Revision Hints Curtin University Maths 120 Revision

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Marks will be posted on Blackboard, Friday study week. Online quiz (20%), individual & total Tut (20%) & Lab (10%), plus total out of 50%

Exam : Monday 18th November (1st Week) 12.00noon, Curtin Stadium (Building 111)

Available - Study Week: Mon 9:30-2:00, Tues 11:30-3:30, Wed 12:30-3:30, Thurs 9:30-3:30, Fri 9:30-12:00

Room: 314.358

Email: i.loosen@curtin.edu.au

Tutors: Qian Sun, Peter, Christopher

Help: Maths Clinics – Maths Department 11-12 & 1-2pm (Wednesday 12-2pm) – EFY

Go over: Previous exams (EM120 Sem 2: 2006, 2008, 2009, 2010, 2011, 2012 EM140 Sem 1: 2009, 2010, 2011, 2012) Exercises at start of lecture notes Tutorial exercises Online quizzes

5 Questions, 20 marks each - Mishmash (Precalculus) - Differentiation & Application - Limits & Approximation - Integration & Complex Numbers - Matrices

Standard Integrals sheet is attached to exam A4 Sheet must be handwritten Only allowed HP 10s Scientific Calculator Show all working!

REVISION Domain & Range (Lecture 4)

- Domain: Don’t divide by 0 or don’t take square root of a negative number Limits of Functions (Lecture 5 & 22)

- Indeterminate forms 0 0 (factorize, cancel common term from numerator &

denominator) or ∞ ∞⁄ (divide through by highest power term in denominator),

- L’Hopital’s Rule ∞ ∞⁄ or 0 0 form

lim→

⇒ lim→

Composite of Functions (Lecture 6)

- Given and find ∘ , ∘ , etc… - Domain of a composite, evaluate the composite at

Inverse of Functions (Lecture 6) - Given . Two steps: Rearrange so ; then interchange the and

variables.

Trigonometric Functions (Lecture 7) - Exact values, i.e. sin 45°

√ , cos 30° √ , tan 60° √3 , etc…

Symmetry of Functions (Lecture 8)

- Odd function , even function , or neither

Solving Unknown Triangles (Lecture 7 & 9) - Trigonometric Ratios (for right angle triangle)

sin , cos and tan

- Sine Rule

sin sin sin - Cosine Rule

cos ⇒ 2 cos

Wave Functions (Lecture 9) - Identify amplitude, period, frequency, time displacement

Solving Trigonometric Equations (Lecture 9 & 10)

- Solve a trigonometric equation for ∈ ,

Polar Coordinates (Lecture 10) - Convert to and from Cartesian

cos and sin or

and tan check correct quadrant, either

Trigonometric Identities (Lecture 10) - sin sin , cos cos , tan tan

An Important Identity (Lecture 10)

- Express cos sin as a single cosine cos where √ and tan

Differentiation - Elementary Rules (Lecture 11) - Product Rule (Lecture 11) - Quotient Rule (Lecture 11) - Trigonometric Functions (Lecture 12) - Chain Rule (Lecture 12) - Implicit (Lecture 13) - Logarithmic (Lecture 20)

Tangent & Normal Lines (Lecture 13)

- where , ,

- Normal line has slope 1

Taylor Series (Lecture 14) - Quadratic Approximation at :

!

- −th order Taylor Polynomial at :

2! ⋯ !

Differentials (Lecture 15) - Find the differential of the function i.e . Use to estimate

change/error in given a change/error in Roots of Equations (Lecture 16) Bisection Method

- Given interval , which contains root & told number of steps/iterations Newton’s Method

- Given interval , which contains root & told number of steps/iterations as well as starting point

Elementary Curve Sketching (Lecture 17) - Stationary points/critical values, points of inflection on an interval , - Is stationary point a maximum or minimum, use first derivative test (sign

diagram) or second derivative test (if ′′ 0 then max, if ′′ 0 then min)

- Intervals of concavity, concave up on interval if ′′ 0 and concave down on interval if ′′ 0

Integration by Substitution (Lecture 18 & 19) - State & make an appropriate substitution , evaluate the anti-derivate over the

lower & upper bounds (change upper & lower bounds in terms of ) Integration (Lecture 19)

- Fundamental Theorem of Calculus

Second:

Exponential & Logarithm Functions (Lecture 20 & 21) - Properties i.e. ln ln ln , ln ln , , etc... Used in

Logarithmic differentiation; to solve equations; etc… - Derivatives i.e. ⇒ ,

⇒ ln , ln ⇒ ,

log ⇒ etc..

Area Under Curves (Lecture 19 & 23) - To determine total area between curve and -axis between , : Plot function (if

feasible), determine if function cuts -axis (i.e. 0) over given interval , , if so (i.e. at ) partition , into appropriate subintervals over

which you then integrate,

Complex Numbers (Lecture 24, 25, 26 & 27)

- Basic arithmetic: add, multiply and divide - Conjugate ̅ - Convert between the different forms (Cartesian, polar & exponential) - If converting from Cartesian form to polar form check argument is in correct

quadrant, if not ±π - De Moivre’s Theorem: ∠ - Identify and sketch a region in the complex plane from an inequality - Determine the -th roots of a complex number

∠2

, 0, 1, 2, … , 1

Matrices (Lecture 29, 30, 31, 32, 33 & 34) - Gaussian Elimination to solve a system of linear equations. Set up augmented

matrix | , get it into row-echelon form. When is there a unique, infinite or no solution. Understand concept of rank

- Solve a system of linear equations using the inverse matrix - Calculate the determinant of a matrix. If is 2 2 then det . If A

is 3 3 then do cofactor expansion along row or column with the most zeros - Find inverse of a matrix using Gauss-Jordan method. Set up matrix | , reduce

on RHS to then on LHS becomes . - Find inverse of a 2 2 matrix,

⇒ 1

- Find the eigenvalues λ of a matrix , i.e. solve det 0, and eigenvectors , 0

You need to obtain at least 40% in the exam to pass the unit! Supplementary Exam – 19-21st February 2014