Calculus: notes and practice questions

Calculus: branch of mathematics dealing with rates of change. Limits: value a function approaches as $x$ approaches a particular value.

Differentiation is the process of finding the gradient function. The Power Rule applies to functions where $x$ is raised to any rational power.

Tangent: Straight line touching a function's graph at exactly one point. Normal: Straight line passing through the point, perpendicular to the tangent.

Definite integral calculates exact accumulation or area between limits $a$ (lower) and $b$ (upper). Integrand is $f(x)$, antiderivative is $F(x)$.

Stationary points: points on a curve where the gradient is zero ($f'(x) = 0$ or $\frac{dy}{dx} = 0$). Turning points: stationary points where the curve changes direction.

Optimisation: Finding the absolute maximum or minimum value of a quantity. Method: Model the quantity as a function of one variable, then find its turning points.

Trapezoidal Rule: numerical method to approximate area under a curve by dividing it into `n` trapezoids (strips). `n` strips correspond to `n+1` function values ($y_0, ..., y_n$).

Special functions include trigonometric ($\sin x, \cos x, \tan x$), exponential ($e^x$), and natural logarithmic ($\ln x$) functions. Chain Rule: Differentiates composite functions ($y = f(g(x))$).

Second order derivative: $f''(x)$ or $\frac{d^2y}{dx^2}$, obtained by differentiating the first derivative $f'(x)$. Local minimum/maximum points: Stationary points where the gradient $f'(x) = 0$.

Integration is the reverse of differentiation. An indefinite integral finds a general antiderivative and always includes a constant of integration ($+c$).

Integration is anti-differentiation. Definite integral calculates exact accumulation with specific boundary values (limits $a$ and $b$).

Kinematics: mathematical modeling and analysis of object motion. Displacement ($s$ or $x$): vector, position relative to start.

Differential Equations: Equations relating a function with its derivatives, used for modeling. Separation of Variables: Analytical method for first-order differential equations where variables can be separated.

Differential equation: An equation involving derivatives, typically in the form $\frac{dy}{dx} = g(x, y)$. Slope field: Graphical representation of differential equation solutions using short tangent line segments, where each segment's gradient is $\frac{dy}{dx}$ at that point.

Euler's Method: A numerical method for approximating solutions to differential equations. It works by treating derivatives as constant over short, discrete steps ($h$).

Coupled differential equations: System where rates of change of variables depend on each other, typically linear in matrix form $\mathbf{\dot{x}} = M\mathbf{x}$. Phase portrait: Diagram illustrating how $x$ and $y$ values change over time, displaying typical solution trajectories.

Euler's Method: Numerical approximation for differential equations using step-by-step calculations. Step Size ($h$): Constant increment for the independent variable ($t$).