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Grade 12 · Calculus
Optimisation & Rates of Change (Grade 12)
MARKING GUIDELINE
Marks
31
Duration
50 minutes
Questions
3
Name:
Class:
Date:
Mark
/ 31
Instructions and Information
- Answer ALL the questions in this question paper.
- Answer QUESTION 1 by circling the letter (A–D) in the answer grid at the end of that section.
- Show ALL calculations clearly.
- Show all units where applicable.
- Number the answers correctly according to the numbering system used in this question paper.
- A non-programmable calculator may be used, unless stated otherwise.
- Write neatly and legibly.
Question 1
[10 MARKS]Four options are given as possible answers to the following questions. Choose the correct answer and circle the letter (A–D) in the grid at the end of this section. If you want to change your choice, put a cross through the wrong letter and circle your new choice.
- 1.1To MAXIMISE or MINIMISE a quantity, you …(1)A)differentiate the expression and set the derivative equal to 0✓B)set the expression itself equal to 0C)substitute the largest allowed valueD)set the second derivative equal to 0Answer: A — A maximum or minimum occurs where the rate of change is zero.
- B — that finds where the quantity is zero, not where it is largest
- C — the optimum need not lie at an endpoint
- D — setting f'' to zero finds the point of INFLECTION
- 1.2A rectangle has a perimeter of 40 m. Its greatest possible AREA is …(1)A)100 m²✓B)400 m²C)40 m²D)200 m²Answer: A — A = l(20 − l), and dA/dl = 20 − 2l = 0 gives l = 10, so A = 100 m².
- B — squared half the perimeter instead of optimising
- C — used the perimeter as though it were the area
- D — halved the perimeter and multiplied by 10 without optimising
- 1.3A closed box has a square base of side x and a volume of 500 cm³. Its height is …(1)A)500x²✓B)500xC)500x²D)x²500Answer: A — V = x²h = 500, so h = 500 ÷ x².
- B — the base is a SQUARE, so its area is x², not x
- C — multiplied instead of dividing
- D — inverted the expression
- 1.4If s(t) = 5t² + 2t metres, the VELOCITY at t = 3 seconds is …(1)A)32 m per second✓B)51 m per secondC)10 m per secondD)30 m per secondAnswer: A — v = s'(t) = 10t + 2, so v(3) = 32.
- B — substituted into s(t) instead of s'(t): 45 + 6 = 51
- C — gave the coefficient without substituting t = 3
- D — forgot to add the 2
- 1.5ACCELERATION is the derivative of …(1)A)velocity with respect to time✓B)distance with respect to timeC)velocity with respect to distanceD)time with respect to velocityAnswer: A — a = dv/dt, which is the second derivative of displacement.
- B — that is VELOCITY
- C — acceleration is measured against TIME
- D — that inverts the relationship
- 1.6The sum of two positive numbers is 30. Their product is largest when the numbers are …(1)A)15 and 15✓B)1 and 29C)10 and 20D)0 and 30Answer: A — P = x(30 − x), and dP/dx = 30 − 2x = 0 gives x = 15, so the product is 225.
- B — 1 × 29 = 29, far below 225
- C — 10 × 20 = 200, still below 225
- D — 0 × 30 = 0, the smallest product possible
- 1.7A cylinder of volume 1 000 cm³ has radius r. Its height is …(1)A)1000πr²✓B)1000πrC)1000r²D)1 000πr²Answer: A — V = πr²h = 1 000, so h = 1 000 ÷ (πr²).
- B — the base area is πr², not πr
- C — left out the π
- D — multiplied instead of dividing
- 1.8At a MAXIMUM turning point, f''(x) is …(1)A)negative✓B)positiveC)zeroD)undefinedAnswer: A — A maximum is concave down, so the second derivative is negative there.
- B — a positive second derivative means a MINIMUM
- C — zero indicates a point of inflection
- D — the second derivative of a polynomial exists everywhere
- 1.9A stone's height is h = 30t − 5t² metres after t seconds. Its greatest height is …(1)A)45 m✓B)90 mC)30 mD)3 mAnswer: A — h'(t) = 30 − 10t = 0 gives t = 3, and h(3) = 90 − 45 = 45 m.
- B — used 30t alone and forgot to subtract the 5t²
- C — gave the coefficient 30 instead of a height
- D — that is the TIME at the maximum, not the height
- 1.10In an optimisation problem, the CONSTRAINT is used to …(1)A)write the quantity in terms of a single variable✓B)find the derivative directlyC)check the answer at the very end onlyD)avoid using calculus at allAnswer: A — An expression in two variables cannot be differentiated at this level, so the constraint removes one of them.
- B — the derivative can only be taken once one variable is left
- C — the constraint is needed BEFORE differentiating, not only afterwards
- D — calculus is still what finds the optimum
Answer grid — marking guideline
| 1.1 | A | B | C | D |
| 1.2 | A | B | C | D |
| 1.3 | A | B | C | D |
| 1.4 | A | B | C | D |
| 1.5 | A | B | C | D |
| 1.6 | A | B | C | D |
| 1.7 | A | B | C | D |
| 1.8 | A | B | C | D |
| 1.9 | A | B | C | D |
| 1.10 | A | B | C | D |
Question 2
[13 MARKS]Given f(x) = x2. Determine the MINIMUM distance between the point (10 ; 2) and a point on f.
- 2.1Write down an expression for the square of the distance d between (10 ; 2) and the point (x ; x2) on f.(3)d2 = (x − 10)2 + (y − 2)2 (1)
y = x2 on f (1)
d2 = (x − 10)2 + (x2 − 2)2 (1) - 2.2Explain why minimising d2 gives the same answer as minimising d.(2)d is never negative (1), and squaring preserves the order of non-negative numbers, so both are smallest at the same x (1)
- 2.3Determine the value of x for which the distance is a minimum.(5)d2 = x4 − 3x2 − 20x + 104 (1)
d(d2)dx = 4x3 − 6x − 20 (2)
Set it equal to 0 and test factors: x = 2 gives 32 − 12 − 20 = 0 (1)
x = 2 (1) - 2.4Hence calculate the minimum distance, in simplest surd form.(3)At x = 2 the point on f is (2 ; 4) (1)
d2 = (−8)2 + 22 = 68 (1)
d = 217 units (1)
Question 3
[8 MARKS]A container is being filled with water. After t seconds the volume, in litres, is given by V(t) = 3t2 + 2t.
- 3.1Determine the AVERAGE rate at which the volume increases over the first 5 seconds.(3)V(0) = 0 and V(5) = 85 (1)
85 − 05 − 0 (1) = 17 litres per second (1) - 3.2Determine the rate at which the volume is increasing AT t = 5 seconds.(3)V′(t) = 6t + 2 (2)
V′(5) = 32 litres per second (1) - 3.3Explain why the two answers differ.(2)The first is the average over the whole interval (1); the second is the instantaneous rate at one moment, and the flow is speeding up (1)
TOTAL: 31 marks
This question paper consists of 3 questions.