Math 1060 — Calculus of One Variable I

The arclength of a curve may be approximated by the sum of the lengths of line segments between points on the curve.

Due: Wednesday, November 5, 2014.

In the problems below, you must mathematically justify all of your answers. That is, you must use theorems (e.g., the mean value theorem, the first derivative test, the second derivative test, etc.) to justify your answers. You can not, for example, simply graph a function to determine its maxima or minima.

1. Find the differential of each function below.
   1. $y = x^2 \sin(2x)$
   2. $y = \sqrt{1 + \ln(x)}$
   3. $y = e^{\cosh(x)}$
   4. $y = \tanh\left(x^2\right)$
2. One side of a right triangle is known to be 20 cm long, and the opposite angle is measured as $30^\circ$, with a possible error of $\pm 1^\circ$. Use differentials to estimate the error in computing the length of the hypotenuse. What is the percentage error?
3. Suppose a five-sided box (i.e., a box with no lid) is made from a rectangular piece of cardboard which measures one meter by 3 meters. The box is constructed by cutting a square from each corner of the box, and folding the sides of the box up. (Each square cut from the corners has the same size.) What is the volume of the largest box that can be constructed in this way?
4. Is $x = 5$ a local extremum of the function $(x - 5)^3 + 2$? Why or why not? (Graphing the function is not an acceptable answer.)
5. Suppose that $f(x)$ is a function which is differentiable on the interval $(a, b)$, and suppose also that $f'(x) < 0$ for all $a < x < b$. Show that $f(x)$ is a decreasing function. (Notice that you must show this for a general differentiable function $f$ satisfying the condition $f'(x) < 0$. You can not pick a single function $f$ and show the result for that one function.)
6. Suppose a function $f(x)$ is defined and continuous on the interval $[-4, 5]$, and differentiable in the interval $(-4, 5)$. Suppose also that $f$ has the property that $f(-4) = -1$ and $f'(x) \leq 3$ for all $-4 < x < 5$. What is the largest possible value of $f(5)$?
7. Suppose a function $g(x)$ is defined and continuous on the interval $[\pi, 2\pi]$, and differentiable in the interval $(\pi, 2\pi)$. Suppose also that $g$ has the property that $g(\pi) = 2$ and $g'(x) \geq 5$ for all $\pi < x < 2\pi$. What is the smallest possible value of $g(2\pi)$?
8. Find all of the intervals on which the function $f(x) = \frac{x}{x^2 + 1}$ is increasing, and all the intervals where the function is decreasing. Find, and classify, all the local maxima and minima of the function.
9. Consider the function $g(z) = \cos^2(z) - 2\sin(z)$ defined on the interval $[0, 2\pi]$. Find all of the intervals, inside $[0, 2\pi]$, on which $g(z)$ is increasing or decreasing. Find and classify all extrema as local maxima or minima, or global maxima or minima.
10. Determine where each of the functions below are concave up and concave down.
    1. $\frac{x^2 - 4}{x^2 + 4}$
    2. $\frac{e^x}{1 - e^x}$
    3. $\ln\left(1 + \ln\left(1 + x^2\right)\right)$
11. Suppose that $f$ and $g$ are two differentiable functions which are concave up on an interval $[a, b]$.
    1. Show that $f + g$ is also concave up on $I$.
    2. Is it necessarily true that $f \cdot g$ is concave up on $I$? If so, prove it. If not, find specific examples of $f$ and $g$ which are both concave up, but whose product is concave down.
12. Suppose that $f$ and $g$ are decreasing, differentiable functions.
    1. Is $f + g$ necessarily decreasing as well? If so, prove it. If not, provide a specific examples of $f$ and $g$ which are decreasing, but where $f + g$ is not decreasing.
    2. Is $f - g$ necessarily decreasing as well? If so, prove it. If not, provide a specific examples of $f$ and $g$ which are decreasing, but where $f - g$ is not decreasing.
    3. Is $f \cdot g$ necessarily decreasing as well? If so, prove it. If not, provide a specific examples of $f$ and $g$ which are decreasing, but where $f \cdot g$ is not decreasing.