Problems

01

L’Hopital practice

Compute these limits using L’Hopital’s Rule:

• (a) ${\lim }_{x\to 1}\left(\frac{x}{x-1}-\frac{1}{\ln x}\right)$
• (b) ${\lim }_{x\to \infty }\ln \left({\left(\frac{x}{5}\right)}^{2/x}\cdot \frac{4}{5}\right)$

02

Maximum area, fixed perimeter of a rectangle

Find the dimensions of the rectangle of greatest area whose perimeter is given by a specific wire of length $410\mathrm{in}$.

03

Closest point on a graph

Consider the graph of $f(x)=\sqrt{x}$. Find the closest point $p$ on the graph of $f$ to the given point $(9,0)$.

(Hint: minimizing a distance $d$ is an equivalent problem to minimizing the square distance $d^2$, and the latter is often easier!)

04

L’Hopital practice

Compute these limits using L’Hopital’s Rule:

• (a) ${\lim }_{x\to \infty }x\sin \left(\frac{7\pi }{x}\right)$
• (b) ${\lim }_{x\to 0}\sin (7x)\csc (3x)$

05

Largest volume of open box

Suppose that we set up an open-topped cardboard box from square of cardboard $3\mathrm{ft}$ on each side. The box is created by cutting equal-length squares out of the corners and bending up the remaining sides. What is the largest possible volume of such a box? You may follow these steps:

1. Draw a diagram of the cardboard and the box.
2. Write $x$ for the side lengths of the cutout squares, and $y$ for the side lengths of the resulting box.
3. Find a formula for the resulting box volume $V$.
4. Find an equation relating $x$ and $y$ and use it to write $V$ in terms of $x$ alone.
5. Maximize the function $V(x)$ on an appropriate interval. (What is your interval?)

06

Largest rectangle bounded by a parabola

Consider the region above the $x$-axis and below the curve $y=9-x^2$ for $x\in [-3,3]$.

What is the area of the largest rectangle contained in this region, with lower base on the $x$-axis and upper corners on the parabola?