sample questions for test - 3

We should use these packages:

using MTH229

Newton's method

• Find the value of the first step of Newton's method when $f(x) = x^2 - x - 1$ and $x_0=2$.

• Find the value after 5 steps of Newton's method when $f(x) = x^3 - 2x - 5$ and $x_0=2$.

• Run Newton's method until convergence starting at $x=2$ for the function $f(x) = x^6 - x^5 - x^4 - x^3 - x^2 - x - 1$.

• Newton's method is guaranteed to converge when $f''(x) > 0$ for all $x$ but it may take some time. How many steps are needed for convergence for the function $f(x) = x^{20}-1$ starting at $x=2$? (Use newton(f, x0, verbose=true).)

• Use Newton's method to find the largest intersection point of $f(x) = e^x$ and $g(x) = x^6$.

extrema

• On $[1,5]$ find the absolute extrema of $f(x) = 3x^4 - 26 x^3 + 60x^2 -11$ Do they happen at endpoints? Critical points? (From Lamar)

• Build a rectangular pen with three parallel partitions using 500 feet of fencing. What dimensions will maximize the total area of the pen? (From UCDavis )

• A sheet of cardboard 3 ft. by 4 ft. will be made into a box by cutting equal-sized squares from each corner and folding up the four edges. What will be the dimensions of the box with largest volume? (Ditto)

• Construct a window in the shape of a semi-circle over a rectangle. If the distance around the outside of the window is 12 feet, what dimensions will result in the rectangle having largest possible area? (Again from UCDavis)

• Find the point $(x, y)$ on the graph of $ y=\sqrt{x} $ nearest the point $(4, 0)$. (Again from UCDavis)

Integration

• For the integral $\int_0^1 x^2 dx$ compare the right riemann sum with $n=3$ to the left riemann sum with $n=3$. How big is the difference? Do this "by hand". For example, the left riemann sum is found with:

f(x) = x^2
delta = (1-0)/3
f(0)*delta + f(1/3)*delta + f(2/3)*delta

• For the integral $\int_0^1 x^2 dx$ compare the value of the "trapezoid" rule to that of "Simpsons" rule using $n=3$. How big is the difference? (You should use the riemann function from the MTH229 package.)

• Which of the last four answers (right, left, trapezoid, simpsons) is closest to $1/3$?

• For the function f(x) = airy(x) we wish to estimate the area under its curve between 0 and 2. Use a right-Riemann sum with $n=10$ to do so.

• For the function f(x) = airy(x) we wish to estimate the area under its curve between 0 and 2. Use a Julia's quadgk function do so.

• Use quadgk to approximate the integral $\int_0^1 \pi \cdot (2 - x^2)^2 dx$.

• Use quadgk to approximate the integral $\int_0^1 \pi \cdot (2 - \sin(x)^2)^2 dx$. Compare the error to the last answer. Which is greater?

• The volume of a glass whose radius is described in terms of its height by $r(h)$ is $V=\int_a^b \pi \cdot r(h)^2 dh$. (picture ) A fluted glass has radius given by $(1 + h)^{3/2}$. Find the volume if the height of the glass is $10$?

• A wavy sided glass has radius given by $r(h) = 1 - 1/2 \cdot e^{h/20} \cdot \sin(3\pi/20 \cdot h)$. Find the volume if the height of the glass is $10$?