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Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
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📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
The Maclaurin series for a composite function f(g(x)) is
Basically, we replace xn with [g(x)]n.
We can add or subtract Maclaurin polynomials by adding or subtracting the coefficients of each power of x from the original polynomials.
We can integrate a function by integrating its Maclaurin polynomial term by term.
We can differentiate a Maclaurin series using the power rule on each term.
We can find the first few terms in the Maclaurin Series for a product f(x)×g(x) by multiplying out the first few terms in the Maclaurin Series for f and g.
We can perform "division" of Maclaurin series by assuming that the result of the division is a series with
where an​ are the coefficients of the nth power term. Then, we multiply g(x)f(x)​ and a0​+a1​x+a2​x2+... by g(x). Finally, we solve for the unknown coefficients using the Maclaurin series of f(x).
Certain limits may be evaluated with L’Hôpital's rule but require difficult or repeated differentiation. Sometimes, expanding a function within the limit using its Maclaurin series and simplifying is an easier method for evaluating these limits.
This kind of simplification often works because we know that after a certain point, every term will tend to 0, allowing us to focus on the few terms that do not.
Maclaurin series allow us to approximate arbitrary functions as polynomials.
The Maclaurin series for a function f is given by
In summation form:
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The Maclaurin series for ex is
The Maclaurin series for sinx is
The Maclaurin series for cosx is
The Maclaurin series for ln(x+1) is
The Maclaurin series for arctanx is
The binomial theorem can be extended to expansions with rational exponents (n∈Q):