Differential Calculus I (A-Class)

HSC STYLE HOMEWORK
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SEMINAR DAY & TIME: ........................
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Preliminary Mathematics Extension 1
Differential Calculus I (A-Class)
General Instructions
• Reading time – 5 minutes
• Working time – 2 hours
• Write using black or blue pen
• Draw diagrams in pencil
• Board-approved calculators may
be used
• A table of standard integrals may
be used
• All necessary working should be
shown in every question
Total marks – 60
• Attempt Questions 1-7
• All questions are of equal value

PMX Homework Differential Calculus I © TALENT 100
Total marks – 60
Attempt Questions 1–5
All questions are of equal value
Question 1 (12 marks)
Marks
The Second Derivative
is the second derivative of , i.e. we differentiate twice
(a)
For each of the following functions, find
.
(i) = + 3 + 2 + 1 2
(ii) = + 2 4 − 3 3
(iii)
=
3
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PMX Homework Differential Calculus I © TALENT 100
Concavity
(b) Concavity is a measure of the curvature of the slope, i.e whether it is curving upwards or curving
downwards.
When a curve is concave down, < 0
When a curve is concave up, > 0
(b) Determine the concavity of the following curves at the point where = 5.
(i) = + 2 − 5 + 3 − 1 2
(ii) = 3 − 5 2
End of Question 1
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PMX Homework Differential Calculus I © TALENT 100
Marks
Question 2 (12 marks)
Stationary Points
The second derivative will be positive for a min, i.e. > 0 for a max
The second derivative will be negative for a max, i.e. < 0 for a max
If = 0, the result is inconclusive and points will need to be tested
Points of Inflexion
STEP 1: Find the values of for which = 0
STEP 2: Use a table of values to determine to determine the sign of around its zeroes. The
second derivative must change sign for a point of inflexion.
(a) For a particular function, it is known that = − √ − 2.
(i) Suppose a stationary point exists at = 6. Determine the nature of 2
the stationary point.
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PMX Homework Differential Calculus I © TALENT 100
(ii) Find the values of for which has a point of inflexion. 2
(iii) Find the values of for which is concave down. 2
A Horizontal Point of Inflexion is a point around which the sign of does not change, but the
sign of does change
The tangent crosses the curve
At a Horizontal Point of Inflexion, = 0 and = 0
NOTE: = = 0 is not enough to prove a horizontal point of inflexion. Always
remember to test points
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PMX Homework Differential Calculus I © TALENT 100
(b) Consider the function = − − 8 + 32 + 1.
(i) Find the coordinates of any stationary points and determine their nature. 3
(ii) Find the coordinates of any points of inflexion. 2
(iii) Briefly sketch the graph of = . 1
End of Question 2
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PMX Homework Differential Calculus I © TALENT 100
Question 3 (12 marks)
Marks
(a) Consider the function =
√ .
(i) State the domain of . 1
(ii) Show that is concave down for all in its domain. 3
(iii) Hence, find the coordinates of any stationary points of and 2
determine their nature.
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PMX Homework Differential Calculus I © TALENT 100
(iv) Neatly sketch the graph of = , showing all important features. 2
(b) (i) Show that 0 = 0 = 0 given that = . 2
(ii) Hence or otherwise, determine the nature of the stationary point at = 0. 2
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PMX Homework Differential Calculus I © TALENT 100
Question 4 (12 marks)
Marks
(a)
The diagram below shows the graph of = .
−2,3
0,1
−4,0
(i) On separate axis, sketch the graphs of = and = . 3
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PMX Homework Differential Calculus I © TALENT 100
(ii) On separate axis, sketch possible graphs of = and = given 4
that = , and that has a stationary point at 0,0.
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PMX Homework Differential Calculus I © TALENT 100
(b)
A particular function = + + + , has a horizontal point of inflexion
at the point (1,3), where , and are constants.
(i) Find expression for and . 2
(ii) Find the values of , and . 3
End of Question 4
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PMX Homework Differential Calculus I © TALENT 100
Question 5 (12 marks)
Marks
(a) A particular function is defined by =
.
(i) State the domain of . 1
(ii) Find the coordinates of any turning points, and determine their nature. 4
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PMX Homework Differential Calculus I © TALENT 100
(iii) Find the coordinates of any points of inflexion 2
(iv) Explain why − ≤ ≤ . 2
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PMX Homework Differential Calculus I © TALENT 100
(v) Hence, neatly sketch the graph of = , clearly showing all turning 3
points, points of inflexion and intercepts with the coordinate axis.
End of paper
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