Calculus

• What Happens When Infinity Is Multiplied by an Infinitesimal?
  Infinity times an infinitesimal has no single value: in limits the product may tend to zero, a finite number, or infinity.
• How to Integrate cos(1/x)
  The antiderivative of cos(1/x) is x cos(1/x) + Si(1/x) + C, where Si is the sine integral special function.
• How to Integrate This Double Integral in Polar Coordinates
  Convert the circular region to polar bounds, include the Jacobian, and evaluate the double integral to obtain π/16.
• How to Use the Tabular Method for Integration by Parts
  The tabular method organizes repeated integration by parts into derivative and antiderivative columns, with alternating signs and a clear stopping rule.
• How to Solve Integrals Using Series
  Use a power series to replace a difficult integrand with simpler terms, then integrate term by term after checking convergence on the interval.
• How to Differentiate f(x) = x√(2x − 3)
  Differentiate x√(2x − 3) using the product and chain rules, simplify the result, and see where the derivative is defined.
• How to Calculate an Ellipsoid’s Volume with a Triple Integral
  Set up a triple integral for an ellipsoid, use scaled spherical coordinates and the Jacobian, and derive its volume from the semiaxes.
• How to Find the Gradient Using Differentiation
  Find the gradient of a curve by differentiating its function and evaluating the derivative at the required point; see a worked example.
• How to Find a Derivative Using the Difference Quotient
  Use the difference quotient [f(x+h) − f(x)]/h, simplify before taking its limit as h approaches zero, and interpret the result as the derivative.
• How to Prove That a Function Is Riemann Integrable
  Prove Riemann integrability by showing upper and lower sums can be made arbitrarily close, or use a standard sufficient condition.
• Expressions for sin(arctan x) and cos(arctan x) Without Trig Functions
  Using the principal range of arctan, sin(arctan x) = x/√(1+x²) and cos(arctan x) = 1/√(1+x²) for every real x.
• How to Find the Rates of Change of a Rectangle’s Area, Perimeter, and Diagonal
  Find the instantaneous rates for a rectangle with changing sides: area rate, perimeter rate, and the common rate of its two equal diagonals.
• How to Solve a First-Order ODE with an Integrating Factor
  Solve a first-order linear differential equation by finding its integrating factor, turning the left side into a product derivative, and integrating.
• How to Find the Area in a Ladder Related-Rates Problem
  For a sliding ladder, model the wall, floor, and ladder as a right triangle, then differentiate its area and the fixed-length constraint.
• Can You Find a Function from Its Tangent Line?
  A tangent line fixes a function's value and derivative at one point, but usually does not determine the function without extra conditions.
• How to Find the Acute Angle Between Two Curves at Their First Intersection
  Find where two curves first intersect, calculate their tangent slopes there, and use the tangent-line formula to determine their acute angle.
• Evaluate the Integral of the Piecewise Function from −2 to 4
  Split the integral at the piecewise breakpoint, evaluate each part, and add the signed results to obtain −4/3.
• Examples Showing When Rolle’s Theorem Does Not Apply
  Rolle’s Theorem does not fail when its hypotheses hold. These examples show what can happen when continuity, differentiability, or equal endpoint values are missing.
• What Do You Get When You Differentiate a Function Twice?
  Differentiating twice gives the second derivative in one variable and the Hessian matrix for a scalar function of several variables.
• How to Differentiate a Piecewise Function
  Differentiate each formula on its interval, then test every breakpoint with continuity and matching one-sided derivatives to determine whether the derivative exists.
• Does $f(0)=0$ Imply That $f'(x)=0$?
  Knowing that f(0)=0 tells you only that the graph passes through the origin; it does not determine whether f' exists or what its value is.
• How to Prove a Graph Has No Horizontal Tangent Lines
  To prove a differentiable graph has no horizontal tangents, show its derivative is never zero on the relevant domain; the function is needed for a specific proof.
• What Quantity Is Conserved in the Lotka–Volterra System?
  For the standard Lotka–Volterra predator–prey equations, a conserved quantity is d x − c ln x + b y − a ln y; here is its derivation.
• How Do You Convert a Discrete Function Into a Continuous One?
  A discrete function has no unique continuous conversion. Learn how to build a continuous extension with linear interpolation and when other methods are needed.
• How to Differentiate the Cross Product of Vector Functions
  Differentiate a cross product of vector-valued functions with the product rule: u' × v + u × v', with a worked component-by-component example.
• How to Calculate the Limit of a Vector
  A vector limit is found component by component: compute each scalar limit, then combine the results, provided every component limit exists.
• How Do You Find the Slope of a Function?
  Find a function’s slope by differentiating it, then evaluate the derivative at the input you care about. The result is the tangent-line slope.
• How to Solve Calculus Word Problems
  Learn a reliable method for calculus word problems: identify the goal, define variables, model the situation, differentiate or integrate, and check units.
• Can a Function Have First and Second Derivatives Equal to Zero?
  Yes. For example, f(x) = x^3 has f'(0) = 0 and f''(0) = 0; the same conditions can also occur at a local minimum, as with x^4.
• What Is the Graph of max(|x|, |y|, |z|) = 1?
  The equation max(|x|, |y|, |z|) = 1 describes the six square faces on the boundary of a cube centered at the origin.
• What Is the Area Bounded by $r=|\sin\theta|$?
  For the polar curve r = |sin θ|, integrating over 0 ≤ θ ≤ 2π gives total area π/2; each lobe encloses π/4.
• What Is the Laplace Transform of $t e^{at}\sin(bt)$?
  The Laplace transform of t e^{at} sin(bt) is 2b(s−a)/([(s−a)^2+b^2]^2), for real a, nonzero b, and Re(s)>a.
• How to Differentiate a Double Summation Using Dot Notation
  For a finite double sum with fixed limits, differentiate each term; dot notation means a derivative with respect to time, not a special summation rule.
• What Is the Derivative of the Sigmoid Function?
  For the logistic sigmoid σ(x)=1/(1+e^−x), the derivative is σ(x)(1−σ(x)); see the derivation, chain rule, and worked examples.
• How Do You Differentiate a Double Summation?
  Differentiate finite double sums term by term; for infinite sums, check uniform convergence of the derivative series before interchanging differentiation and summation.
• How Do Limits Resolve Zeno’s Achilles Paradox?
  Zeno’s Achilles paradox is resolved mathematically by showing that infinitely many shrinking time intervals sum to a finite limit, when motion is modeled continuously.
• How to Minimize a Sum-of-Squares Error Function
  Minimize a sum-of-squares error by setting its gradient to zero; for linear models, solve the least-squares equations and check uniqueness.
• Proof of Frullani’s Theorem
  A proof of Frullani’s theorem using differentiation under a scale parameter, with sufficient convergence conditions and a worked rational-function example.
• How to Find the Derivative of the Logistic Function
  Differentiate the logistic sigmoid using the chain rule, obtain σ′(x) = σ(x)(1 − σ(x)), and adapt the result to scaled logistic functions.
• How to Find Horizontal and Vertical Asymptotes Using Limits
  Horizontal asymptotes come from limits as x → ±∞; vertical asymptotes from infinite one-sided limits at a point. Step-by-step method with worked examples.