FAD1014 Exam Focus — Leak Topics

[!warning] Based on exam tips/leaks These three topics — Maclaurin Series, Ellipse, and Differential Equations — were highlighted as likely exam focus areas. Use this as a targeted revision guide alongside full lecture notes.

1. Maclaurin Series

What to Expect

  • Deriving the Maclaurin series for a function from the definition (computing successive derivatives at $x=0$)
  • Using standard expansions with substitution to find series for composite functions (e.g., $e^{-3x^2}$, $\cos x^3$, $\sin(2x^2)$)
  • Term-by-term multiplication of two known series (e.g., $(\sin 3x^2)e^{2x}$, $x^4 e^{-3x^2}$)
  • Term-by-term integration of a series to approximate definite integrals with no elementary antiderivative (e.g., $\int \frac{\sin x}{x},dx$, $\int e^{x^2},dx$)
  • Approximating function values and comparing with calculator values
  • Taylor series about a point $a \neq 0$ may also appear (e.g., $\sin x$ about $x = \pi/4$)

Key Formulas

Maclaurin series (Taylor at $x=0$): $$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots$$

Taylor series (about $x = a$): $$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$$

Must-Know Expansions

Function Series Valid for
$e^x$ $\displaystyle \sum_{n=0}^\infty \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$ All $x$
$\sin x$ $\displaystyle \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$ All $x$
$\cos x$ $\displaystyle \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$ All $x$
$\ln(1+x)$ $\displaystyle \sum_{n=1}^\infty \frac{(-1)^{n+1} x^n}{n} = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots$ $-1 < x \leq 1$
$\sinh x$ $\displaystyle \sum_{n=0}^\infty \frac{x^{2n+1}}{(2n+1)!} = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \cdots$ All $x$
$\cosh x$ $\displaystyle \sum_{n=0}^\infty \frac{x^{2n}}{(2n)!} = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \cdots$ All $x$
$\arctan x$ $\displaystyle \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{2n+1} = x - \frac{x^3}{3} + \frac{x^5}{5} - \cdots$ $-1 \leq x \leq 1$
$(1+x)^n$ $\displaystyle \sum_{r=0}^\infty \binom{n}{r} x^r = 1 + nx + \frac{n(n-1)}{2!}x^2 + \cdots$ $

Problem Types Likely on Exam

Type Example Method
Derivation from definition $f(x) = e^x$, $\sin x$, $\ln(1+x)$ Compute $f^{(n)}(0)$, build $\sum \frac{f^{(n)}(0)}{n!}x^n$
Substitution $e^{-3x^2}$, $\cos(x^3)$, $\ln(1+2x)$ Replace $x$ with the argument in a standard series
Multiplication $x^4 e^{-3x^2}$, $(\sin 3x^2)e^{2x}$ Multiply two known series, collect terms up to required order
Integration $\int \frac{\sin x}{x},dx$ Replace integrand with its series, integrate term-by-term
Taylor about $a \neq 0$ $\sin x$ about $x=\pi/4$, $\sqrt{x}$ about $x=4$ Compute derivatives at $x=a$, build $\sum \frac{f^{(n)}(a)}{n!}(x-a)^n$
Approximation Approximate $\sin 48^\circ$, $\sqrt{5}$ using Taylor Truncate series, substitute value, compare with exact

Revision Checklist

  • [ ] Derive Maclaurin series from definition (compute $f(0), f'(0), f''(0), \ldots$)
  • [ ] Recognise patterns: factorials in denominator, alternating signs in $\sin x$, $\cos x$
  • [ ] Use standard expansions with substitution (e.g., $e^{x^2}$, $\sin(2x)$)
  • [ ] Multiply two known series and collect like terms
  • [ ] Integrate a series term-by-term to approximate a definite integral
  • [ ] Taylor series about a point $a$ (if examinable)
  • [ ] Approximate a value and compare to calculator result

Common Exam Traps

Trap Fix
Forgetting to divide by $n!$ in the general term Each term is $\frac{f^{(n)}(0)}{n!}x^n$, not just $f^{(n)}(0)x^n$
Mixing up $\sin x$ and $\cos x$ series $\sin x$ has odd powers only; $\cos x$ has even powers only
Forgetting alternating signs $\sin x$ and $\cos x$ alternate: $(-1)^n$ factor
Substitution in $\ln(1+x)$: forgetting domain $\ln(1+x)$ converges for $-1 < x \leq 1$; check validity when substituting
Term-by-term integration: forgetting the constant Always include $+C$ for indefinite integrals

Related Resources

2. Ellipse

What to Expect

  • Identifying ellipse features (vertices, foci, centre, major/minor axes) from standard or general equations
  • Completing the square to convert general form $Ax^2 + By^2 + Cx + Dy + K = 0$ into standard form
  • Finding the equation of an ellipse given geometric information (centre, foci, vertices)
  • Sketching an ellipse from its equation
  • Distinguishing horizontal vs vertical orientation by comparing $a$ and $b$
  • Possibly combined with hyperbola identification in a conic-section comparison question

Key Concepts

Definition: Set of all points $P(x,y)$ such that the sum of distances to two foci is constant: $d_1 + d_2 = 2a$.

Standard equation (centre at origin, horizontal): $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \quad a > b$$

Standard equation (centre at origin, vertical): $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \quad b > a$$

Properties (horizontal):

  • Foci: $(\pm c, 0)$ where $c^2 = a^2 - b^2$
  • Vertices: $(\pm a, 0)$
  • Major axis length: $2a$, Minor axis length: $2b$
  • Latus rectum length: $\frac{2b^2}{a}$

Properties (vertical):

  • Foci: $(0, \pm c)$ where $c^2 = b^2 - a^2$
  • Vertices: $(0, \pm b)$
  • Major axis length: $2b$, Minor axis length: $2a$
  • Latus rectum length: $\frac{2a^2}{b}$

Centre at $(h,k)$ (horizontal): $$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$$

  • Vertices: $(h \pm a, k)$, Foci: $(h \pm c, k)$

Centre at $(h,k)$ (vertical): $$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$$

  • Vertices: $(h, k \pm b)$, Foci: $(h, k \pm c)$

General equation: $$Ax^2 + By^2 + Cx + Dy + K = 0, \quad A \neq B, \text{ same sign}$$

Problem Types

Type What to Do
Given standard equation, find features Read $a^2$, $b^2$; compute $c^2 =
Given general equation, find features Complete the square for $x$ and $y$ → standard form → read off features
Write equation from given info Use centre → determine $a$, $c$ from vertices/foci → compute $b^2 = a^2 - c^2$ → write standard form
Sketch ellipse Plot centre, vertices, co-vertices; draw smooth oval through them
Determine orientation Compare $a$ and $b$: larger denominator under the $x^2$ term → horizontal; under $y^2$ → vertical

Revision Checklist

  • [ ] Derive the standard ellipse equation from the distance-sum definition
  • [ ] Recall $c^2 = a^2 - b^2$ (major axis denominator squared minus minor axis denominator squared)
  • [ ] Complete the square for both $x$ and $y$ terms
  • [ ] Distinguish horizontal vs vertical orientation
  • [ ] Find foci, vertices, co-vertices, latus rectum
  • [ ] Write the equation from geometric data (centre, vertex, focus)
  • [ ] Sketch an ellipse given its equation

Common Exam Traps

Trap Fix
Forgetting $c^2 = a^2 - b^2$ (not $c^2 = a^2 + b^2$ like hyperbola) Ellipse subtracts; hyperbola adds
Mixing up $a$ and $b$ for vertical ellipses For vertical: major axis is $2b$, so $c^2 = b^2 - a^2$
Completing the square: forgetting to divide RHS by the constant Standard form must equal 1; if RHS $\neq 1$, divide through
Confusing centre $(h,k)$ sign when completing square $(x-h)^2$ means the centre $x$-coordinate is $h$, not $-h$
Thinking $a$ is always under $x^2$ $a$ is always the semi-major axis — it goes with the larger denominator

Related Resources

3. Differential Equations

What to Expect

  • Identifying the type of a given DE — this is the single most important skill
  • Solving separable first-order DEs: separate variables, integrate, apply initial condition
  • Solving first-order linear DEs using the integrating factor method
  • Solving Bernoulli DEs by substituting $v = y^{1-n}$ to reduce to linear form
  • Solving exact DEs by checking $\partial M/\partial y = \partial N/\partial x$ and integrating
  • Solving non-homogeneous DEs (both linearly independent and dependent cases) via coordinate translation or substitution
  • Mixing problems as applications of first-order linear DEs (constant volume and variable volume)
  • Verifying that a given function is a solution to a DE
  • Finding general and particular solutions given initial conditions

Key Types

1. Separable DE

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

Method: Separate → Integrate → Solve for $y$.

Example: $\displaystyle \frac{dy}{dx} = \frac{2y}{x^2-1}$, $y(2) = 1$ → $y = \frac{3(x-1)}{x+1}$

2. First-Order Linear DE

$$\frac{dy}{dx} + P(x)y = Q(x)$$

Method: Compute integrating factor $I = e^{\int P(x),dx}$, then: $$y \cdot I = \int Q(x) \cdot I , dx + C$$

3. Bernoulli DE

$$\frac{dy}{dx} + P(x)y = Q(x)y^n, \quad n \neq 0,1$$

Method: Substitute $v = y^{1-n}$ → reduces to a first-order linear DE in $v$ → solve using integrating factor.

4. Exact DE

$$M(x,y),dx + N(x,y),dy = 0$$

Test for exactness: $\displaystyle \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$

Method: Find $F(x,y)$ such that $\frac{\partial F}{\partial x} = M$ and $\frac{\partial F}{\partial y} = N$. Solution: $F(x,y) = C$.

5. Non-Homogeneous DE (Linearly Independent)

$$(a_1 x + b_1 y + c_1),dx + (a_2 x + b_2 y + c_2),dy = 0, \quad a_1 b_2 - a_2 b_1 \neq 0$$

Method: Translate coordinates: $x = u + h$, $y = v + k$. Solve for $h,k$ to eliminate constant terms → reduces to homogeneous DE.

6. Non-Homogeneous DE (Linearly Dependent)

$$(a_1 x + b_1 y + c_1),dx + (a_2 x + b_2 y + c_2),dy = 0, \quad a_1 b_2 - a_2 b_1 = 0$$

Method: Substitute $u = a_1 x + b_1 y$ → reduces to a separable DE in $u$ and $x$.

7. Mixing Problems (Application)

$$\frac{dA}{dt} = \text{rate in} - \text{rate out}$$

  • Constant volume: inflow = outflow → separable or linear ODE
  • Variable volume: inflow $\neq$ outflow → $V(t) = V_0 + (r_{in} - r_{out})t$ → first-order linear ODE

Problem-Type Identification Flowchart

flowchart TD
    A[Given DE] --> B{Can variables be<br/>separated?}
    B -->|Yes| C[Separable: ∫ g(y) dy = ∫ f(x) dx]
    B -->|No| D{Has form<br/>dy/dx + P(x)y = Q(x)y^n?}
    D -->|n = 0,1| E[First-Order Linear:<br/>I = e^∫P dx]
    D -->|n ≠ 0,1| F[Bernoulli:<br/>v = y^(1-n)]
    D -->|No| G{Is it M dx + N dy = 0<br/>with ∂M/∂y = ∂N/∂x?}
    G -->|Yes| H[Exact: find F(x,y) = C]
    G -->|No| I{Is it of form<br/>(a₁x+b₁y+c₁)dx + (a₂x+b₂y+c₂)dy = 0?}
    I -->|a₁b₂ - a₂b₁ ≠ 0| J[Non-Homogeneous Independent:<br/>translate x = u+h, y = v+k]
    I -->|a₁b₂ - a₂b₁ = 0| K[Non-Homogeneous Dependent:<br/>substitute u = a₁x + b₁y]

Revision Strategy

  1. Master identification first. Before solving any DE, practise identifying its type in under 10 seconds. This is the skill that separates students who can solve from those who freeze.
  2. Memorise the integrating factor $I = e^{\int P,dx}$ — it appears in linear, Bernoulli (after reduction), and mixing problems.
  3. Separable drill: practise separating and integrating quickly. Watch for partial fractions when integrating rational functions.
  4. Bernoulli drill: recognise $y^n$ on RHS, apply $v = y^{1-n}$, differentiate, substitute, solve linear.
  5. Exact drill: always test $\partial M/\partial y = \partial N/\partial x$ first. If it fails, the DE may be non-exact (not covered here).
  6. Non-homogeneous drill: compute $a_1 b_2 - a_2 b_1$ to determine independent vs dependent, then apply the appropriate method.
  7. Mixing problems: draw a diagram. Identify inflow/outflow rates and concentrations. Set up $\frac{dA}{dt}$ equation. Solve with integrating factor if volume varies.

Common Exam Traps

Trap Fix
Trying to separate a non-separable DE Check: can you write it as $g(y),dy = f(x),dx$? If not, it's not separable
Forgetting the $+C$ after integrating Always include the constant — it's what makes it a general solution
Mixing up Bernoulli $v = y^{1-n}$ $v$ is $y$ to the power of $(1-n)$, not $y^n$
Skipping exactness test Always compute $\partial M/\partial y$ and $\partial N/\partial x$ before proceeding
Forgetting to apply initial condition The particular solution uses the IC to determine the constant
Mixing problems: confusing rate in vs rate out Rate in = (concentration in) × (flow rate in); Rate out = (concentration in tank) × (flow rate out)

Related Resources

Quick Reference

Topic Weight (est.) Key Skill Related Lecture
Maclaurin Series 30% Derivation from definition / substitution / term-by-term operations L25-L26
Ellipse 30% Completing the square / finding foci, vertices, centre L29-L30
Differential Equations 40% Identifying type / applying correct solution method L15-L16 + standalone DE topics

[!tip] Exam Strategy Start with Differential Equations — it carries the most weight (est. 40%) and requires methodical step-by-step work. Identify each DE's type before solving.

For Maclaurin Series, memorise the five standard expansions cold ($e^x$, $\sin x$, $\cos x$, $\ln(1+x)$, $(1+x)^n$). Most problems are substitutions or combinations of these — you don't need to derive from scratch if you know the series.

For Ellipse, practise completing the square until it's automatic. The most common mistake is getting the centre sign wrong. Remember: ellipse $c^2 = a^2 - b^2$ (subtract), hyperbola $c^2 = a^2 + b^2$ (add).

Time management: If stuck on a DE type-identification, move on and come back. Don't spend more than 8 minutes on any single sub-question.

Final sanity check: Does your Maclaurin series match the expected pattern (alternating signs for trig, all positive for $e^x$)? Is your ellipse $c^2$ positive? Does your particular solution actually satisfy the original DE? Plug it back in to verify.