Overview (Leaving Cert Applied Maths): Revision Notes

Overview

Calculus and differential equations form a fundamental part of applied mathematics, providing powerful tools for solving real-world problems involving rates of change, areas under curves, and relationships between variables. This overview covers the essential techniques you'll need to master for your Leaving Certificate examinations.

Integration fundamentals

Integration is the reverse process of differentiation and allows us to find areas under curves, volumes, and solve problems involving accumulation. Understanding the basic integration rules forms the foundation for more complex applications.

Basic integration rules

The power rule is your most frequently used integration technique. For any function of the form $x^n$, the integral becomes $\frac{x^{n+1}}{n+1} + c$, where $c$ represents the constant of integration.

The natural logarithm function has a special integration rule. When integrating $\frac{1}{x}$, the result is $\ln x + c$. This rule is particularly important because it appears frequently in applied problems and must be memorised as it's not provided in examination tables.

Definite integrals

Definite integrals represent the exact area between a curve and the x-axis over a specified interval. The fundamental theorem of calculus tells us that:

$\int\_a^b f(x)dx = F(b) - F(a)$

where $F(x)$ is the antiderivative of $f(x)$.

Advanced integration techniques

Integration by parts uses the formula $\int u , dv = uv - \int v , du$ and requires careful selection of $u$ and $dv$.

Integration by substitution simplifies complex integrals by changing variables. The key steps involve identifying a suitable substitution ($u$ = part of the function), finding $dx$ in terms of $du$, substituting into the integral, solving the simplified integral, and finally substituting back to the original variable.

Rates of change and motion

Differentiation reveals how quantities change with respect to one another, making it essential for understanding motion, growth rates, and optimisation problems.

The relationship between distance, velocity, and acceleration

These three kinematic quantities are intimately connected through calculus. Distance ($s$) represents position, velocity ($v$) is the rate of change of distance with respect to time, and acceleration ($a$) is the rate of change of velocity with respect to time.

Starting with distance as a function of time, $s(t)$, velocity becomes $v = \frac{ds}{dt}$ (the first derivative of distance). Acceleration is $a = \frac{dv}{dt}$ (the first derivative of velocity, or the second derivative of distance).

Vector differentiation

When working with vectors, each component can be differentiated separately. This principle applies to both i and j components in two-dimensional problems, allowing you to handle vector quantities using familiar differentiation rules.

Work and variable forces

In physics applications, work represents energy transfer and often involves variable forces that change with position.

Calculating work done by variable forces

When a force varies with position, the work done cannot be calculated using the simple formula $W = Fd$. Instead, integration becomes necessary to account for the changing force throughout the displacement.

The formula $W = \int\_a^b F(x)dx$ calculates work by summing infinitesimally small amounts of work ($F(x)dx$) over the entire displacement from position $a$ to position $b$.

Differential equations

Differential equations contain derivatives and describe relationships between functions and their rates of change. They appear frequently in modelling natural phenomena and engineering problems.

First-order differential equations with general solutions

These equations involve only first derivatives and produce general solutions containing arbitrary constants. The solution process typically involves separating variables by collecting all terms containing one variable on one side and terms containing the other variable on the opposite side.

After separation, integrate both sides independently. The resulting solution includes a constant of integration, making it a general solution that represents a family of curves.

First-order differential equations with definite values

When initial conditions are provided, you can determine the specific value of the constant of integration. Follow the same separation and integration process as for general solutions, then use the given initial condition to evaluate the constant.

Second-order separable differential equations

These equations contain second derivatives but can often be simplified using substitution techniques. Let $v$ represent $\frac{dy}{dx}$, transforming the second derivative $\frac{d^2y}{dx^2}$ into $v\frac{dv}{dy}$. This substitution can convert a second-order equation into a first-order equation in terms of $v$ and $y$.

Proportionality relationships

When two quantities $P$ and $Q$ are proportional, we write $P = kQ$, where $k$ is the constant of proportionality. This relationship frequently leads to differential equations when rates of change are involved, such as $P$ being proportional to the rate of change of $Q$.

Examination strategies

Success in calculus requires methodical approaches and attention to detail. Know your integration methods thoroughly and practise identifying which technique applies to different function types. When separating variables in differential equations, work carefully to avoid algebraic errors.

Force diagrams require proper scaling and attention to sign conventions - forces in opposite directions should have opposite signs. Choose intelligently between different possible expressions for acceleration, considering which form will lead to easier integration.