Engineering and Science Admissions Test (ESAT) Mathematics 2 (Advanced Mathematics Module) Syllabus

Mathematics 2 (Advanced Mathematics Module) syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Mathematics 2 (Advanced Mathematics Module) in Engineering and Science Admissions Test (ESAT), not a summary of it.

1. 1

   Algebra, Functions and Proof

   4 topics

   • Advanced Algebraic Techniques
     • Partial fractions with repeated and quadratic factors
     • Algebraic and modulus inequalities
   • The Binomial Expansion
     • Expansion for positive integer powers
     • Expansion for fractional and negative indices; range of validity
   • Mathematical Proof
     • Proof by deduction and exhaustion
     • Disproof by counterexample
     • Proof by contradiction
   • Functions and Transformations
     • Composite and inverse functions
     • Modulus function equations and graphs
2. 2

   Differentiation

   3 topics

   • Differentiation from First Principles and Standard Results
     • Derivatives of powers, exponentials, logarithms and trig functions
   • Rules of Differentiation
     • Product and quotient rules
     • Chain rule and related rates
     • Implicit and parametric differentiation
   • Applications of Differentiation
     • Tangents, normals and stationary points
     • Maxima, minima and points of inflection
     • Optimisation problems
3. 3

   Integration

   4 topics

   • Indefinite and Definite Integration
     • Standard integrals of powers, exponentials and trig functions
     • Definite integrals and area under a curve
   • Techniques of Integration
     • Integration by substitution
     • Integration by parts
     • Integration using partial fractions and trig identities
   • Applications of Integration
     • Areas between curves
     • Volumes of revolution
   • Differential Equations
     • First-order separable differential equations
     • Modelling with differential equations
4. 4

   Trigonometry and Further Functions

   3 topics

   • Trigonometric Identities and Formulae
     • Compound and double-angle formulae
     • The R cos(x+a) / R sin(x+a) form
     • Reciprocal and inverse trigonometric functions
   • Parametric Equations
     • Converting between parametric and Cartesian forms
   • Sequences, Series and Iteration
     • Recurrence relations
     • Sigma notation
     • Numerical solution of equations by iteration
5. 5

   Vectors, Graphs and Numerical Methods

   3 topics

   • Vectors in Two and Three Dimensions
     • Vector arithmetic and magnitude
     • Position vectors and geometric applications
     • Unit vectors and direction
   • Graphs of Functions
     • Curve sketching using calculus
     • Asymptotes and behaviour at infinity
   • Numerical Methods
     • Locating roots by sign change
     • Trapezium rule for numerical integration

Mathematics 2 (Advanced Mathematics Module) flashcards for Engineering and Science Admissions Test (ESAT)

21 of 51 cards from the Mathematics 2 (Advanced Mathematics Module) deck — real questions with worked answers.

1. State the binomial expansion of $(1+x)^{n}$ for any real $n$, valid for $|x|<1$.

   $$(1+x)^{n}=1+nx+\frac{n(n-1)}{2!}x^{2}+\frac{n(n-1)(n-2)}{3!}x^{3}+\cdots$$ valid for $|x|<1$ when $n$ is not a non-negative integer.

2. What is the general term in the expansion of $(a+b)^{n}$ for positive integer $n$?

   The term is $\binom{n}{r}a^{n-r}b^{r}$, where $\binom{n}{r}=\frac{n!}{r!(n-r)!}$, for $r=0,1,\dots,n$.

3. How do you split $\frac{4x+1}{(x-1)(x+2)}$ into partial fractions (state the form)?

   Write it as $\frac{A}{x-1}+\frac{B}{x+2}$, then solve for $A$ and $B$ by equating numerators. Here $A=\frac{5}{3}$, $B=\frac{7}{3}$.

4. What partial-fraction form is used for a repeated linear factor, e.g. $\frac{f(x)}{(x-2)^{2}}$?

   $$\frac{A}{x-2}+\frac{B}{(x-2)^{2}}$$ A separate term is needed for each power up to the multiplicity.

5. In a proof by contradiction, what is the logical structure?

   Assume the negation of the statement is true, derive a logical contradiction (an impossibility), and conclude the original statement must be true.

6. What does proof by exhaustion involve?

   Breaking the statement into a finite number of cases and verifying the result holds in every case.

7. What single counterexample is needed to disprove the statement 'for all integers $n$, $n^{2}+n+41$ is prime'?

   Take $n=41$: then $41^{2}+41+41=41\times43$, which is composite. A single counterexample disproves a universal claim.

8. Differentiate $f(x)=x^{2}$ from first principles.

   $$f'(x)=\lim_{h\to 0}\frac{(x+h)^{2}-x^{2}}{h}=\lim_{h\to 0}(2x+h)=2x$$

9. State the formal (first principles) definition of the derivative of $f(x)$.

   $$f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$$

10. State the standard derivatives of $\sin x$, $\cos x$ and $e^{x}$.

    $\frac{d}{dx}\sin x=\cos x$, $\quad\frac{d}{dx}\cos x=-\sin x$, $\quad\frac{d}{dx}e^{x}=e^{x}$.

11. What is the derivative of $\ln x$ and of $\tan x$?

    $\frac{d}{dx}\ln x=\frac{1}{x}$ and $\frac{d}{dx}\tan x=\sec^{2}x$.

12. State the product rule for differentiation.

    If $y=uv$ then $$\frac{dy}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}$$

13. State the quotient rule for differentiation.

    If $y=\dfrac{u}{v}$ then $$\frac{dy}{dx}=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^{2}}$$

14. State the chain rule for $y=f(g(x))$.

    $$\frac{dy}{dx}=f'(g(x))\,g'(x),\quad\text{or}\quad\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$$

15. How do you find the derivative $\frac{dy}{dx}$ when $x$ and $y$ are given parametrically as functions of $t$?

    $$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}=\frac{\dot{y}}{\dot{x}}\quad(\dot{x}\neq 0)$$

16. How are stationary points classified using the second derivative?

    At a stationary point ($f'(x)=0$): if $f''(x)>0$ it is a local minimum; if $f''(x)<0$ a local maximum; if $f''(x)=0$ the test is inconclusive (check the sign of $f'$).

17. What condition identifies a point of inflection?

    $f''(x)=0$ and $f''(x)$ changes sign through that point (the concavity changes).

18. What is the equation of the tangent to $y=f(x)$ at the point $(a,f(a))$?

    $$y-f(a)=f'(a)(x-a)$$

19. How are the gradients of perpendicular lines (e.g. a tangent and its normal) related?

    Their product is $-1$; the normal gradient is $-\frac{1}{f'(a)}$ where $f'(a)$ is the tangent gradient.

20. State the standard result $\int x^{n}\,dx$ and note the exception.

    $$\int x^{n}\,dx=\frac{x^{n+1}}{n+1}+C\quad(n\neq -1)$$ When $n=-1$, $\int x^{-1}\,dx=\ln|x|+C$.

21. State $\int e^{x}\,dx$, $\int \sin x\,dx$ and $\int \cos x\,dx$.

    $\int e^{x}\,dx=e^{x}+C$, $\quad\int\sin x\,dx=-\cos x+C$, $\quad\int\cos x\,dx=\sin x+C$.

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