Analysis
Problems related to calculus, limits, series, etc.
Problems in this category
a. Find the equation of the tangent plane to $z = \ln(2x+y)$ at the point $(1, 2)$. b. Let $\sum_{x=...
Multivariable CalculusPartial DerivativesTangent PlaneInfinite Seriesp-seriesConvergence TestsIntegral Test
2025/4/21
We are asked to solve the following problems: a. Find the equation of the tangent plane to $z = \ln(...
Tangent PlaneSeries ConvergencePartial DerivativesRatio TestIntegrationMidpoint RuleSimpson's RuleRadius of ConvergenceMultivariable Calculus
2025/4/21
The problem asks to differentiate the function $y = (2x+1)^{-1}$.
DifferentiationChain RuleQuotient RuleCalculus
2025/4/20
We are asked to evaluate the limit: $\lim_{x\to\infty} \frac{e^x + 2x - 1}{2e^x + 1}$
LimitsL'Hopital's RuleExponential Functions
2025/4/20
The question asks how the domains of the pieces of a piecewise-defined function relate to the entire...
Piecewise FunctionsDomainFunction Definition
2025/4/18
Given the function $f(x) = \frac{x^2 + 2x + 1}{x-1}$, we are asked to: 1) Find real numbers $a$, $b$...
LimitsDerivativesTable of VariationsRational Functions
2025/4/17
The problem asks us to analyze the function $f(x) = 5 \cdot (\frac{1}{2})^x$ and graph it.
Exponential FunctionsFunction AnalysisGraphing
2025/4/17
We are given two functions: $f(x) = x + 2 + \frac{4}{x-1}$ and $g(x) = x \ln x$. We need to study th...
CalculusDerivativesFunction AnalysisIncreasing/Decreasing FunctionsCritical Points
2025/4/16
We need to evaluate the definite integral of $\frac{1}{1+x^{60}}$ from $0$ to $\infty$. That is, we ...
Definite IntegralIntegrationCalculusSpecial Functions
2025/4/16
We are asked to evaluate the limits: (1) $ \lim_{x \to 0} (\frac{1}{x} \cdot \sin x) $ (2) $ \lim_{x...
LimitsTrigonometryL'Hopital's RuleTaylor Series
2025/4/15