This document is one of More SageMath Tutorials. You may edit it on github. \(\def\NN{\mathbb{N}}\) \(\def\ZZ{\mathbb{Z}}\) \(\def\QQ{\mathbb{Q}}\) \(\def\RR{\mathbb{R}}\) \(\def\CC{\mathbb{C}}\)

Calculus, plotting & interact

Some differentiating and plotting

Exercises

  1. Let \(f(x) = x^4 + x^3 - 13 x^2 - x + 12\). Define \(f\) as a symbolic function.

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  2. Plot \(f\) on the domain \(-4.5 \leq x \leq 3.5\).

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  3. Find numerical approximations for the critical values of \(f\) by taking the derivative of \(f\) and using the find_root method. (Hint: plot the derivative.)

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  4. Find numerical approximations for the critical values of \(f\) by taking the derivative of \(f\) and using the roots(ring=RR) method. (Here, RR stands for the real numbers.) Are there any roots over the ring of rationals (QQ)?

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  5. Compute the equation \(y = mx +b\) of the tangent line to the function \(f\) at the points \(x=-1\) and \(x=2\).

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  6. Write a function that takes \(x\) as an argument and returns the equation of the tangent line to \(f\) through the point \(x\).

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  7. Write a function that takes \(x\) as an argument and plots \(f\) together with the the tangent line to \(f\) through the point \(x\). Make the line red.

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  8. Convert the function you created above into an @interact object. Turn the argument \(x\) into a slider . (Hint: see the documentation for interact for examples on creating sliders.)

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Differential Equations

Using symbolic functions and the command desolve in Sage, we can define and solve differential equations. Here is an example.

We will solve the following differential equation:

\[y'(t) + y(t) = 1\]

First we define the variable \(t\):

sage: var('t')
t

Next, we define the symbolic function \(y\):

sage: y = function('y', t)
sage: y
y(t)

We can now create the differential equation:

sage: diff_eqn = diff(y,t) + y - 1
sage: diff_eqn
diff(y(t), t, 1) + y(t) - 1

We can use the show command to typeset the above equation to make it easier to read:

sage: show(diff_eqn)

Finally, we use the desolve command to solve the differential equation:

sage: soln = desolve(diff_eqn, y)
sage: soln
e^(-t)*(e^t + c)

sage: show(soln)

Exercises

  1. Find and plot the solution to the following differential equation with the intial condition \(y(0) = -2\).

    \[y'(t) = y(t)^2 - 1\]

    (Hint: see the documentation of the desolve command for dealing with initial conditions.)

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  2. Find and plot the solution to the differential equation

    \[t y'(t) + 2 y(t) = \frac{e^t}{t}\]

    with initial conditions \(y(1) = -2\). (Hint: see the documentation of the desolve command for dealing with initial conditions.) [Introductory Differential Equations using SAGE, David Joyner]

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Problem

Let \(a>b>0\) be fixed real numbers and form a triangle with one vertex on the line \(y=x\), one vertex on the line \(y=0\) and the third vertex equal to \((a,b)\).

Find the coordinates of the vertices that minimize the perimeter of the triangle (remember that (a,b) is fixed!). What is the perimeter?

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