∑ DANEMATHICS
FREE CAPS MATHS RESOURCES
Grade 11 · Functions and Graphs
Average Gradient on a Curve (Grade 11)
EXAM-STYLE CLASS TEST
Marks
33
Duration
50 minutes
Questions
3
Name:
Class:
Date:
Mark
/ 33
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.
- 1.1The average gradient of f between x = a and x = b is …(1)A)(f(b) − f(a))/(b − a)B)(f(b) + f(a))/(b − a)C)(b − a)/(f(b) − f(a))D)f(b) − f(a)
- 1.2For f(x) = x2, the average gradient between x = 1 and x = 4 is …(1)A)15B)3C)8D)5
- 1.3For f(x) = x2 − 2x, the average gradient between x = 0 and x = 2 is …(1)A)2B)−2C)0D)4
- 1.4For g(x) = 6/x, the average gradient between x = 1 and x = 3 is …(1)A)2B)−2C)−4D)4
- 1.5A NEGATIVE average gradient over an interval means that the curve …(1)A)ends lower than it startedB)falls all the wayC)is a straight lineD)has no turning point in between
- 1.6As the two points slide together, the chord becomes the …(1)A)asymptoteB)axis of symmetryC)diameterD)tangent at that point
- 1.7For f(x) = x2, the average gradients from x = 3 to x = 3,1 ; 3,01 ; 3,001 get closer and closer to …(1)A)9B)3C)6D)0
- 1.8An object's distance is s(t) = 2t2 metres after t seconds. Its average speed from t = 1 to t = 3 is …(1)A)16 m/sB)8 m/sC)18 m/sD)4 m/s
- 1.9For f(x) = 2x, the average gradient between x = 0 and x = 3 is …(1)A)7/3B)8/3C)7D)2
- 1.10For f(x) = x2, which statement is TRUE?(1)A)The average gradient is the same over every intervalB)The average gradient is always 2C)The average gradient from x = −1 to x = 1 is 2D)The average gradient from x = 1 to x = 2 is greater than from x = 0 to x = 1
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
[12 MARKS]Given f(x) = −x2 + 4x.
- 2.1Calculate f(1) and f(3).(2)
- 2.2Determine the average gradient of f between x = 1 and x = 3, and explain what your answer tells you about the curve.(3)
- 2.3Determine the average gradient of f between x = 0 and x = 2.(3)
- 2.4Determine the average gradient of f between x = 1 and x = 1 + h, where h ≠ 0. Simplify your answer.(4)
Question 3
[11 MARKS]A stone is thrown upwards. Its height after t seconds is h(t) = 20t − 5t2 metres.
- 3.1Calculate the average rate of change of the height between t = 0 and t = 1, and give its units.(3)
- 3.2Calculate the average rate of change between t = 2 and t = 4, and explain the sign of your answer.(4)
- 3.3Use the average rates of change from t = 2 to t = 2,1 and from t = 2 to t = 2,01 to estimate the rate of change at t = 2, and explain your estimate.(4)
TOTAL: 33 marks
This question paper consists of 3 questions.