taylor

Description

taylor(f) computes the Taylor series expansion of f up to the fifth order. The expansion point is 0.

taylor(f,Name,Value) uses additional options specified by one or more Name,Value pair arguments.

taylor(f,v) computes the Taylor series expansion of f with respect to v.

taylor(f,v,Name,Value) uses additional options specified by one or more Name,Value pair arguments.

taylor(f,v,a) computes the Taylor series expansion of f with respect to v around the expansion point a.

taylor(f,v,a,Name,Value) uses additional options specified by one or more Name,Value pair arguments.

Input Arguments

Name-Value Pair Arguments

Examples

Compute the Maclaurin series expansions of these functions:

syms x
taylor(exp(x))
taylor(sin(x))
taylor(cos(x))

ans =
x^5/120 + x^4/24 + x^3/6 + x^2/2 + x + 1
 
ans =
x^5/120 - x^3/6 + x
 
ans =
x^4/24 - x^2/2 + 1

Compute the Taylor series expansions around x = 1 for these functions. The default expansion point is 0. To specify a different expansion point, use ExpansionPoint:

syms x
taylor(log(x), x, 'ExpansionPoint', 1)

ans =
x - (x - 1)^2/2 + (x - 1)^3/3 - (x - 1)^4/4 + (x - 1)^5/5 - 1

Alternatively, specify the expansion point as the third argument of taylor:

taylor(acot(x), x, 1)

ans =
pi/4 - x/2 + (x - 1)^2/4 - (x - 1)^3/12 + (x - 1)^5/40 + 1/2

Compute the Maclaurin series expansion for this function. The default truncation order is 6. Taylor series approximation of this function does not have a fifth-degree term, so taylor approximates this function with the fourth-degree polynomial:

syms x
f = sin(x)/x;
t6 = taylor(f)

t6 =
x^4/120 - x^2/6 + 1

Use Order to control the truncation order. For example, approximate the function up to the orders 8 and 10:

t8 = taylor(f, 'Order', 8)
t10 = taylor(f, 'Order', 10)

t8 =
- x^6/5040 + x^4/120 - x^2/6 + 1
 
t10 =
x^8/362880 - x^6/5040 + x^4/120 - x^2/6 + 1

Plot the original function f and its approximations t6, t8, and t10. Note how the accuracy of the approximation depends on the truncation order.

plotT6 = ezplot(t6, [-4, 4]);
hold on
set(plotT6,'Color','red')

plotT8 = ezplot(t8, [-4, 4]);
set(plotT8,'Color','magenta')

plotT10 = ezplot(t10, [-4, 4]);
set(plotT10,'Color','cyan')

plotF = ezplot(f, [-4, 4]);
set(plotF,'Color','blue','LineWidth', 2)

legend('approximation of sin(x)/x up to O(x^6)',...
'approximation of sin(x)/x up to O(x^8)',...
'approximation of sin(x)/x up to O(x^1^0)',...
'sin(x)/x',...
'Location', 'South')

title('Taylor Series Expansion')
hold off

Compute the Taylor series expansion of this expression. By default, taylor uses an absolute order, which is the truncation order of the computed series.

taylor(1/(exp(x)) - exp(x) + 2*x, x, 'Order', 5)

ans =
-x^3/3

To compute the Taylor series expansion with a relative truncation order, use OrderMode. For some expressions, a relative truncation order provides more accurate approximations.

taylor(1/(exp(x)) - exp(x) + 2*x, x, 'Order', 5, 'OrderMode', 'Relative')

ans =
- x^7/2520 - x^5/60 - x^3/3

Compute the Maclaurin series expansion of this multivariate function. If you do not specify the vector of variables, taylor treats f as a function of one independent variable.

syms x y z
f = sin(x) + cos(y) + exp(z);
taylor(f)

ans =
x^5/120 - x^3/6 + x + cos(y) + exp(z)

Compute the multivariate Maclaurin expansion by specifying the vector of variables:

syms x y z
f = sin(x) + cos(y) + exp(z);
taylor(f, [x, y, z])

ans =
x^5/120 - x^3/6 + x + y^4/24 - y^2/2 + z^5/120 + z^4/24 + z^3/6 + z^2/2 + z + 2

Compute the multivariate Taylor expansion by specifying both the vector of variables and the vector of values defining the expansion point:

syms x y
f = y*exp(x - 1) - x*log(y);
taylor(f, [x, y], [1, 1], 'Order', 3)

ans =
x + (x - 1)^2/2 + (y - 1)^2/2

If you specify the expansion point as a scalar a, taylor transforms that scalar into a vector of the same length as the vector of variables. All elements of the expansion vector equal a:

taylor(f, [x, y], 1, 'Order', 3)

ans =
x + (x - 1)^2/2 + (y - 1)^2/2