148 questions · 31 question types identified
A question is this type if and only if it asks to find a limit as x→0 or x→∞ using series expansions or l'Hôpital's rule.
The differential equation is given in explicit form dy/dx = f(x,y) or d²y/dx² = f(x,y,dy/dx), and successive derivatives are found by direct differentiation and substitution.
Questions that require multiplying two known Maclaurin series directly (e.g., e^x cos x, (1+x) sin x, e^x ln(1+3x), sin x sin 2x sin 4x).
Finding Maclaurin series for logarithmic functions involving trigonometric expressions such as ln(cos x), ln(cos ax), ln(sin x + cos x), ln(1 + sin x), or ln(1 + tan x). These require using trigonometric derivatives and often involve finding non-zero terms.
Questions that ask to write down or find a Maclaurin series by directly substituting an expression (like x², 2x, or x³) into a single standard series (e^x, ln(1+x), sin x, cos x, etc.).
A question is this type if and only if it asks for a series expansion of a logarithmic function (ln of some expression) by differentiation or standard results.
A question is this type if and only if it asks to write down or deduce a series for a related function (e.g., ln(1-x) from ln(1+x)) without full derivation.
Questions asking to find Maclaurin series by differentiation for functions involving roots or exponentials of trigonometric/exponential expressions, such as √(a+e^x), √(a+sin x), or e^(sin x).
The differential equation is given in implicit form (e.g., y·d²y/dx² + (dy/dx)² + y = 0 or (1+x²)·d²y/dx² + x·dy/dx = y), requiring algebraic manipulation to find successive derivatives.
Questions asking for Taylor series expansion about x = π/4, typically involving tan x, sec²x, or cos 2x.
Finding Maclaurin series for logarithmic functions of the form ln(a+bx) or ln(1/(1-bx)) using differentiation. These are straightforward applications where successive derivatives follow a clear pattern.
A question is this type if and only if it asks about the error in a series approximation, percentage error, or range of validity of a series.
A question is this type if and only if it gives partial series information and asks which function or what constants match that series.
Questions where the base function is exponential, trigonometric, or their products (e.g., e^x sin x, e^(cos²x), tan²x), requiring standard differentiation rules to establish the derivative relationship.
Questions where the base function is an inverse trigonometric or inverse hyperbolic function (e.g., tan⁻¹(2x), sin⁻¹(2x), sech⁻¹(x+1/2), tanh⁻¹(x)), requiring implicit differentiation or chain rule with inverse function derivatives.
Questions asking for Taylor series expansion about x = π/3 or x = π/6, typically involving sin, cos, tan, sec, or cot functions.
A question is this type if and only if it asks to use a series expansion to find an approximate value for a definite integral.
Questions involving hyperbolic functions raised to powers (e.g., cosh^n x, sinh x sin x) where the derivative relationship involves recurrence relations or products of hyperbolic and trigonometric functions.
A question is this type if and only if it asks for a Maclaurin series (expansion about x = 0) of a function that has been horizontally shifted, such as f(x + a).
Questions that require substituting one Maclaurin series into another standard series (e.g., e^(e^x - 1), ln(1 + sin x), cosh(ix)) involving nested functions.
Questions that require adding, subtracting, or using algebraic identities to combine multiple standard series or derive one series from another (e.g., finding cos²x from cos 2x, finding sin²x from cos 2x, or adding e^(1+x²) + e^(1-x)).
Finding Maclaurin series for logarithmic functions involving exponential expressions such as ln(e^x + c), ln(e^(ax) cos x), or ln(1 + e^x). These often involve simplifying using logarithm properties and may combine exponential and trigonometric derivatives.
A question is this type if and only if it asks to substitute a specific value into a series to approximate a number (not an integral).
A question is this type if and only if it requires combining a binomial expansion with an exponential series to find terms of a product.
Questions asking to find Maclaurin series by differentiation for inverse trigonometric functions such as arcsin(x), arctan(x), or arccos(x).
A question is this type if and only if it asks for a Taylor series expansion about the point x = 1, typically involving logarithmic or algebraic functions.
Questions asking to find Maclaurin series by differentiation for inverse hyperbolic functions such as tanh^(-1)(x).
Questions asking to find Maclaurin series by differentiation for products like e^x·tan(x) or sec(x) where the function structure requires product/quotient rule techniques.
Questions that provide a differential equation relating derivatives (e.g., (x²+1)y'' = 2y² + (1-2x)y') and ask to derive higher-order relationships by differentiating the equation itself, then find the series.
Questions that require finding the series for 1/f(x) by multiplying the reciprocal series with the original series to get 1 (e.g., sec x from cos x).
Questions that require finding series for functions of series, such as (ln(1+x))^2 or ln(e^(2x) + e^(-2x)), where the outer function must be applied to an inner series.