Analysis
Problems related to calculus, limits, series, etc.
Problems in this category
The problem asks us to determine whether the function $f(x)$ defined as $f(x) = \begin{cases} x+5, &...
Function AnalysisEven FunctionOdd FunctionPiecewise FunctionFunction Properties
2025/5/14
We are given a second-order linear homogeneous differential equation $y'' + 4y = 0$ with boundary co...
Differential EquationsSecond-Order Linear Differential EquationsBoundary Value ProblemTrigonometric Functions
2025/5/14
Let $P_2$ be the space of polynomials of degree at most 2. An inner product on this space is defined...
Inner ProductPolynomialsIntegrationLinear AlgebraOrthogonal Polynomials
2025/5/14
The problem asks to identify the correct definition of the derivative from first principles. The opt...
CalculusDerivativesLimitsFirst Principles
2025/5/14
We are asked to evaluate the limit of the expression $\frac{x-4}{x^2 - 16}$ as $x$ approaches 4.
LimitsAlgebraic ManipulationRational Functions
2025/5/14
We need to evaluate the limit of the function $\frac{x^2 - 4}{x^2 - 16}$ as $x$ approaches 4. That ...
LimitsAlgebraRational FunctionsFactorization
2025/5/14
We are asked to evaluate the following limit: $Q = \lim_{x \to 0} \frac{1 - \cos x \cos 2x \cos 3x \...
LimitsTrigonometrySmall Angle ApproximationSummation
2025/5/14
The function $f(x) = x^2$ is not injective when its domain is the set of all real numbers, but it is...
FunctionsInjective FunctionsDomain and RangeReal Numbers
2025/5/14
The image presents a set of math problems involving limits, probability, complex numbers, integrals,...
LimitsCalculusContinuityTrigonometric LimitsL'Hopital's RuleGrowth Rates
2025/5/14
The problem asks us to determine the volume of a sphere with radius $r$ using integration.
IntegrationVolume CalculationSphereCalculus
2025/5/13