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Algorithm Homework:

print(diff(sin(x^2), x, 1)) print(diff(sin(x^2), x, 2))

A person is walking away from a streetlight at a constant speed of 1 meter per second. The person’s height is 1.8 meters, and there is a streetlight that is 5 meters above the ground. The person casts a shadow on the ground.

  1. At what rate is the person’s shadow length changing when they are 10 meters away from the streetlight?
  2. Find the rate of change of the angle of elevation of the person’s line of sight to the streetlight when they are 10 meters away from the streetlight.

Use a linear approximation to estimate the given number, and use concavity to determine if your estimate is an over or under estimate.

  1. (1.999)^4
  2. 1/4.002

The circumference of a sphere was measured to be 84 cm with a possible error of 0.5 cm.

  1. Use differentials to estimate the maximum error in the calculated surface area. What is the relative error?
  2. Use differentials to estimate the maximum error in the calculated volume. What is the relative error?

Show that if $f$ is a continuous function on an open interval contaning $[a,b]$ with $f(a)f(b)<0$ and $f’(x)>0$ for all $x\in [a,b]$, then $f$ has a unique zero (solution of $f(x)=0$) in $[a,b]$.

If $f$ is differentiable on an open interval containing $[0,1]$ with $f(0)=1$ and $f(1)=0$, then there exists $c\in (0,1)$ such that $f’(c)=-\frac{f(c)}{c}$.

Evaluate $\displaystyle\lim_{x\to \infty}\sin\left(\frac{1}{x}\right)^{\tan^{-1}(\frac{1}{x})}$.

Answer the following questions about $f(x)=x\left(\ln(x)\right)^2$:

  1. Is $f$ an odd function or even function?
  2. Find all critical points of $f$, and determine local maximal and local minimal values of $f(x)$.
  3. Find the intervals of increase/decrease of $f$.
  4. Find the interval on which f(x) is concave upward/downward.
  5. Find the asymptotes (vertical, horizontal, or slant) of $y=f(x)$
  6. Sketch the graph of $f(x)$

Newton’s method is an iterative numerical technique used to approximate solutions for equations, often employed to find the roots of real-valued functions. Below is a pseudocode representation of Newton’s method:

Goal: Find an approximate solution x1 to the equation f(x) = 0.
Input: A function (f), an initial value (x0), a tolerance error (err), and a maximum number of iterations (m).
Return: An approximate solution x1 with |f(x1)| less than err or reaching the maximum number of iterations.
Code:
def NewtonMethod(f, x0, err, m):
    x = x0
    iteration = 0
    while iteration < m:
        if abs(f(x)) < err:
            return x
        if f'(x) == 0:
            return "No solution found"
        x = x - f(x) / f'(x)  # Find the x-intercept of the tangent line to the curve y = f(x) at (x, f(x))
        iteration = iteration + 1
    return "Exceeded the maximum number of iterations"

Your task is to implement the provided pseudocode in Sage/Python or any language you are used to use and use your code to approximate the solution of the equation $\cos(x) = x^2 - 4$ to six decimal places.