# 2000 SMT/Calculus Problems

## Contents

## Problem 1

Find the slope of the tangent at the point of inflection of .

## Problem 2

Karen is attempting to climb a rope that is not securely fastened. If she pulls herself up feet at once, then the rope slips 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 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 = divides the area of the square in half. What is C?

## Problem 4

For what value of does achieve its minimum?

## Problem 5

For let . 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 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 = , with ranging over the positive real numbers?

## Problem 8

Find

## Problem 9

Let If , find .

## Problem 10

A mirror is constructed in the shape of equals for , and for . A ray of light enters at (10,1) with slope 1. How many times does it bounce before leaving?