Test 3 review

Review for test 3 in MTH229

Test 3 will be on December 15th. It will cover 3 projects.

As with the first test, you can use the computer and the internet, but you can not use your phone during the test or communicate with others in any way during the test.

As with the first test, some questions will require you to show your Julia commands to receive full credit.

As in class, we use the following packages:

using MTH229
using Plots

(Unfortunately, plotly() doesn’t work within this HTML page, so our graphics are static.)

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Here are some sample questions. Any similarity to actual questions is not intended and should not be inferred.

Newton’s method

Question

This figure shows a visualization of Newton’s method starting at \(x_1 = 2\). What is the value of \(x_4\) (after 3 iterations). Read this from the graph.

The value \(x_1=2\) is a bad starting point. Does the algorithm still converge?

Yes, it bunches up near \(-2.1\)

Question

Use Newton’s method to find a zero of \(\sin(\sin(x)) - 1/2\) starting at \(x_1=1\).

Question

Use Newton’s method to find where \(f(x) = \cos(x)\) intersects \(g(x) = x/10\) starting at \(\pi/2\):

Do you get the same answer if you started at \(-\pi/2\)?

Starting at different points can lead to different answers or even have a difference if the algorithm converges. In this case, the algorithm for this problem started at \(-\pi/2\) does converge, just elsewhere.

Question

The graph of \(f(x) = \sin(\sin(x)) - \cos(x)\) shows a critical point near \(2.5\). Use Newton’s method to find a numerically precise value.

Question

How many iterations does Newton’s method take to converge to an approximate zero of \(f(x) = \sin(\sin(x)) - \cos(x)\) starting at \(x_1 = 1\)?

 🎁

Extrema

Question

Our goal: find the dimensions of the rectangle of largest area that has its base on the \(x\)-axis and its other two vertices above the x-axis and lying on the parabola \(y = 12 − x^2\).

What is the constraint for this problem?

The point \((x,y)\) lies on the parabola, so \(y = 12 - x^2\) is the constraint.

What is the objective for this problem?

The point \((x,y)\) lies on the parabola, the base of the rectangle stretches for \(-x\) to \(x\) so the area is \((2x)y\).

What is the \(x\) value for the maximum area?

Question

In an elliptical sport field we want to design a rectangular soccer field with the maximum possible area. The sport field is given by the graph of

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1. \]

Assume \(a = 200\) and \(b=150\), find the length \(2x\) and width \(2y\) of the pitch that maximizes the area of the pitch.

What is the constraint for this problem?

One vertex of the rectangular field is \((x,y)\), a point on the ellipse.

What is the objective for this problem?

One vertex of the rectangular field is \((x,y)\), a point on the ellipse. The length of each side is \(2x\) and \(2y\), so area is \((2x)(2y)\).

What is the value for \(2x\)?

What is the value for \(2y\)?

Question

Each rectangular page of a book must contain \(30 cm^2\) of printed text, and each page must have \(2 cm\) margins at top and bottom, and \(1 cm\) margin at each side. What is the minimum possible area of such a page?

What is the constraint for this problem?

We use \(w\) and \(h\) for the size of the printed text, so \(30 = wh\) is the constraint.

What is the objective for this problem?

We use \(w\) and \(h\) for the size of the printed text, so the page size is \(w + 2(1)\) and \(h+2(2)\).

What is the minimum area?

Integration

Question

Let \(f(x) = \sin(x^2)\) What is the right-Riemann sum with \(n=10\) for \(\int_0^{\pi/4} f(x) dx\)?

If \(n=10\) and the left-Riemann sum is chosen, what is the value?

If \(n=10\) and Simpson’s method is used for riemann, what is the value?

Using quadgk what is found for the definite integral?

Question

It is known that \(\int_0^\pi \sin(x) dx = 2\). Does a right Riemann sum have 5 digits of accuracy when \(n=100\)?

Question

Based on the plot of \(f(x) = \cos(x)\) and \(g(x) = 1 - x^2/2\) which integral is larger?

@syms x
plot(cos(x), -sqrt(2), sqrt(2); label="cos")
plot!(1-x^2/2;                  label = "1-x^2/2")

The graph shows (just barely) that \(\cos(x) \geq 1 - x^2/2 \geq 0\) on this interval, so the integrals have similar inequalities, as they represent areas under a curve.

Question

An ellipse is given by

\[ \frac{x^2}{200^2} + \frac{y^2}{100^2} = 1 \]

Find the perimeter.

Hint: Solve for \(y\) using the non-negative solution to the square-root; find the arc-length for \(x \in [-200, 200]\); double to get your answer.

Question

The function \(r(x) = (1 - x^4)^{(1/3)}\) between \([-1, 1]\) is rotated around the \(x\) axis. What is the resulting volume?

Hint: use \(V = \int_a^b \pi r(x)^2\).