∫ Calculus

104 topics

• Idea of a Limit (Advanced) – The value a function approaches as the input approaches a point.
• One-Sided Limits (Advanced) – Left and right limits must agree for the limit to exist.
• Limit Laws (Advanced) – Limits of sums, products and quotients are computed from the parts.
• Factoring and Rationalizing Limits (Advanced) – Remove 0/0 indeterminacy with algebra.
• Limits at Infinity (Advanced) – End behaviour of functions.
• Infinite Limits and Asymptotes (Advanced) – Function values grow without bound near a vertical asymptote.
• Squeeze Theorem (Advanced) – If g ≤ f ≤ h and g, h → L then f → L.
• Continuity (Advanced) – No breaks, jumps or holes: limit equals function value.
• Types of Discontinuity (Advanced) – Removable, jump and infinite discontinuities.
• Intermediate Value Theorem (Advanced) – A continuous function takes every value between f(a) and f(b).
• Average Rate of Change (Advanced) – Slope of a secant line.
• Definition of the Derivative (Advanced) – Instantaneous rate of change as a limit of secant slopes.
• Tangent Lines (Advanced) – The line touching a curve with slope equal to the derivative.
• Derivative as Velocity (Advanced) – Derivative of position is velocity; of velocity is acceleration.
• Power Rule (Advanced) – Derivative of xⁿ.
• Constant and Sum Rules (Advanced) – Derivative of constant is 0; derivatives add.
• Product Rule (Advanced) – Derivative of a product.
• Quotient Rule (Advanced) – Derivative of a quotient.
• Chain Rule (Advanced) – Derivative of a composite function.
• Derivatives of Trig Functions (Advanced) – Derivatives of sin, cos, tan.
• Derivative of eˣ (Advanced) – The exponential function is its own derivative.
• Derivative of ln x (Advanced) – Derivative of the natural log.
• Derivatives of Inverse Trig (Expert) – Derivatives of arcsin, arccos, arctan.
• Implicit Differentiation (Expert) – Differentiate both sides of F(x,y)=0 treating y as a function of x.
• Logarithmic Differentiation (Expert) – Take logs before differentiating products and powers.
• Higher-Order Derivatives (Advanced) – Differentiate repeatedly to get f″, f‴.
• Related Rates (Expert) – Link rates of change of connected quantities.
• Linear Approximation (Expert) – Approximate f near a using its tangent line.
• Differentials (Expert) – Use dy = f′(x)dx for small changes.
• Critical Points (Advanced) – Points where f′ = 0 or does not exist.
• Increasing and Decreasing Functions (Advanced) – Sign of f′ tells the direction.
• First Derivative Test (Advanced) – Classify extrema by sign changes of f′.
• Second Derivative Test (Expert) – f″ > 0 means concave up; f″ < 0 concave down.
• Inflection Points (Expert) – Where concavity changes.
• Absolute Extrema (Expert) – Check critical points and endpoints on a closed interval.
• Optimization Problems (Expert) – Find the best value of a quantity using derivatives.
• Mean Value Theorem (Expert) – Some point has tangent slope equal to average slope.
• Rolle’s Theorem (Expert) – If f(a)=f(b) there is a point where f′=0.
• L’Hôpital’s Rule (Expert) – Resolve 0/0 or ∞/∞ with derivatives.
• Curve Sketching (Expert) – Use limits, derivatives and asymptotes to draw graphs.
• Newton’s Method (Expert) – Iteratively find roots using tangent lines.
• Antiderivatives (Advanced) – A function whose derivative is f.
• Indefinite Integral Rules (Advanced) – Power, constant multiple and sum rules.
• Riemann Sums (Advanced) – Approximate area with rectangles.
• Definite Integral (Advanced) – Signed area under a curve.
• Fundamental Theorem of Calculus I (Expert) – Differentiation undoes integration.
• Fundamental Theorem of Calculus II (Expert) – Evaluate integrals using antiderivatives.
• Properties of Definite Integrals (Advanced) – Linearity, additivity and reversal of limits.
• Substitution Rule (Expert) – Reverse of the chain rule.
• Integration by Parts (Expert) – Reverse of the product rule.
• Trigonometric Integrals (Expert) – Integrals of powers of sin and cos.
• Trigonometric Substitution (Expert) – Use x = a sinθ for √(a²−x²).
• Partial Fractions Integration (Expert) – Decompose rational functions then integrate.
• Improper Integrals (Expert) – Integrals with infinite limits or discontinuities.
• Area Between Curves (Advanced) – Integrate the difference of two functions.
• Volumes by Disks and Washers (Expert) – Revolve an area around an axis.
• Volumes by Shells (Expert) – Cylindrical shell method.
• Arc Length (Expert) – Length of a curve.
• Surface Area of Revolution (Expert) – Area of a surface formed by rotating a curve.
• Average Value of a Function (Advanced) – Mean height of a curve over an interval.
• Work and Fluid Force (Expert) – Integrals compute work and hydrostatic force.
• Trapezoid Rule for Integrals (Expert) – Approximate area with trapezoids.
• Simpson’s Rule (Expert) – Approximate area with parabolas.
• Sequences and Limits (Advanced) – Convergence of infinite lists of numbers.
• Infinite Series (Expert) – Sum of infinitely many terms.
• Geometric Series Convergence (Advanced) – Converges iff |r| < 1.
• Divergence Test (Expert) – If aₙ does not tend to 0 the series diverges.
• Integral Test and p-Series (Expert) – Σ 1/nᵖ converges iff p > 1.
• Comparison Tests (Expert) – Compare with known series.
• Ratio and Root Tests (Expert) – Convergence via limit of aₙ₊₁/aₙ.
• Alternating Series Test (Expert) – Alternating decreasing terms tending to 0 converge.
• Absolute vs Conditional Convergence (Expert) – Σ|aₙ| converging is stronger than Σaₙ converging.
• Power Series (Expert) – Series Σ cₙ(x−a)ⁿ with a radius of convergence.
• Taylor Series (Expert) – Represent functions with infinite polynomials.
• Common Maclaurin Series (Expert) – Series for eˣ, sin x, cos x, ln(1+x).
• Taylor’s Remainder (Expert) – Error bound of Taylor approximations.
• Parametric Derivatives (Expert) – dy/dx from x(t), y(t).
• Polar Calculus (Expert) – Area and slope in polar coordinates.
• Vectors and Space Curves (Expert) – Position vectors r(t) and tangent vectors.
• Partial Derivatives (Expert) – Derivatives with respect to one variable of several.
• Gradient (Expert) – Vector pointing in the direction of steepest increase.
• Directional Derivatives (Expert) – Rate of change in any direction.
• Multivariable Chain Rule (Expert) – Derivatives through intermediate variables.
• Tangent Planes (Expert) – Linear approximation of surfaces.
• Lagrange Multipliers (Expert) – Optimise with constraints.
• Multivariable Extrema (Expert) – Use the Hessian to classify critical points.
• Double Integrals (Expert) – Volume under a surface over a region.
• Double Integrals in Polar (Expert) – Use dA = r dr dθ.
• Triple Integrals (Expert) – Integrate over a 3D region.
• Cylindrical and Spherical (Expert) – Coordinate systems for 3D integration.
• Change of Variables and Jacobian (Expert) – Transform integrals with the Jacobian determinant.
• Vector Fields (Expert) – Assign a vector to every point in space.
• Line Integrals (Expert) – Integrate along a curve.
• Conservative Fields (Expert) – Fields with potential functions; path independent.
• Green’s Theorem (Expert) – Relates line integrals and double integrals.
• Divergence and Curl (Expert) – Measures of source strength and rotation.
• Surface Integrals (Expert) – Integrate over a surface.
• Stokes’ Theorem (Expert) – Relates surface integral of curl to line integral.
• Divergence Theorem (Expert) – Flux through a closed surface equals integral of divergence.
• Differentiating Under the Integral (Expert) – Swap derivative and integral under conditions.
• Gamma Function (Expert) – Extends factorials to real numbers.
• Beta Function (Expert) – Related to Gamma.
• Gaussian Integral (Expert) – Area under the bell curve.
• Calculus of Variations Intro (Expert) – Find functions that optimise integrals; Euler–Lagrange equation.