Analytical Plotting with Symbolic Math Toolbox

Plot Explicit Functions $$y=f(x)$$ Using fplot

Plot the function $$\sin (\exp (x))$$.

syms x
fplot(sin(exp(x)))

Plot the trigonometric functions $$\sin (x)$$, $$\cos (x)$$, and $$\tan (x)$$ simultaneously.

fplot([sin(x),cos(x),tan(x)])

Plot a Function Defined by $$y=f(x,a)$$ for Various Values of $$a$$

Plot the function $$\sin (\exp (x/a))$$ for $$a=1,2$$, and $$4$$.

syms x a
expr = sin(exp(x/a));
fplot(subs(expr,a,[1,2,4]))
legend show

Plot the Derivative and Integral of a Function

Plot a function $$f(x)=x(1+x)+2$$, its derivative $$df(x)/dx$$, and its integral $$\int f(x)dx$$.

syms f(x)
f(x) = x*(1 + x) + 2

f(x) = $$x\hspace{0.17em}\left(x+1\right)+2$$

f_diff = diff(f(x),x)

f_diff = $$2\hspace{0.17em}x+1$$

f_int = int(f(x),x)

f_int = 
$$\frac{x\hspace{0.17em}\left(2\hspace{0.17em}{x}^2+3\hspace{0.17em}x+12\right)}{6}$$

fplot([f,f_diff,f_int])
legend({'$f(x)$','$df(x)/dx$','$\int f(x)dx$'},'Interpreter','latex','FontSize',12)

Plot a Function $$y=g(x_0,a)$$ with $$a$$ as the Horizontal Axis

Find the $$x_0$$ that minimizes a function $$g(x,a)$$ by solving the differential equation $$dg(x,a)/dx=0$$.

syms g(x,a);
assume(a>0);
g(x,a) = a*x*(a + x) + 2*sqrt(a)

g(x, a) = $$2\hspace{0.17em}\sqrt{a}+a\hspace{0.17em}x\hspace{0.17em}\left(a+x\right)$$

x0 = solve(diff(g,x),x)

x0 = 
$$-\frac{a}{2}$$

Plot the minimum value of $$g(x_0,a)$$ for $$a$$ from 0 to 5.

fplot(g(x0,a),[0 5])
xlabel('a')
title('Minimum Value of $g(x_0,a)$ Depending on $a$','interpreter','latex')

Plot an Implicit Function $$f(x,y)=c$$ Using fimplicit

Plot circles defined by $$x^2+y^2=r^2$$ with radius $$r$$ as the integers from 1 to 10.

syms x y
r = 1:10;
fimplicit(x^2 + y^2 == r.^2,[-10 10])
axis square;

Plot Contours of a Function $$f(x,y)$$ Using fcontour

Plot contours of the function $$f(x,y)=x^3-4x-y^2$$ for contour levels from –6 to 6.

syms x y f(x,y)
f(x,y) = x^3 - 4*x - y^2;
fcontour(f,[-3 3 -4 4],'LevelList',-6:6);
colorbar
title 'Contour of Some Elliptic Curves'

Plot an Analytic Function and Its Approximation Using Spline Interpolant

Plot the analytic function $$f(x)=x\exp (-x)\sin (5x)-2$$.

syms f(x)
f(x) = x*exp(-x)*sin(5*x) -2;
fplot(f,[0,3])

Create a few data points from the analytic function.

xs = 0:1/3:3;
ys = double(subs(f,xs));

Plot the data points and the spline interpolant that approximates the analytic function.

hold on
plot(xs,ys,'*k','DisplayName','Data Points')
fplot(@(x) spline(xs,ys,x),[0 3],'DisplayName','Spline interpolant')
grid on
legend show
hold off

Plot Taylor Approximations of a Function

Find the Taylor expansion of $$\cos (x)$$ near $$x=0$$ up to 5th and 7th orders.

syms x
t5 = taylor(cos(x),x,'Order',5)

t5 = 
$$\frac{x^4}{24}-\frac{x^2}{2}+1$$

t7 = taylor(cos(x),x,'Order',7)

t7 = 
$$-\frac{x^6}{720}+\frac{x^4}{24}-\frac{x^2}{2}+1$$

Plot $$\cos (x)$$ and its Taylor approximations.

fplot(cos(x))
hold on;
fplot([t5 t7],'--')
axis([-4 4 -1.5 1.5])
title('Taylor Series Approximations of cos(x) up to 5th and 7th Order')
legend show
hold off;

Plot the Fourier Series Approximation of a Square Wave

A square wave of period $$2\pi$$ and amplitude $$\pi /4$$ can be approximated by the Fourier series expansion

Plot a square wave with period $$2\pi$$ and amplitude $$\pi /4$$.

syms t y(t)
y(t) = piecewise(0 < mod(t,2*pi) <= pi, pi/4, pi < mod(t,2*pi) <= 2*pi, -pi/4);
fplot(y)

Plot the Fourier series approximation of the square wave.

hold on;
n = 6;
yFourier = cumsum(sin((1:2:2*n-1)*t)./(1:2:2*n-1));
fplot(yFourier,'LineWidth',1)
hold off

The Fourier series approximation overshoots at a jump discontinuity and the "ringing" does not die out as more terms are added to the approximation. This behavior is also known as the Gibbs phenomenon.

Plot a Parametric Curve $$(x(t),y(t),z(t))$$ Using fplot3

Plot a helix that is defined by $$(\sin (t),\cos (t),t/4)$$ for $$t$$ from –10 to 10.

syms t
fplot3(sin(t),cos(t),t/4,[-10 10],'LineWidth',2)
view([-45 45])

Plot a Surface Defined by $$z=f(x,y)$$ Using fsurf

Plot a surface defined by $$\log (x)+\exp (y)$$. Analytical plotting using fsurf (without generating numerical data) shows the curved areas and asymptotic regions near $$x=0$$.

syms x y
fsurf(log(x) + exp(y),[0 2 -1 3])
xlabel('x')

Plot a Multivariate Surface $$(x(u,v),y(u,v),z(u,v))$$ Using fsurf

Plot a multivariate surface defined by

where $$f(u)=\exp (-u^2/3)\sin (u)+3/2$$.

Set the plot interval of $$u$$ from –5 to 5 and $$v$$ from 0 to 2$$\pi$$.

syms f(u) x(u,v) y(u,v) z(u,v)
f(u) = sin(u)*exp(-u^2/3)+1.5;
x(u,v) = u;
y(u,v) = f(u)*sin(v);
z(u,v) = f(u)*cos(v);
fsurf(x,y,z,[-5 5 0 2*pi])

Plot a Multivariate Surface $$(x(s,t),y(s,t),z(s,t))$$ Using fmesh

Plot a multivariate surface defined by

where $$r=8+\sin (7s+5t)$$. Show the plotted surface as meshes by using fmesh. Set the plot interval of $$s$$ from 0 to 2$$\pi$$ and $$t$$ from 0 to $$\pi$$.

syms s t
r = 8 + sin(7*s + 5*t);
x = r*cos(s)*sin(t);
y = r*sin(s)*sin(t);
z = r*cos(t);
fmesh(x,y,z,[0 2*pi 0 pi],'Linewidth',2)
axis equal

Plot an Implicit Surface $$f(x,y,z)=c$$ Using fimplicit3

Plot the implicit surface $$1/x^2-1/y^2+1/z^2=0$$.

syms x y z
f = 1/x^2 - 1/y^2 + 1/z^2;
fimplicit3(f)

Plot the Contours and Gradient of a Surface

Plot the surface $$\sin (x)+\sin (y)-(x^2+y^2)/20$$ using fsurf. You can show the contours on the same graph by setting 'ShowContours' to 'on'.

syms x y
f = sin(x)+sin(y)-(x^2+y^2)/20

f = 
$$\sin \left(x\right)+\sin \left(y\right)-\frac{x^2}{20}-\frac{y^2}{20}$$

fsurf(f,'ShowContours','on')
view(-19,56)

Next, plot the contours on a separate graph with finer contour lines.

fcontour(f,[-5 5 -5 5],'LevelStep',0.1,'Fill','on')
colorbar

Find the gradient of the surface. Create 2-D grids using meshgrid and substitute the grid coordinates to evaluate the gradient numerically. Show the gradient using quiver.

hold on
Fgrad = gradient(f,[x,y])

Fgrad = 
$$\left(\begin{array}{c}\cos \left(x\right)-\frac{x}{10}\\ \cos \left(y\right)-\frac{y}{10}\end{array}\right)$$

[xgrid,ygrid] = meshgrid(-5:5,-5:5);
Fx = subs(Fgrad(1),{x,y},{xgrid,ygrid});
Fy = subs(Fgrad(2),{x,y},{xgrid,ygrid});
quiver(xgrid,ygrid,Fx,Fy,'k')
hold off