Questions
WORKSHEET - Mathematics (Advanced)
1. Geometry and Calculus, 2UA 2013 HSC 12a
10. Geometrical Applications of Differentia Differentiation tion Curve Sketching and T he Primitive Function Teacher: Abi Teacher: Abi Parsons Exam Equivalent Time: 58.5 Time: 58.5 minutes (based on HSC allocation of 1.5
The cubic Show that
has a point of inflexion at .
.
(2 marks)
minutes approx. per mark)
2. Geometry and Calculus, 2UA 2005 HSC 4b A function
is defined by
.
(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) (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)
HISTORICAL CONTRIBUTION
3. Geometry and Calculus, 2UA 2014 HSC 11f
T10 Geometry and Calculus is is the single largest topic within the Mathematics course, contributing 15.8% contributing 15.8% to each paper, on average, since 2003.
The gradient function of a curve passes through the point .
This topic has been split into three sub-categories: 1-Maxima and Mini ma (5.7%), 2-Curve Sketching and The P rimitive Function (8.0%), and 3-Tangents and Normals (2.1%).
Find the equation of the curve.
is given by
. The curve
(2 marks)
PAST HSC ANALYSIS - What to expect and common pitfalls Curve Sketching and The Primitive Function (8.0%). The (8.0%). The most popular curve is easily the cubic , asked 7 times since 2003, including 2015. Note that cubic sketches have been examined in consecutive years in 2004/5/6 and 2010/11. Sketches of polynomial s of degree 4 are the next most popular, asked in 2007/8 and 2012 . An examiner favourite in this topi c 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 2015, in questions w orth 1-5 marks. A definite focus area. Many students do not have a clear understanding of concavity Markers' Comments of note: Many and the second derivative. When drawing a graph within a given domain, clearly identify the extremes!
4. Geometry and Calculus, 2UA 2014 HSC 14a Find the coordinates of the stationary point on the graph nature. (3 marks)
, and determine its
5. Geometry and Calculus, 2UA 2004 HSC 4b Consider the function
7. Geometry and Calculus, 2UA 2012 HSC 14a
.
A function is given by
(i) Find the coordinates of the stationary points of the curve their nature. (3 marks) (ii) Sketch the curve showing w here it meets the axes. (2 marks)
and determine
(iii) Find the values of
(2 marks)
for which the curve
is concave up.
(i) Find the coordinates of the stationary points of
.
The graph has a horizontal point of inflexion at turning point at .
, a point of inflexion at
and determine their nature.
(ii) Hence, sketch the graph
showing the stationary points.
(iii) For what values of
is the function increasing?
(iv) For what values of
will
(2 marks)
(1 mark)
have no solution?
(1 mark)
and a maximum
8. Geometry and Calculus, 2UA 2010 HSC 8d Let Find the values of
, where for which
is a constant.
is an increasing function.
(2 marks)
9. Geometry and Calculus, 2UA 2014 HSC 15c The line
is a tangent to the curve
(i) Sketch the line a nd the curve on one diagram. (ii) Find the coordinates of (iii) Find the value of Sketch the graph of the derivative
.
(3
marks)
6. Geometry and Calculus, 2UA 2014 HSC 14e The diagram shows the graph of a function
.
.
.
at a point
.
(1 mark)
(3 marks)
(1 mark)
(3 marks)
Copyright © 2004-16 The State of New South Wales (Board of Studies, Teaching and Educational Standards NSW)
Worked Solutions 1. Geometry and Calculus, 2UA 2013 HSC 12a
(iii)
2. Geometry and Calculus, 2UA 2005 HSC 4b (i)
(ii)
(iv)
3. Geometry and Calculus, 2UA 2014 HSC 11f
4. Geometry and Calculus, 2UA 2014 HSC 14a
(ii)
5. Geometry and Calculus, 2UA 2004 HSC 4b (i) (iii)
6. Geometry and Calculus, 2UA 2014 HSC 14e
(ii)
7. Geometry and Calculus, 2UA 2012 HSC 14a (i)
(iii)
(iv)
Mean mark 42% MARKER'S COMMENT: Be careful to use the correct inequality signs, and not carelessly include or by mistake. ♦
Mean mark 12%. This part was the second most poorly answered question in the 2012 exam.
9. Geometry and Calculus, 2UA 2014 HSC 15c
Mean mark 28%. MARKER'S C OMMENT OMMENT:: The arithmetic required to solve proved the undoing of too many students students in this question. TAKE CARE!
(ii)
♦♦♦
(i)
8. Geometry and Calculus, 2UA 2010 HSC 8d
♦♦
Mean mark 40% COMMENT: Given ♦
, it follows . Make sure you understand the arithmetic behind this (NB. Simply take the of both sides).
(iii)
♦♦
Mean mark 30%.
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