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) [2]
(ii) [3]
(b) Find in each of the following.
(i) y = (x2 - 1)e1 - 3x (Do not simplify) [2]
(ii) y = (Leave with positive indices.) [2]
(iii) y = (Do not simplify) [2]
(iv) y = using Leibniz notation where u = 2e2x + 3 [2]
Question 2 [4 and 4 marks]
Let f(x) = 5 + and g(x) =
(a) State the domain and range of g(f(x)). [4]
(b) State the domain and range of f(g(x)). [4]
Question 3. [1, 1, 2 and 1 marks]
The graph of 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]
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]
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]
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]
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) = S0e-kt
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]
Solutions
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) [2]
Ö
Ö
(ii) [3]
Ö
Ö Ö
(b) Find in each of the following.
(i) y = (x2 - 1)e1 - 3x (Do not simplify) [2]
= 2x e1 - 3x + (x2 - 1). (-3). e1 - 3x Ö Ö
(ii) y = (Leave with positive indices.) [2]
Ö Ö
(iii) y = (Do not simplify) [2]
Ö Ö
(iv) y = using Leibniz notation where u = 2e2x + 3 [2]
Ö
Ö
Question 2 [4 and 4 marks]
Let f(x) = 5 + and g(x) =
(a) State the domain and range of g(f(x)). [4]
Ö Dx = {x | x ≥ 0) Ö Ry {y | 0 < y < 0.2} Ö Ö
(b) State the domain and range of f(g(x)). [4]
Ö Dx = {x | x > 0) Ö Ry {y | y > 6} Ö Ö
Question 3. [1, 1, 2 and 1 marks]
The graph of 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 Ö
(b) Determine the approximate range for y. [1]
0 ≤ y≤ 0.03 Ö
(c) When x = 2, is it possible to determine the tangent line? Explain your answer. [2]
No. Ö Tangent is not possible at a closed point. Ö
(d) Explain, using first derivative principles, why a local maximum point occurs. [1]
f’(x) > 0 to the left of x ≈ 0.7 and f’(x) < 0 to the right of x ≈ 0.7
Þ max TP Ö
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 Ö x ≈ 2.5207 Ö
(b) Find the value of p given that the three roots are -3, 0.5 and 4. [2]
Þ f(x) = (x + 3)(2x - 1)(x - 4) Ö
\ p = 12 Ö
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 Ö
(b) the stationary points of f (x). [2]
x = 0, Ö x = 1.5 Ö
(c) the points of inflection of f (x). [1]
x = 0 Ö
Sketch f (x) on the axes below, highlighting each of the points found in (a), (b) and (c) on the previous page. [3]
Bounds Ö
Shape Ö
Main points Ö
Use this sketch, or otherwise, to determine the x value(s) where:
(d) f (x) > 0 [2]
-1 < x < 0, Ö 2 < x ≤ 2.5 Ö
(e) f¢ (x) > 0 [2]
1.5 < x < 2.5 Ö Ö
(f) f¢¢ (x) > 0 [2]
0 < x < 2.5 Ö Ö
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 Ö Ö
SA = Ö
Min at x = 4.3089 Ö
Verify minimum from first derivative or second derivative test or a sketch. Ö
Dimensions are 4.3089cm by 4.3089cm by 2.1544cm Ö
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) = S0e-kt
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
Ö
Hence S0 = 200 and k = 0.20 Ö Ö
(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 - 200e-0.20´20 Ö
= 196.3369 ML Ö
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