This study guide covers advanced calculus techniques required for VCE Specialist Mathematics Unit 3, focusing on differentiation of inverse circular functions, implicit differentiation, and sophisticated integration techniques including substitution, partial fractions, and integration by parts.
The derivatives of the inverse sine, inverse cosine, and inverse tangent functions are fundamental to Specialist Mathematics.
| Function $f(x)$ | Derivative $f’(x)$ | Domain |
|---|---|---|
| $\arcsin\left(\frac{x}{a}\right)$ | $\frac{1}{\sqrt{a^2 - x^2}}$ | $x \in (-a, a)$ |
| $\arccos\left(\frac{x}{a}\right)$ | $-\frac{1}{\sqrt{a^2 - x^2}}$ | $x \in (-a, a)$ |
| $\arctan\left(\frac{x}{a}\right)$ | $\frac{a}{a^2 + x^2}$ | $x \in \mathbb{R}$ |
General Chain Rule Form:
If $u$ is a function of $x$:
$$\frac{d}{dx}(\arcsin(u)) = \frac{1}{\sqrt{1-u^2}} \frac{du}{dx}$$
EXAM TIP: When differentiating $\arctan(f(x))$, ensure you simplify the final expression. VCAA Exam 1 often requires these derivatives to be evaluated at specific values of $x$ (e.g., $x=1$ or $x=\sqrt{3}$).
Used when a relation is defined by an equation where $y$ cannot be easily isolated (e.g., $x^2 + y^2 = 25$ or $e^y + xy = 2$).
The Process:
1. Differentiate both sides of the equation with respect to $x$.
2. Apply the Chain Rule to terms involving $y$: $\frac{d}{dx}(g(y)) = g’(y) \frac{dy}{dx}$.
3. Use the Product Rule for terms like $xy$: $\frac{d}{dx}(xy) = x\frac{dy}{dx} + y$.
4. Rearrange the resulting equation to solve for $\frac{dy}{dx}$.
COMMON MISTAKE: Forgetting to differentiate the constant on the right-hand side of the equation. For example, in $x^2 + y^2 = 9$, the derivative of $9$ is $0$, not $9$.
The second derivative $\frac{d^2y}{dx^2}$ (or $f’‘(x)$) determines the shape of the graph:
* If $f’‘(x) > 0$, the graph is concave up.
* If $f’‘(x) < 0$, the graph is concave down.
* A point of inflection occurs where $f’‘(x) = 0$ (or is undefined) and the concavity changes.
Used to simplify an integral by changing the variable of integration. Usually, we let $u = g(x)$, then $du = g’(x) dx$.
Steps for Definite Integrals:
1. Identify a suitable $u$ (usually the “inner” function).
2. Calculate $du$ in terms of $dx$.
3. Change the limits of integration to be in terms of $u$.
4. Substitute all terms and integrate with respect to $u$.
REMEMBER: When performing a definite integral using substitution, if you change the limits of integration to $u$-values, you do not need to substitute $x$ back in at the end.
Used for rational functions $\frac{P(x)}{Q(x)}$ where the degree of $P(x)$ is less than $Q(x)$. If the degree of $P(x) \geq Q(x)$, perform polynomial long division first.
Common Decompositions:
* Linear factors: $\frac{1}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}$
* Repeated linear factors: $\frac{1}{(x-a)^2} = \frac{A}{x-a} + \frac{B}{(x-a)^2}$
* Irreducible quadratic factors: $\frac{1}{x^2+a^2}$ (leads to $\arctan$ results).
VCAA FOCUS: Partial fractions are frequently combined with logarithmic integration: $\int \frac{1}{ax+b} dx = \frac{1}{a} \ln|ax+b| + c$.
Based on the product rule for differentiation.
$$\int u \frac{dv}{dx} dx = uv - \int v \frac{du}{dx} dx$$
Choosing $u$ (LIATE Rule):
Choose $u$ in order of priority:
1. Logarithmic functions ($\ln x$)
2. Inverse trigonometric functions ($\arcsin x$)
3. Algebraic functions ($x^n$)
4. Trigonometric functions ($\sin x$)
5. Exponential functions ($e^x$)
KEY TAKEAWAY: Integration by parts is essential for integrals like $\int x e^x dx$ or $\int \ln(x) dx$ (where you let $u = \ln(x)$ and $\frac{dv}{dx} = 1$).
Specialist Mathematics requires the use of identities to integrate powers of trig functions:
* $\int \cos^2(ax) dx = \int \frac{1}{2}(1 + \cos(2ax)) dx$
* $\int \sin^2(ax) dx = \int \frac{1}{2}(1 - \cos(2ax)) dx$
* For odd powers (e.g., $\sin^3(x)$), use $\sin^2(x) = 1 - \cos^2(x)$ and then use substitution.
The area $A$ between $f(x)$ and $g(x)$ from $x=a$ to $x=b$:
$$A = \int_{a}^{b} |f(x) - g(x)| dx$$
Always determine which function is “upper” and which is “lower” on the interval.
When a region is rotated $360^\circ$ about an axis.
STUDY HINT: Always draw a sketch of the region before setting up the integral. For rotation about the $y$-axis, you must rearrange your equation to make $x^2$ the subject.
The length $s$ of a curve $y = f(x)$ from $x=a$ to $x=b$ is given by:
$$s = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx$$
For a parametric curve $x=x(t), y=y(t)$ from $t=t_1$ to $t=t_2$:
$$s = \int_{t_1}^{t_2} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt$$
| Function | Integral |
|---|---|
| $\frac{1}{\sqrt{a^2 - x^2}}$ | $\arcsin\left(\frac{x}{a}\right) + c$ |
| $\frac{a}{a^2 + x^2}$ | $\arctan\left(\frac{x}{a}\right) + c$ |
| $\frac{1}{x^2 + a^2}$ | $\frac{1}{a} \arctan\left(\frac{x}{a}\right) + c$ |
| $\sin^2(kx)$ | $\frac{x}{2} - \frac{\sin(2kx)}{4k} + c$ |
| $\cos^2(kx)$ | $\frac{x}{2} + \frac{\sin(2kx)}{4k} + c$ |
APPLICATION: These techniques are the building blocks for solving Differential Equations, which model real-world phenomena like population growth, cooling objects, and chemical reactions.
