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Grade 12 · Calculus
Optimisation & Rates of Change (Grade 12)
EXAM-STYLE CLASS TEST
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)set the expression itself equal to 0B)differentiate the expression and set the derivative equal to 0C)substitute the largest allowed valueD)set the second derivative equal to 0
- 1.2A rectangle has a perimeter of 40 m. Its greatest possible AREA is …(1)A)400 m²B)100 m²C)40 m²D)200 m²
- 1.3A closed box has a square base of side x and a volume of 500 cm³. Its height is …(1)A)500/xB)500x²C)x²/500D)500/x²
- 1.4If s(t) = 5t² + 2t metres, the VELOCITY at t = 3 seconds is …(1)A)30 m per secondB)51 m per secondC)10 m per secondD)32 m per second
- 1.5ACCELERATION is the derivative of …(1)A)velocity with respect to timeB)velocity with respect to distanceC)time with respect to velocityD)distance with respect to time
- 1.6The sum of two positive numbers is 30. Their product is largest when the numbers are …(1)A)15 and 15B)1 and 29C)0 and 30D)10 and 20
- 1.7A cylinder of volume 1 000 cm³ has radius r. Its height is …(1)A)(1 000)/πrB)1 000πr²C)(1 000)/πr²D)(1 000)/r²
- 1.8At a MAXIMUM turning point, f″(x) is …(1)A)undefinedB)positiveC)negativeD)zero
- 1.9A stone's height is h = 30t − 5t² metres after t seconds. Its greatest height is …(1)A)30 mB)3 mC)90 mD)45 m
- 1.10In an optimisation problem, the CONSTRAINT is used to …(1)A)write the quantity in terms of a single variableB)avoid using calculus at allC)find the derivative directlyD)check the answer at the very end only
Answer grid — circle your answers for Question 1
| 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)
- 2.2Explain why minimising d2 gives the same answer as minimising d.(2)
- 2.3Determine the value of x for which the distance is a minimum.(5)
- 2.4Hence calculate the minimum distance, in simplest surd form.(3)
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)
- 3.2Determine the rate at which the volume is increasing AT t = 5 seconds.(3)
- 3.3Explain why the two answers differ.(2)
TOTAL: 31 marks
This question paper consists of 3 questions.