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(1) Calculate the rst derivative f′(x)for each of the following functions. (10 marks, 2 each) (a) f(x)=(5x−4)3, (b) f(x)=√3x−2, (c) f(x)=(7−5x)4/3, (d) f(x)= sin(x) xcos ...

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Answered

Hand your completed assignment in to the correct box in the Student Resource Centre (G40 Building
301) before the due date. Please use a Mathematics Department cover sheet available from the Student
Resource Centre. Don't forget to write down your name (given name as on your ID card), UPI, ID
number. Show all working; answers that do not show their work will typically receive reduced or
zero marks. Late assignments cannot be marked under any circumstances.
This assignment was written by Igor' Kontorovich ([email protected]) and Jesse Hart
([email protected]). The tutors in the Mathematics Assistance Room (in Building 302 Room
170) can give you help and advice, but all work submitted must be your own.
Best of luck!
1. (a) The intervals/statements in (i)-(iii) feature mistakes that are commonly seen. Explain what is
wrong with each of them.
(i) (1;4): (ii) [7;1]: (iii) [8; 9] contains only two numbers.
(b) Simplify the following.
(i) (1; 5) \ f1; 2g: (ii)
(1; 5)n[2; 4]

[ f1; 2g :
[5 Marks]
2. The aim of this problem is to familiarise you with proving and disproving, which are fundamental
ideas in mathematics. For this problem you are given a list of statements and you must determine for
each of them whether the statement is true or false. In mathematics, a statement is true if it holds
for all possible cases that it addresses.
To prove a statement you need to show that it is true in the most general case. You may need to
use variables for this. A statement is false if it does not hold for at least one case. To disprove the
statement you need to provide a single concrete example for which the statement does not hold. You
can nd two worked examples on Canvas!Modules!Course Information.
(a) If a function has a horizontal asymptote, then the function and the asymptote have no intersection.
(b) If a function has a vertical asymptote, then the function and the asymptote have no intersection.
(c) u, v and w are nonzero vectors in R2. If u  v = 0 and v w = 0 then u and w are parallel.
(d) u, v and w are nonzero vectors in R3. If u  v = 0 and v w = 0 then u and w are parallel.
[8 Marks]
3. Sketch the graph for each of the following by hand. Use a separate Cartesian plane (i.e. (x; y)-plane)
for each graph. The x-axis should be the interval [; ] and the y-axis should be chosen appropriately
so that the features of the graph are clear.
(a) f1(x) = (
p
x)2: (b) f2(x) =
p
x2:
(c) f3(x) = xjxj: (d) f4(x) = tan(x) sin(x) cot(x):
Note: similar problems may appear in the test/exam, it is important that you are able to produce such graphs
on your own.

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Tellus molestie nunc non blandit massa enim nec dui. Tellus molestie nunc non blandit massa enim nec dui. Ac tortor vitae purus faucibus ornare suspendisse sed nisi. Pharetra et ultrices neque ornare aenean euismod. Pretium viverra suspendisse potenti nullam ac tortor vitae. Morbi quis commodo odio aenean sed. At consectetur lorem donec massa sapien faucibus et. Nisi quis eleifend quam adipiscing vitae proin sagittis nisl rhoncus. Duis at tellus at urna condimentum mattis pellentesque. Vivamus at augue eget arcu dictum varius duis at. Justo donec enim diam vulputate ut. Blandit libero volutpat sed cras ornare arcu. Ac felis donec et odio pellentesque diam volutpat commodo. Convallis a cras semper auctor neque. Tempus iaculis urna id volutpat lacus. Tortor consequat id porta nibh.

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