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MATHEMATICS 3C TEST Time Allowed 30 minutes

Total Marks 26

Calculator Free Section.

Equipment Allowed pencils, pens ruler

Question 1. [5 and 8 marks](a) Simplify each of the following.

(i)

€

x2 − 1x

÷ x2 + 2x + 1

3x2 − 6x[2]

(ii)

€

5

y2 − 9 +

6y − 3

− 7

y + 3 [3]

1

(b) Find

€

dydx

in each of the following.

(i) y = (x2 1)e1 3x (Do not simplify) [2]

(ii) y =

€

2x − x + 3π3 + 4

x2 (Leave with positive indices.) [2]

(iii) y =

€

2x3

5 − 3x4 ⎛ ⎝ ⎜

⎞ ⎠ ⎟2 (Do not simplify) [2]

(iv) y =

€

u2 − 3 using Leibniz notation

€

dydx

= dydu

× dudx

where u = 2e2x + 3 [2]

2

Question 2 [4 and 4 marks]

Let f(x) = 5 +

€

x and g(x) =

€

1x

(a) State the domain and range of g(f(x)). [4]

(b) State the domain and range of f(g(x)). [4]

3

Question 3. [1, 1, 2 and 1 marks]

The graph of

€

y = x

e x2 is drawn below over the domain 0 x 2.

(a) Approximate the values of x such that y is an increasing function. [1]

(b) Determine the approximate range for y. [1]

(c) When x = 2, is it possible to determine the tangent line? Explain your answer. [2]

(d) Explain, using first derivative principles, why a local maximum point occurs. [1]

4

MATHEMATICS 3C TEST Time Allowed 30 minutes

Total Marks 28

Calculator Section.

Equipment Allowed CAS Calculator, pencils, pens ruler

Question 1. [2 and 2 marks]

Consider the function f (x) = 2x3 3x2 23x + p where p is a constant.

(a) Determine where the local (relative) extrema points occur. [2]

(b) Find the value of p given that the three roots are 3, 0.5 and 4. [2]

5

Question 2. [1, 2, 1, 3, 2, 2 and 2 marks]

On your CAS calculator, sketch the curve f (x) = x4 – 2x3 over the domain –1 x 2.5.

Determine

(a) the roots of f (x). [1]

(b) the stationary points of f (x). [2]

(c) the points of inflection of f (x). [1]

Sketch f (x) on the axes below, highlighting each of the points found in (a), (b) and (c) on the previous page. [3]

6

Use this sketch, or otherwise, to determine the x value(s) where:

(d) f (x) > 0 [2]

(e) f (x) > 0 [2]

(f) f (x) > 0 [2]

7

Question 3. [6 marks]

A small box with a square base and open at the top is to occupy 40mm3 for a diamond display. Ignoring waste and material thickness, find the dimensions of the box if the least amount of material is required in its construction. [6]

Question 4. [3 and 2 marks]

An oil tanker has hit a reef during a cyclone. The impact tore a hole in the hull that resulted in

crude oil escaping into the ocean. The quantity of crude oil S(t), (in megalitres (ML)), remaining in

the ship at time t hours is approximated by the exponential relationship

S(t) = S0ekt

After 5 and 8 hours, 73.55ML and 40.37ML of crude oil respectively remained in the ship.

(a) Determine the values of S0 (nearest unit) and k (to 2 decimal places). [3]

(b) Determine the quantity of oil that has escaped from the tanker after 20 hours. [2]

8

Solutions

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MATHEMATICS 3C TEST Time Allowed 30 minutes

Total Marks 26

Calculator Free Section.

Equipment Allowed pencils, pens ruler

Question 1. [5 and 8 marks](a) Simplify each of the following.

(i)

€

x2 − 1x

÷ x2 + 2x + 1

3x2 − 6x[2]

€

(x − 1)(x + 1)x

× 3x(x − 2)(x + 1)(x − 1)

€

3(x − 2)(x − 1)(x + 1)

(ii)

€

5

y2 − 9 +

6y − 3

− 7

y + 3 [3]

€

5 + 6(y + 3) − 7(y − 3)

(y2 − 9)

€

44 − y

y2 − 9

9

(b) Find

€

dydx

in each of the following.

(i) y = (x2 1)e1 3x (Do not simplify) [2]

€

dydx

= 2x e1 3x + (x2 1). (3). e1 3x

(ii) y =

€

2x − x + 3π3 + 4

x2 (Leave with positive indices.) [2]

€

dydx

= 2 − 1

2 x − 8

x3

(iii) y =

€

2x3

5 − 3x4 ⎛ ⎝ ⎜

⎞ ⎠ ⎟2 (Do not simplify) [2]

€

dydx

= 6x. 5 − 3x4 ⎛

⎝ ⎜ ⎞ ⎠ ⎟2 − 2x3 .2.(5 − 3x4 ).(−12x 3 )

5 − 3x4 ⎛ ⎝ ⎜

⎞ ⎠ ⎟4

(iv) y =

€

u2 − 3 using Leibniz notation

€

dydx

= dydu

× dudx

where u = 2e2x + 3 [2]

€

dydx

= 2u

2 u2 − 3.4e 2x

€

dydx

= (2e 2x + 3)

2e 2x + 3 ⎛ ⎝ ⎜

⎞ ⎠ ⎟2 − 3

.4e 2x

10

Question 2 [4 and 4 marks]

Let f(x) = 5 +

€

x and g(x) =

€

1x

(a) State the domain and range of g(f(x)). [4]

€

g(5 + x ) = 1

5 + x Dx = {x | x ≥ 0) Ry {y | 0 < y < 0.2}

(b) State the domain and range of f(g(x)). [4]

€

f 1x

⎛ ⎝ ⎜

⎞ ⎠ ⎟ = 5 + 1

x Dx = {x | x > 0) Ry {y | y > 6}

11

Question 3. [1, 1, 2 and 1 marks]

The graph of

€

y = x

e x2 is drawn below over the domain 0 x 2.

(a) Approximate the values of x such that y is an increasing function. [1]

0 < x < 0.7

(b) Determine the approximate range for y. [1]

0 ≤ y ≤ 0.03

(c) When x = 2, is it possible to determine the tangent line? Explain your answer. [2]

No. Tangent is not possible at a closed point.

(d) Explain, using first derivative principles, why a local maximum point occurs. [1]

12

f’(x) > 0 to the left of x ≈ 0.7 and f’(x) < 0 to the right of x ≈ 0.7

max TP

MATHEMATICS 3C TEST Time Allowed 30 minutes

Total Marks 28

Calculator Section.

Equipment Allowed CAS Calculator, pencils, pens ruler

Question 1. [2 and 2 marks]

Consider the function f (x) = 2x3 3x2 23x + p where p is a constant.

(a) Determine where the local (relative) extrema points occur. [2]

x ≈ 1.5207 x ≈ 2.5207

(b) Find the value of p given that the three roots are 3, 0.5 and 4. [2]

f(x) = (x + 3)(2x 1)(x 4)

p = 12

13

Question 2. [1, 2, 1, 3, 2, 2 and 2 marks]

On your CAS calculator, sketch the curve f (x) = x4 – 2x3 over the domain –1 x 2.5.

Determine

(a) the roots of f (x). [1]

x = 0, x = 2

(b) the stationary points of f (x). [2]

x = 0, x = 1.5

(c) the points of inflection of f (x). [1]

x = 0

Sketch f (x) on the axes below, highlighting each of the points found in (a), (b) and (c) on the previous page. [3]

Bounds

Shape

Main points

14

Use this sketch, or otherwise, to determine the x value(s) where:

(d) f (x) > 0 [2]

1 < x < 0, 2 < x ≤ 2.5

(e) f (x) > 0 [2]

1.5 < x < 2.5

(f) f (x) > 0 [2]

0 < x < 2.5

15

Question 3. [6 marks]

A small box with a square base and open at the top is to occupy 40cm3 for a diamond display. Ignoring waste and material thickness, find the dimensions of the box if the least amount of material is required in its construction. [6]

V = x2h = 40 SA = x2 + 4xh

SA =

€

x2 + 4x.40

x2

Min at x = 4.3089

Verify minimum from first derivative or second derivative test or a sketch.

Dimensions are 4.3089cm by 4.3089cm by 2.1544cm

Question 4. [3 and 2 marks]

An oil tanker has hit a reef during a cyclone. The impact tore a hole in the hull that resulted in

crude oil escaping into the ocean. The quantity of crude oil (in megalitres (ML)), remaining in the

ship at time t hours is approximated by the exponential relationship

S(t) = S0ekt

After 5 and 8 hours, 73.55ML and 40.37ML of crude oil respectively remained in the ship.

(a) Determine the values of S0 (nearest unit) and k (to 4 decimal places). [3]

Use the simultaneous solver in calculator

€

73.55 = xe−5y

40.37 = xe−8y x, y

⎧ ⎨ ⎪ ⎩ ⎪

Hence S0 = 200 and k = 0.20

(b) Using the equation found in (a), determine the quantity of oil that has escaped from the

tanker after 20 hours. [2]

S(t) = 200 200e0.2020

= 196.3369 ML

16

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