Lecture 32: Review of Lectures 18 to 31

Below is a first, preliminary list of equations to review in preparation of the second midterm exam. Consider also the quizzes and homework assignments.

1. Count the number of points with integer coordinates \((x,y)\), in the region defined by the inequalities \(0 \leq x < 20\), \(0 \leq y < 20\), \(y \geq x/2\), and \(y \leq 3x\). Give the Sage commands (not the output) for the three stages below.

   1. Generate a list \(L\) of integer points \((i,j)\) for \(i\) and \(j\) ranging between 0 and 19.

   2. Select from the list \(L\) those points in the cone \(y \geq x/2\) and by \(y \leq 3 x\).

   3. Count the number of points in the cone. Write also the number below.

2. For some parameter \(t\), consider the sequence recursively defined as:

   \[F_n = (1-t) F_{n-1} + t F_{n-2}, \mbox{ for } n > 1,\]

   where \(F(0) = a\) and \(F(1) = b\). Using the recursive definition write an efficient Sage function \(F\) to compute \(F_n\) as F(a, b, t, n). What is the result of F(a, b, 0.3, 100)?

3. Consider the function \({\displaystyle f(t) = \int_0^t (1 - e^x) dx}\), for \(t \geq 0\). Define this function in Sage. What is \(f'(1)\)?

4. The function \({\displaystyle g(x,t) = \frac{1-t^2}{1 - 2xt + t^2}}\) is a generating function for the Chebyshev polynomials. Compute a Taylor series approximation for \(g(x,t)\) around \(t = 0\) of order 10. Select the coefficient of \(t^8\) and compare with the output of chebyshev_T(8, x). Is there a difference between the two?

5. Consider the point \((1,1)\) on the curve \(f(x,y) = x^2 - y^3 - x + y = 0\).

   1. Give the Sage command(s) to compute a Taylor series about the point \((1,1)\) where the term of the error is of second order.

   2. Compute the slope of the tangent line of the curve at the point \((1,1)\) and use the slope to determine the tangent line. Write the equation of the tangent line.

   Verify that the equation for the tangent line corresponds to the first-order Taylor series at \((1,1)\).

6. Consider the curve \(x^4 - 3 x y + y^4 = 0\). Give all Sage commands to

   1. to make a plot for \(x\) and \(y\) both ranging between \(-2\) and \(+2\);

   2. to convert the curve into polar coordinates; and

   3. to plot the curve in polar coordinates.

7. Consider \(p = 5 x^2 a^2 + 61 x^2 a + 66 x^2 + 10 x a^2 + 121 x a + 121 x + a^2 + 15 a + 44\), as a polynomial in \(x\) with parameter \(a\).

   1. Find the roots of \(p\).

   2. For which values of the parameter \(a\) is the answer valid?

   3. Give the Sage command(s) to treat the special case(s).

   4. As you can see the polynomial \(p\) is shown in expanded form. Give the Sage command(s) to ‘’un-expand’’, i.e.: what is the command which reveals better the structure of \(p\)?

8. Let \(a\) and \(b\) be positive numbers. Consider \(f = x^2/a + y/b\) and the unit circle \(x^2 + y^2 = 1\). Give all Sage commands to determine the number of candidate extremal values of \(f\) on the unit circle. Use a lexicographic Groebner basis to compute a triangular form of the equations for this problem.

9. Give the Sage commands for the following tasks. Create a 5-by-5 matrix \(A\) over the rationals where the \((i,j)\)-the element is \(1/(i+j)\) (for \(i\) and \(j\) both from 1 to 5). Define b as a vector of length 5 of ones. Solve the system defined by \(A x = b\). Verify that \(b - A x\) equals zero.

10. Consider the intial value problem \(dy/dt = 2 - 6y\), \(y(0) = -1\).

    (a) Solve this problem and plot the solution trajectory for \(t \in [0,2]\).

    (b) Plot the slope field for \(t \in [0,2]\) and \(y \in [-1,0.5]\). Place also the particular solution computed in (a) on the plot.

11. Minimize \(x+3y\)

    subject to \(x \geq 2\), \(y \geq 1\), \(x + 2y \leq 8\), \(x+y \leq 6\).

    Formulate the linear programming problem and solve it.