Calculus: notes and practice questions

The limit of a function describes its behavior as $x$ approaches a certain value: $\lim_{x \to a} f(x)$.

The derivative of $f(x) = ax^n$ with respect to $x$ is: $$.

The tangent to a curve at a point is a line that touches the curve without crossing it. Its slope is the derivative $f'(x)$ at that point. Tangent equation:

The definite integral of a function $f(x)$ from $a$ to $b$ is: $$.

Chain Rule: If $y = f(g(x))$, then $\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$. Product Rule: $(uv)' = u'v + uv'$.

Points of Inflection: Occur when $f''(x) = 0$ or undefined, and the concavity changes (verify by checking sign changes in $f''(x)$). Testing Max/Min Points:

Kinematics involves the motion of objects described using displacement ($s$), velocity ($v$), acceleration ($a$), and time ($t$): Velocity: $v = \frac{ds}{dt}$.

Indefinite integral: The reverse process of differentiation, represented as: $$.

A definite integral calculates the area under a curve $f(x)$ between $x = a$ and $x = b$: $$.

Strategy: Visualize, define variables and constraints, construct constraint and optimization equations (in one variable), differentiate, solve f'(x)=0, and justify max/min using sign diagrams or the second derivative test. Justification: Use the first derivative test (sign change of f'(x)) or the second derivative test (sign of f''(x)) to confirm stationary points as local maxima or minima.

Kinematics relates displacement ($s$), velocity ($v$), and acceleration ($a$) via differentiation and integration. Differentiating displacement gives velocity; differentiating velocity gives acceleration.

Integration by substitution is the reverse of the chain rule, used when an integrand contains a composite function and the derivative of its inner function. The process involves setting $u$ equal to the inner function, differentiating to find $du$, substituting into the integral to eliminate $x$, integrating with respect to $u$, and finally back-substituting to express the answer in terms of $x$.

L'Hôpital's rule applies to indeterminate forms $\frac{0}{0}$ or $\frac{\infty}{\infty}$. Other indeterminate forms ($0 \times \infty$, $\infty - \infty$, $1^\infty$, $0^0$, $\infty^0$) must be algebraically manipulated into a suitable form.

Implicit differentiation is used for equations where $y$ is not explicitly defined as a function of $x$. When differentiating a term with $y$ with respect to $x$, always multiply by $\frac{dy}{dx}$ (Chain Rule).

Related rates problems involve finding the rate of change of one quantity in terms of the rate of change of another quantity. The Chain Rule is fundamental: $\frac{dy}{dt} = \frac{dy}{dx} \times \frac{dx}{dt}$.

Optimization problems involve finding maximum/minimum values of a quantity. Steps: Diagram, formulate objective function (in one variable), define domain, find derivative, find stationary points (f'(x)=0), test nature of stationary points (sign diagram or 2nd derivative test), check endpoints if domain is closed.

Derivatives: Includes standard forms for exponential functions ($e^x, a^x$), logarithmic functions ($\ln x, \log_a x$), reciprocal trigonometric functions ($\sec x, \csc x, \cot x$), and inverse trigonometric functions ($\arcsin x, \arccos x, \arctan x$). Chain rule applications are essential for composite functions. Integrals: Covers integrals of exponential and logarithmic forms, including those with linear arguments. A key pattern is recognizing when the numerator is the derivative of the denominator for logarithmic results.

Partial fractions are used to integrate rational functions where direct substitution fails. First, ensure the fraction is proper (degree of numerator < degree of denominator) using polynomial long division if necessary.

Integration by parts uses the product rule in reverse to integrate products of functions. The formula is $\int u \frac{dv}{dx} dx = uv - \int v \frac{du}{dx} dx$.

Calculate the area between curves by integrating the difference between the upper and lower functions with respect to x or y. If limits are not provided, find them by solving for intersection points.

A first-order differential equation is separable if it can be written as $\frac{dy}{dx} = f(x)g(y)$ or $\frac{dy}{dx} = \frac{f(x)}{h(y)}$. The method involves separating variables: $\int \frac{1}{g(y)} dy = \int f(x) dx$.

A first-order differential equation is homogeneous if it can be expressed as $\frac{dy}{dx} = f\left(\frac{y}{x}\right)$. Use the substitution $y = vx$, which implies $\frac{dy}{dx} = v + x \frac{dv}{dx}$.

The integrating factor method solves first-order linear differential equations of the form $\frac{dy}{dx} + P(x)y = Q(x)$. The integrating factor is calculated as $I(x) = e^{\int P(x) dx}$. A constant of integration is not needed here.

The Maclaurin series is a Taylor series expansion of a function $f(x)$ centered at $a=0$. It expresses $f(x)$ as an infinite sum of terms involving its derivatives evaluated at $x=0$.

Euler's method approximates solutions to first-order differential equations $\frac{dy}{dx} = f(x, y)$ with an initial condition $(x_0, y_0)$. It uses an iterative formula: $y_{n+1} = y_n + h \times f(x_n, y_n)$, where $h$ is the step size.