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
π = included in formula booklet β’ π« = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
π = included in formula booklet β’ π« = 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
π = included in formula booklet β’ π« = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
π = included in formula booklet β’ π« = not in formula booklet
Maclaurin series allow us to approximate arbitrary functions as polynomials.
The Maclaurin series for a function f is given by
In summation form:
Powered by Desmos
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 (nβQ):
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)Γg(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 anβ are the coefficients of the nth power term. Then, we multiply g(x)f(x)β and a0β+a1βx+a2βx2+... by g(x). Finally, we solve for the unknown coefficients using the Maclaurin series of f(x).
Certain limits may be evaluated with LβHΓ΄pital'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.