Math 221

5.1. Acceleration as the rate at which velocity changes. As you are driving in your car your velocity does not stay constant, it changes with time. Suppose v(t) is your velocity at time t (measured in miles per hour). You could try to figure out how …

The two previous examples have much in common. If we ignore all the details about geometry, graphs, highways and motion, the following happened in both examples: We had a function y = f(x), and we wanted to know how much f(x) changes if x changes. …

If you try to define “instantaneous velocity” you will again end up trying to divide zero by zero. Here is how it goes: When you are driving in your car the speedometer tells you how fast your are going, i.e. what your velocity is. What is this v …

To work with derivatives you have to know what a limit is, but to motivate why we are going to study limits let’s first look at the two classical problems that gave rise to the notion of a derivative: the tangent to a curve, and the instantaneous v …

For many functions the rule which tells you how to compute it is not an explicit formula, but instead an equation which you still must solve. A function which is defined in this way is called an “implicit function.” 4.1. Example. One can d …

Functions 3.1. Definition. To specify a function f you must give a rule which tells you how to compute the value f(x) of the function for a given real number x, and: say for which real numbers x the rule may be applied. The set of …

What is the 2007th digit after the period in the expansion of 1/7 ? Which of the following fractions have finite decimal expansions? a = 2/3 , b = 3 /25 , c = 276937 /15625 . Draw the following sets of real numbers. Each of these sets is the …

1. What is a number? 1.1. Different kinds of numbers. The simplest numbers are the positive integers 1, 2, 3, 4, · · · the number zero 0, and the negative integers · · · , ?4, ?3, ?2, ?1. Together these form the integers or “whol …