ii)

6. Geometry and Calculus, 2UA 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!

5. Geometry and Calculus, 2UA 2010 HSC 8d 7. Geometry and Calculus, 2UA 2014 HSC 14e

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

8. Geometry and Calculus, 2UA 2004 HSC 9c

(i)

(iii)

MARKER'S COMMENT: Very challenging for most students. Successful students recognised the link to part (ii).

(ii)

9. Geometry and Calculus, 2UA 2009 HSC 10

(a)

♦♦ Mean mark 28% for all of Q10 (note that data for each question part is not available).

(b)

MARKER'S COMMENT: When 2 graphs are drawn on the (d) same set of axes, you must label them.

(e)

(c)(i)

(f)

(c)(ii)

(ii)

10. Geometry and Calculus, 2UA 2012 HSC 14a

(i)

♦ Mean mark 42% MARKER'S COMMENT: Be (iii) careful to use the correct inequality signs, and not carelessly include or by mistake.

(iv)

♦♦♦ Mean mark 12%.

11. Geometry and Calculus, 2UA 2010 HSC 9b (iv) ♦♦ Mean mark 28% EXAM TIP: Clearly identify THE EXTREMES when given a (i) defined domain. In this case, the origin is obvious graphically, and the other extreme at , is CLEARLY LABELLED!

(ii)

12. Geometry and Calculus, 2UA 2009 HSC 8a

(i)

♦♦ Although exact data not

♦♦♦ Parts (ii) and (iii) proved (ii) available, part (ii) was poorly particularly difficult for students answered. with mean marks of 12% and 11% respectively.

(iii) (iii) ♦♦ Although exact data not available, part (iii) was also poorly answered. MARKER'S COMMENT: Most graphs were poorly drawn with axes not labelled, and scales inaccurate.