using MTH229
From the graph, estimate the value of the first step of Newton's method starting at $1$
f(x) = x^x - e^x plot(f, 2, 3.2) plot!(zero, 2, 3.2) scatter!([3], [f(3)])
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 $g(x) = x^6$.
Consider the graph. Does the point look like a good initial guess to the zero?
f(x) = x^5 - x - 1 x = 0 plot(f, -1, 1.5) plot!(zero, -1, 1.5) scatter!([0], [f(0)])
What inscribed rectangle in a half circle of radius 1 has maximum area? This figure shows 3 such inscribed rectangles.
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)
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 project.)
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?
Use quadgk and the formula $\int_a^b \sqrt{1 + f'(x)^2} dx$ to approximate the length of the graph of $e^x$ from 0 to 4.
Use quadgk and the formula $\int_a^b \sqrt{1 + f'(x)^2} dx$ to approximate the length of the graph of $\sqrt{1 + x^4}$ from -1 to 1.
Use quadgk and the formula $\int_a^b \sqrt{1 + f'(x)^2} dx$ to approximate the length of the graph of $\sqrt{1 + x^2}$ from -1 to 1.