Question marks
Differentiate the following functions, simplifying your answers as far as possible.
(a)
(b)
(c)
(d)
Question marks
Cedric wants to make a rectangular poster for his birthday party. He has a triangular piece of card from which he wants to cut the largest possible rectangle. The triangular piece of card is a right-angled triangle with base and height . The width and height of the rectangle to be cut out are and cm respectively, as shown in the shaded region below.
(a) Show that
Use this equation, and the known values of and , to express in terms of .
(b) Hence, write an expression for the area of the rectangle in terms of the length .
(c) Use differentiation to determine the width that gives the maximum area of the poster, and show that this is a maximum. What is the maximum area of the poster and what is its height?
Question marks
Find the indefinite integral of the function
(Hint: it may be helpful to simplify first.)
Question marks
You should be able to answer this question after studying Unit
Use integration by substitution to find the indefinite integral in part (a) and to evaluate (to 2 d.p.) the definite integral in part (b).
(a) dx
(b)
Question 5- 10 marks
This question is about the function
(a) Explain why the graph of lies on or above the -axis for all values of in the interval .
(b) Write down an expression, involving a definite integral, that gives the area between the graph of and the -axis, from to .
(c) Use integration by parts to find the area described in part (b), giving both the exact answer and an approximation to three decimal places.
Question marks
(a) Use integration by substitution to find the indefinite integral
(b) Use integration by parts twice to find the indefinite integral
Question marks
Mini examination paper
This paper has TWO sections. You should attempt ALL questions in each section.
Section A has 6 questions, each worth 2 marks.
Section B has 2 questions, each worth 4 marks.
Each question in Section A is multiple-choice, with ONE correct answer
from five options. Answer each question by stating the correct answer. No marks will be deducted for incorrectly answered questions.
For both questions in Section B, write your answers in the boxes provided. Do not include any working; only fully correct answers will be awarded the marks for a question. No marks will be deducted for incorrectly answered questions.
6. SECTION A
Question 1
Which of the following is equivalent to
A
Question 2
Which of the following is equivalent to the inequality
C
D
Question 3
How many values of in radians between and satisfy
Question 4
What is the derivative of
Question 5
On which interval is the function decreasing?
Question 6
Which of the following is equal to
A
B
C
D
13. SECTION B
Question 7
A function has derivative . What are the stationary points of , and their natures?
The -coordinate of one stationary point is at =______.
It is a ____ (options: local maximum, local minimum or horizontal point of inflection).
The -coordinate of the other stationary point is at
It is a_____ (options: local maximum, local minimum or horizontal point of inflection).
Question 8
The velocity (in metres per second) of an object in terms of the time (in seconds) is given by
The displacement of the object at time is . What is the displacement of the object (in metres) at time
The displacement is_____ metres (to 2 d.p.) at .
[END OF QUESTION PAPER]
