What is a truncated Maclaurin series?

A truncated Maclaurin series is a Maclaurin series that is shortened by stopping at a particular power of x. This provides an approximation of the original function.

How do you find the Maclaurin series for a product of two functions ?

To find the Maclaurin series for a product of two functions : start with the Maclaurin series for the individual functions, put brackets around each series and multiply them together, then collect terms and simplify coefficients . How many terms to use for each Maclaurin series will depend on the highest power of x that you need in your final answer.

How do you find the Maclaurin series for a composite function ?

To find the Maclaurin series for a composite function : start with the Maclaurin series for the basic ' outside function ', substitute the ' inside function ' everywhere x appears, then expand and simplify .

How can you use a function's Maclaurin series to find the Maclaurin series of the functions' derivative ?

The Maclaurin series of a function's derivative can be found by differentiating the Maclaurin series of the original function term by term.

True or False? The Maclaurin series solution to a differential equation always gives you the explicit function of x that corresponds to the solution.

False. The Maclaurin series solution to a differential equation does not necessarily give you the explicit function of x that corresponds to the solution. What it does give you is the exact Maclaurin series of that solution.

How can you increase the accuracy of the Maclaurin series approximation for a differential equation solution?

You can increase the accuracy of the Maclaurin series approximation for a differential equation solution by calculating additional terms of the Maclaurin series for higher powers of x .

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Maclaurin Series (DP IB Analysis & Approaches (AA))

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What is a Maclaurin series?
A Maclaurin series is a way of representing a function as an infinite sum of increasing integer powers of x.
E.g. $\sin x = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \dots$
What is a truncated Maclaurin series?
A truncated Maclaurin series is a Maclaurin series that is shortened by stopping at a particular power of x. This provides an approximation of the original function.
True or False?
A truncated Maclaurin series is always exactly equal to the original function for $x = 0$ and $x = 1$ .
False.
A truncated Maclaurin series is always exactly equal to the original function for $x = 0$ .
What is the general Maclaurin series formula?
The general Maclaurin series formula is $f (x) = f (0) + {x f}^{'} (0) + \frac{x^{2}}{2!} f^{''} (0) + . . .$ .
This is in the exam formula booklet.
What is the Maclaurin series expansion for $e^{x}$ ?
The Maclaurin series expansion for $e^{x}$ is $e^{x} = 1 + x + \frac{x^{2}}{2!} + \dots$ .
This is in the exam formula booklet.
What is the Maclaurin series expansion for $\ln (1 + x)$ ?
The Maclaurin series expansion for $\ln (1 + x)$ is $\ln (1 + x) = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \dots$ .
This is in the exam formula booklet.
What is the Maclaurin series expansion for $\sin x$ ?
The Maclaurin series expansion for $\sin x$ is $\sin x = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \dots$ .
This is in the exam formula booklet.
What is the Maclaurin series expansion for $\cos x$ ?
The Maclaurin series expansion for $\cos x$ is $\cos x = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \dots$ .
This is in the exam formula booklet.
What is the Maclaurin series expansion for $\arctan x$ ?
The Maclaurin series expansion for $\arctan x$ is $\arctan x = x - \frac{x^{3}}{3} + \frac{x^{5}}{5} - \dots$ .
This is in the exam formula booklet.
How do you find the Maclaurin series for a product of two functions?
To find the Maclaurin series for a product of two functions:
- start with the Maclaurin series for the individual functions,
- put brackets around each series and multiply them together,
- then collect terms and simplify coefficients.
How many terms to use for each Maclaurin series will depend on the highest power of x that you need in your final answer.
How do you find the Maclaurin series for a composite function?
To find the Maclaurin series for a composite function:
- start with the Maclaurin series for the basic 'outside function',
- substitute the 'inside function' everywhere x appears,
- then expand and simplify.
How can you use a function's Maclaurin series to find the Maclaurin series of the functions' derivative?
The Maclaurin series of a function's derivative can be found by differentiating the Maclaurin series of the original function term by term.
True or False?
Integrating the Maclaurin series for a derivative $f^{'} (x)$ gives the Maclaurin series for the function $f (x)$ without any additional considerations being required.
False.
Integrating the Maclaurin series for a derivative $f^{'} (x)$ gives the Maclaurin series for the function $f (x)$ , but you must also consider the constant of integration.
What is the connection between Maclaurin series expansions and binomial theorem series expansions?
For a function like ${(1 + x)}^{n}$ , the binomial theorem series expansion is exactly the same as the Maclaurin series expansion for the same function.
If you have a differential equation in the form $\frac{d y}{d x} = g (x, y)$ , what other piece of information do you need to know in order to find the Maclaurin series of the solution to the equation?
If you have a differential equation in the form $\frac{d y}{d x} = g (x, y)$ , in order to find the Maclaurin series of the solution to the equation you also need to know $y (0)$ (i.e., the value of $y$ when $x$ is equal to 0).
How many derivatives do you need to find when solving a differential equation using Maclaurin series?
The number of derivatives you need to find when solving a differential equation using Maclaurin series depends on how many terms of the Maclaurin series you want to find.
For example, for terms up to $x^{4}$ , you need to find derivatives up to $y^{(4)}$ .
(Note: only the complete infinitely-long series gives the exact solution to the differential equation.)
What are the steps for finding the Maclaurin series for the solution to a differential equation?
The steps for finding the Maclaurin series for the solution to a differential equation are:
1. Use implicit differentiation to find expressions for $y^{''}, y^{'''}, y^{(4)}$ , etc., in terms of $x, y$ and lower-order derivatives of $y$ .
2. Use the given initial value for $y (0)$ to find the values of $y^{'} (0), y^{''} (0), y^{'''} (0), y^{(4)} (0)$ , etc., one by one.
3. Put the values found in step 2 into the general Maclaurin series formula $f (x) = f (0) + {x f}^{'} (0) + \frac{x^{2}}{2!} f^{''} (0) + . . .$ (where $f (0) = y (0)$ , $f^{'} (0) = y^{'} (0)$ , etc.)
4. Simplify the coefficients for each of the powers of $x$ in the resultant Maclaurin series.
The general Maclaurin series formula is in the exam formula booklet.
True or False?
A Maclaurin series solution to a differential equation can be used to find approximations of the solution $y = f (x)$ for different values of $x$ , even if the exact solution cannot be found explicitly.
True.
A Maclaurin series solution to a differential equation can be used to find approximations of the solution $y = f (x)$ for different values of $x$ , even if the exact solution cannot be found explicitly.
True or False?
The Maclaurin series solution to a differential equation always gives you the explicit function of x that corresponds to the solution.
False.
The Maclaurin series solution to a differential equation does not necessarily give you the explicit function of x that corresponds to the solution.
What it does give you is the exact Maclaurin series of that solution.
How can you increase the accuracy of the Maclaurin series approximation for a differential equation solution?
You can increase the accuracy of the Maclaurin series approximation for a differential equation solution by calculating additional terms of the Maclaurin series for higher powers of x.

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