Find First Three Terms of Maclaurin Series Calculator

Enter function type, coefficient, and x value. Get terms, estimate, data table, and chart instantly. Learn expansions through formulas, examples, exports, and clear steps.

Calculator

Function family

Coefficient a

Evaluate at x

Graph start

Graph end

Decimal places

Plotly Graph

The chart compares the actual function with the three-term Maclaurin approximation over your selected interval.

Example Data Table

Function	a	First three terms	x	Approximate value
e^(ax)	1	1 + x + x^2/2	0.2	1.22
sin(ax)	1	x - x^3/6 + x^5/120	0.5	0.479427
cos(ax)	1	1 - x^2/2 + x^4/24	0.5	0.877604
ln(1 + ax)	1	x - x^2/2 + x^3/3	0.3	0.262
1 / (1 - ax)	1	1 + x + x^2	0.4	1.56

Formula Used

The Maclaurin series expands a function around x = 0. A three-term approximation uses the first three nonzero terms. For many functions, the pattern can be written as:

f(x) ≈ f(0) + f'(0)x + f''(0)x^2 / 2!

Some functions, such as sin(ax) or arctan(ax), do not have a constant term. In those cases, the calculator uses the first three nonzero terms, such as x, x^3, and x^5 terms. That approach matches the standard way series are written in calculus.

Function family	First three terms
e^(ax)	1 + ax + (a^2x^2)/2!
sin(ax)	ax - (a^3x^3)/3! + (a^5x^5)/5!
cos(ax)	1 - (a^2x^2)/2! + (a^4x^4)/4!
ln(1 + ax)	ax - (a^2x^2)/2 + (a^3x^3)/3
1 / (1 - ax)	1 + ax + a^2x^2
1 / (1 + ax)	1 - ax + a^2x^2
arctan(ax)	ax - (a^3x^3)/3 + (a^5x^5)/5
sinh(ax)	ax + (a^3x^3)/3! + (a^5x^5)/5!
cosh(ax)	1 + (a^2x^2)/2! + (a^4x^4)/4!

How to Use This Calculator

Choose a supported function family from the list.
Enter the coefficient a used inside the function.
Enter the x value where you want the estimate.
Set the graph start and graph end values.
Select how many decimal places you want displayed.
Press the calculate button to show the result above the form.
Review the polynomial, approximation, actual value, and error.
Use the CSV or PDF buttons to save the output.

FAQs

1. What does this calculator return?

It returns the first three nonzero Maclaurin terms for a selected function family, then evaluates that polynomial at your chosen x value and compares it with the actual function.

2. Why are some terms odd powers only?

Odd functions like sin(ax), sinh(ax), and arctan(ax) have zero coefficients on even powers around zero. Their Maclaurin expansions naturally keep only odd powers.

3. Why are some terms even powers only?

Even functions like cos(ax) and cosh(ax) are symmetric around zero. Their derivatives make the odd-power coefficients vanish, so only even powers remain.

4. Is the approximation always accurate?

No. Accuracy is usually strongest near x = 0. As x moves farther away, the truncated series may deviate more from the actual function.

5. What does the coefficient a change?

The coefficient changes the rate of growth or oscillation inside the function. It also changes the coefficients of the Maclaurin terms shown in the polynomial.

6. Why can ln(1 + ax) fail for some inputs?

The natural log needs 1 + ax to stay positive for real outputs. The series also behaves best near zero and typically within its convergence interval.

7. What does the graph show?

The graph plots the actual function and the three-term approximation over your selected interval. It helps you see where the series tracks well and where error grows.

8. Can I save my result?

Yes. After calculation, you can download a CSV summary, generate a PDF report, or print the result directly from the page.