Mathematical analysis
In this example, we will demonstrate the solution of several simple problems from the field of mathematical analysis – the construction of tangent lines, symbolic differentiation of equations, and finding the Jacobian of a system of equations.
First, run the preparation cell with the code.
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Pkg.add(["Latexify", "Symbolics"])
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using Latexify
using Symbolics
using Plots
using Printf
plotlyjs();
Building a tangent line
Find the intersection point of the Y-axis and the line tangent to the sine wave at the point .
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a = pi/3
# Graph of a sine wave
plot(sin, -pi:0.1:pi, framestyle = :origin, legend=false)
# Points, graphs through which the secant should pass
scatter!(sin, [a])
# Equation of the tangent at pi/3
tangent(x) = sin(a) + cos(a)*(x - a)
# Graph of the tangent at pi/3
plot!(tangent, -3:2)
# The point where the secant line intersects the y axis
scatter!([0], [tangent(0)], c=:yellow)
# Summary of the result (with a couple of spaces at the end)
annotate!(0, касательная(0), text(@sprintf("%.3f", касательная(0))*" ", :blue, :right, 8))
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println("The point where the tangent intersects the ordinate axis: (0, " * string(касательная(0)) * ")")
Symbolic differentiation
Differentiate the function .
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@Symbolics.variables t
Df = Symbolics.Differential(t); # Let's create a differentiation operator
z = t + t^2; # Let's define a function whose derivative needs to be found
# Let's differentiate the function and simplify the result.
expand_derivatives(Df(z))
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Jacobian of the system of equations
Find the Jacobian of a system of linear functions:
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@Symbolics.variables x_1 x_2 x_3 a_11 a_12 a_13 a_21 a_22 a_23 a_31 a_32 a_33 b_1 b_2 b_3
Symbolics.jacobian([ a_11*x_1 + a_12*x_2 + a_13*x_3 - b_1 ;
a_21*x_1 + a_22*x_2 + a_23*x_3 - b_2 ;
a_31*x_1 + a_32*x_2 + a_33*x_3 - b_3 ], [x_1, x_2, x_3])
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