Analysis
Problems related to calculus, limits, series, etc.
Problems in this category
The problem defines a sequence $(u_n)$ with the initial term $u_0 = 1$ and the recursive formula $u_...
SequencesLimitsArithmetic SequencesRecursive Formula
2025/4/14
We are given a function $f(x)$ defined piecewise as: $f(x) = x + \sqrt{1-x^2}$ for $x \in [-1, 1]$ $...
FunctionsDomainContinuityDifferentiabilityDerivativesVariation TableCurve Sketching
2025/4/14
We are given two sequences $(U_n)$ and $(V_n)$ defined by the following relations: $U_0 = -\frac{3}{...
SequencesGeometric SequencesConvergenceSeries
2025/4/14
We are given a sequence $(U_n)_{n \in \mathbb{N}}$ defined by $U_0 = 1$ and $U_{n+1} = \frac{1}{2} U...
SequencesSeriesGeometric SequencesConvergenceLimits
2025/4/14
We are given a sequence $(U_n)_{n \in N}$ defined by $U_0 = 7$ and $U_{n+1} = \frac{1}{2}(U_n + 5)$....
SequencesSeriesGeometric SequencesConvergenceBoundedness
2025/4/14
The problem asks us to determine the derivative of the function $y = \cos x$.
CalculusDifferentiationTrigonometryDerivatives
2025/4/14
We need to evaluate the definite integral: $\int_{-2}^{3} \frac{(x-2)(6x^2 - x - 2)}{(2x+1)} dx$.
Definite IntegralIntegrationPolynomialsCalculus
2025/4/13
The problem states: If $(x+1)f'(x) = f(x) + \frac{1}{x}$, then $f'(\frac{1}{2}) = ?$
Differential EquationsIntegrationPartial Fraction DecompositionCalculus
2025/4/13
The problem asks us to find the value of $l$ if $\int x^2 \, dx = lx + \frac{x^3}{3} + c$, where $c$...
IntegrationDefinite IntegralsCalculusPower Rule
2025/4/13
We are given a set of questions about functions and continuity. We need to find the answers to these...
ContinuityLimitsTrigonometric FunctionsCalculus
2025/4/13