MAS152
Essential Mathematical Skills & Techniques
Examples 1: Functions of Real Variables
1. Sketch the graphs of the following functions. You should sketch the graph in such a way
that it is clear how to extend your picture and mark on all important poin
MAS152
Essential Mathematical Skills & Techniques
Examples 2: Dierentiation of functions of a single variable
1. Find the derivatives of the following functions by working from rst principles:
1
(iii) tan x.
(ii) ,
(i) xn (where n = 1 is a constant),
x
2.
THE ARGAND DIAGRAM
5 minute review. Review the Argand diagram (aka Argand plane, aka complex
plane), the modulus and argument and the polar form of a complex number, z =
r(cos + i sin ). Also cover the eect of addition and multiplication.
Class warm-up. P
DIFFERENTIATION 1: FIRST RESULTS
df
5 minute review. Recall the denition dx (x0 ) = limh0 f (x0 +h)f (x0 ) by working
h
out the gradient of the line on the curve between (x0 , f (x0 ) and (x0 + h, f (x0 + h).
Perhaps do an example, such as y = x3 at x0 =
DIFFERENTIATION RULES
5 minute review. Remind the students of the addition, product, quotient and
dy
chain rules, the latter as both if y = f (g(x) then dx = g (x)f (g(x) and if
dy
dy
y = f (u) with u = u(x), then dx = du . du .
dx
Class warm-up. Find the
COMPLEX NUMBERS
5 minute review. Remind students what a complex number is, how to add and
multiply them, what the complex conjugate is, and how to do division. They will
also need to know that a root of a polynomial f (x) is a solution to f (x) = 0, and
t
CURVE SKETCHING
5 minute review. Using the graph of y =
as an example, remind students briey
1
x1
(or any other suitably easy curve)
what a graph is;
the dierence between plotting (by calculating) and sketching (by reasoning);
the importance of labelli
DE MOIVRES THEOREM
5 minute review. Recap de Moivres Theorem, cos(n) + i sin(n) = (cos +
i sin )n , and how to solve z n = r(cos +i sin ) for z, perhaps by doing the warm-up
below.
Class warm-up. Find the cube roots of 1 + i.
Problems. Choose from the bel
GENERAL LOGARITHMS AND HYPERBOLIC FUNCTIONS
5 minute review. Remind students
what loga x is for general a > 1 (i.e., loga x is the power of a needed to make
x) and that ln = loge ;
the denitions of sinh, cosh and tanh;
the identities cosh2 (x) sinh2 (x
EULERS RELATION
5 minute review.
Review Eulers relation, ei = cos + i sin , commenting
briey on how it follows from the Maclaurin series of exp, sin and cos. Also cover
the exponential forms of sin and cos, namely
sin =
ei ei
2i
and
cos =
ei + ei
.
2
Clas
INVERSE FUNCTIONS, EXPONENTIALS AND LOGARITHMS
5 minute review.
things like
A brief reminder of inverse functions, exp and ln covering
f 1 (f (x) = x and f (f 1 (x) = x;
y = f (x) and y = f 1 (x) have graphs which are reections in y = x;
range(f ) = do
MAS152
Essential Mathematical Skills & Techniques
Examples 4: Partial Dierentiation
1. For each of the following functions, calculate
f
2f
2f
f
and
. Show that
=
.
x
y
xy
yx
(a) x3 + 3x2 y + xy 2 + 4y 3
(b) x sin(xy)
(c) xy 2 ln(x2 + y 2 )
sin r
(d)
, whe
MAS152
Essential Mathematical Skills & Techniques
Examples 5: Complex Numbers 1
1. Express the following in the form z = a + ib (where a and b are real numbers) and write
down the complex conjugate, modulus and principal argument of each of the resulting
MAS152
Essential Mathematical Skills & Techniques
Examples 3: Maclaurin and Taylor Series: LHpitals Rule
o
1. Find the Maclaurin series for
(a) f (x) = sin(3x + 2) (rst 4 non-zero terms),
(b) f (x) = sin2 x (rst 3 non-zero terms).
2. Show graphically that
MAS152
Essential Mathematical Skills & Techniques
Examples 7: Vectors
1. If a and b are vectors as given in the table below, verify that the scalar products are as
shown. Which of the three pairs of vectors is perpendicular?
a
(1, 1, 0)
(4, 1, 3)
(3, 1, 4
MAS152
Essential Mathematical Skills & Techniques
Examples 6: Complex Numbers 2
1. Find the three roots of the equation (z 1)3 = 8i and plot them on the Argand diagram.
2. Use the result 2i sin = ei ei to establish the identity
1
sin4 = (cos 4 4 cos 2 + 3
Solutions to Vectors
1. (2') (1,1,0).(3,4,5) = 1 x 3+ (1) x 4+0 x 5 = 1
(ii) (4,1,3).(1,3,7) = 4 x (1)+ 1 x 3 + (3) x (7) = 20
(3,1,4).(2,2,1) = 3 x 2+ 1 x (2) +4 ><(1)= 0 => a and b are perpendicular.
2. a=(0,1,1)=>a=|a|=(/02+(1)2+12=\/§
b=(3,4,5)=>b= |
MAS152
Essential Mathematical Skills & Techniques
Examples 1: Functions of Real Variables
Solutions
Solutions for Q1 can be obtained using any graphing software, such as that accessible on
Wolfram Alpha (http:/www.wolframalpha.com).
1
INVERSE FUNCTION RULE AND CURVE SKETCHING
5 minute review.
Briey cover the inverse trigonometric functions and their
1
derivatives (and remind students that that sin1 x does not mean sin x ).
d
1
Class warm-up. Run through the proof that dx (cos1 x) = 1x2