# 2000 SMT/Calculus Problems

## Problem 1

Find the slope of the tangent at the point of inflection of $y = x^3 - 3x^2 + 6x + 2000$.

## Problem 2

Karen is attempting to climb a rope that is not securely fastened. If she pulls herself up $x$ feet at once, then the rope slips $x^3$ feet down. How many feet at a time must she pull herself up to climb with as few pulls as possible?

## Problem 3

A rectangle of length $\frac{1}{4} \pi$ and height 4 is bisected by the x-axis and is in the first and fourth quadrants, with the leftmost edge on the y-axis. The graph of $y$ = $\sin(x) + C$ divides the area of the square in half. What is C?

## Problem 4

For what value of $x$ $(0 < x < \frac{\pi}{2})$ does $\tan x + \cot x$ achieve its minimum?

## Problem 5

For $-1 < x < 1$ let $f(x) = \sum_{i=1}^{\infty} \frac{x^i}{i}$. Find a closed form expression (a closed form expression is one not involving summation)for f.

## Problem 6

A hallway of width 6 feet meets a hallway of width $6\sqrt{5}$ feet at right angles. Find the length of the longest pipe that can be carried horizontally around this corner.

## Problem 7

An envelope of a set of lines is a curve tangent to all of them. What is the envelope of the family of lines y = $\frac{2}{x_0} + x(1 - \frac{1}{x_0^{2}})$, with $x_0$ ranging over the positive real numbers?

## Problem 8

Find $\int_{0}^{\frac{\pi}{2}} ln$ $\sin \theta$ $d\theta$

## Problem 9

Let $f(x) = \sqrt{x+\sqrt{0+\sqrt{x+0+\sqrt{x+\cdots}}}}$ If $f(a)=4$, find $f'(a)$.

## Problem 10

A mirror is constructed in the shape of $y$ equals $\pm\sqrt{x}$ for $0 \le x \le 1$, and $\pm1$ for $1 < x < 9$. A ray of light enters at (10,1) with slope 1. How many times does it bounce before leaving?