taylor

Find the Maclaurin series expansions of the exponential, sine, and cosine functions up to the fifth order.

syms x
T1 = taylor(exp(x))

T1 = 
$$\frac{x^5}{120}+\frac{x^4}{24}+\frac{x^3}{6}+\frac{x^2}{2}+x+1$$

T2 = taylor(sin(x))

T2 = 
$$\frac{x^5}{120}-\frac{x^3}{6}+x$$

T3 = taylor(cos(x))

T3 = 
$$\frac{x^4}{24}-\frac{x^2}{2}+1$$

You can use the sympref function to modify the output order of symbolic polynomials. Redisplay the polynomials in ascending order.

sympref("PolynomialDisplayStyle","ascend");
T1

T1 = 
$$1+x+\frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{24}+\frac{x^5}{120}$$

T2

T2 = 
$$x-\frac{x^3}{6}+\frac{x^5}{120}$$

T3

T3 = 
$$1-\frac{x^2}{2}+\frac{x^4}{24}$$

The display format you set using sympref persists through your current and future MATLAB® sessions. Restore the default value by specifying the "default" option.

sympref("default");

Find the Taylor series expansions at $\mathit{x}=1$ for these functions. The default expansion point is 0. To specify a different expansion point, use ExpansionPoint.

syms x
T = taylor(log(x),x,ExpansionPoint=1)

T = 
$$x-\frac{{\left(x-1\right)}^2}{2}+\frac{{\left(x-1\right)}^3}{3}-\frac{{\left(x-1\right)}^4}{4}+\frac{{\left(x-1\right)}^5}{5}-1$$

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

T = taylor(acot(x),x,1)

T = 
$$\frac{\pi }{4}-\frac{x}{2}+\frac{{\left(x-1\right)}^2}{4}-\frac{{\left(x-1\right)}^3}{12}+\frac{{\left(x-1\right)}^5}{40}+\frac{1}{2}$$

Find the Maclaurin series expansion for f = sin(x)/x. The default truncation order is 6. The Taylor series approximation of this expression does not have a fifth-degree term, so taylor approximates this expression with the fourth-degree polynomial.

syms x
f = sin(x)/x;
T6 = taylor(f,x);

Use Order to control the truncation order. For example, approximate the same expression up to the orders 7 and 9.

T8 = taylor(f,x,Order=8);
T10 = taylor(f,x,Order=10);

Plot the original expression f and its approximations T6, T8, and T10. Note how the accuracy of the approximation depends on the truncation order.

fplot([T6 T8 T10 f])
xlim([-4 4])
grid on
legend("approximation of sin(x)/x with error O(x^6)", ...
       "approximation of sin(x)/x with error O(x^8)", ...
       "approximation of sin(x)/x with error O(x^{10})", ...
       "sin(x)/x",Location="Best")
title("Taylor Series Expansion")

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

syms x
T = taylor(1/exp(x) - exp(x) + 2*x,x,Order=5)

T = 
$$-\frac{x^3}{3}$$

Find the Taylor series expansion with a relative truncation order by using OrderMode. For some expressions, a relative truncation order provides more accurate approximations.

T = taylor(1/exp(x) - exp(x) + 2*x,x,Order=5,OrderMode="relative")

T = 
$$-\frac{x^7}{2520}-\frac{x^5}{60}-\frac{x^3}{3}$$

Find the Maclaurin series expansion of this multivariate expression. 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);
T = taylor(f)

T = 
$$\frac{x^5}{120}-\frac{x^3}{6}+x+\cos \left(y\right)+{\mathrm{e}}^z$$

Find the multivariate Maclaurin series expansion by specifying the vector of variables.

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

T = 
$$\frac{x^5}{120}-\frac{x^3}{6}+x+\frac{y^4}{24}-\frac{y^2}{2}+\frac{z^5}{120}+\frac{z^4}{24}+\frac{z^3}{6}+\frac{z^2}{2}+z+2$$

You can use the sympref function to modify the output order of a symbolic polynomial. Redisplay the polynomial in ascending order.

sympref("PolynomialDisplayStyle","ascend");
T

T = 
$$2+z+\frac{z^2}{2}+\frac{z^3}{6}+\frac{z^4}{24}+\frac{z^5}{120}-\frac{y^2}{2}+\frac{y^4}{24}+x-\frac{x^3}{6}+\frac{x^5}{120}$$

The display format you set using sympref persists through your current and future MATLAB sessions. Restore the default value by specifying the "default" option.

sympref("default");

Find the multivariate Taylor series 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);
T = taylor(f,[x y],[1 1],Order=3)

T = 
$$x+\frac{{\left(x-1\right)}^2}{2}+\frac{{\left(y-1\right)}^2}{2}$$

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.

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

T = 
$$x+\frac{{\left(x-1\right)}^2}{2}+\frac{{\left(y-1\right)}^2}{2}$$

Find the error estimate when approximating a function $f\left(x\right)=\log (x+1)$ using the Taylor series expansion. Here, consider the Taylor approximation up to the 7th order (with the truncation order $\mathit{n}=8$) at the expansion point $\mathit{a}=0$.

The error or remainder in the Taylor approximation is given by the Lagrange form:

The upper bound of the error estimate can be calculated by finding a positive real number $\mathit{M}$ such that $\left|{\mathit{f}}^{\mathit{n}}\left(\mathit{c}\right)\right|\le \mathit{M}$ for all $\mathit{c}$ between $\mathit{a}$ and $\mathit{x}$.

Find the Taylor series expansion of the function $f\left(x\right)=\log (x+1)$ up to the 7th order by specifying Order as 8.

syms x
f = log(x+1)

f = $$\log \left(x+1\right)$$

T = taylor(f,Order=8)

T = 
$$\frac{x^7}{7}-\frac{x^6}{6}+\frac{x^5}{5}-\frac{x^4}{4}+\frac{x^3}{3}-\frac{x^2}{2}+x$$

To estimate the error in the Taylor approximation, first compute the term ${\mathit{f}}^8\left(\mathit{c}\right)$.

syms c
fn(c) = subs(diff(f,8),x,c)

fn(c) = 
$$-\frac{5040}{{\left(c+1\right)}^8}$$

For positive values of $\mathit{x}$, the upper bound of the error estimate can be calculated by using the relation $\left|{\mathit{f}}^8\left(\mathit{c}\right)\right|\le 5040$ (because $\mathit{c}$ must be a positive value between $0$ and a positive $\mathit{x}$). Next, find the upper bound of the error estimate Rupper(x) by using the Lagrange from ${\mathit{R}}_7\left(\mathit{x}\right)$ and the relation $\left|{\mathit{f}}^8\left(\mathit{c}\right)\right|\le 5040$.

Rupper(x) = 5040*x^8/factorial(8)

Rupper(x) = 
$$\frac{x^8}{8}$$

Evaluate the Taylor series expansion at the point $\mathit{x}=0.5$. Find the upper bound of the error estimate in the Taylor approximation.

Teval = subs(T,x,0.5)

Teval = 
$$\frac{909}{2240}$$

Rmax = double(Rupper(0.5))

Rmax = 
4.8828e-04

For comparison, evaluate the exact function at $\mathit{x}=0.5$ and find the remainder in the Taylor approximation.

feval = subs(f,x,0.5)

feval = 
$$\log \left(\frac{3}{2}\right)$$

R = double(abs(feval-Teval))

R = 
3.3846e-04