Chapter 9
Chain rule applied to
related rates and implicit
differentiation
9.1
Applications of the chain rule to related rates
In many applications of the chain rule, we are interested in processes that take place over
time. We ask how the relationships b

escribing the central tendency - an introduction to best-fit analysis
Imagine you have a data set from an experiment consisting of
n number which we label xi, i=1.n. If the
quantity being measured was fairly consistent from measurement to measurement, you

Course notes/Earth's energy balance
<
Note that you are not expected to memorize the terminology or formulae discuss here. They are included only
to make it easier for you to connect what you're learning here with current issues in the news and what you
m

There are a number of problems that do not come out correctly if you use degrees instead of radians. The
reason is that degrees are not a natural unit in a mathematical sense. For example, if you want to calculate the
arc length of a piece of a circle tha

Numerical integration is the process of going background from a given derivative of a function to the function
itself using computational tools. The simplest form of numerical integration takes advantage of the
,approximation to the derivative that we get

Chapter 16
Review Problems
315
316
Chapter 16. Review Problems
Exercises
16.1. Multiple Choice:
(1) : The equation of the tangent line to the function y = f (x) at the point x0 is
(a) y = f ! (x0 ) + f (x0 )(x x0 )
(b) y = x0 + f (x0 )/f ! (x0 )
(c) y = f

Chapter 13
Qualitative methods for
differential equations
Not all differential equations are easily solved analytically. Furthermore, even when we find
the analytic solution, it is not always easy to interpret, graph, or understand. This motivates
a numbe

Chapter 12
Solving differential
equations
12.1 Introduction
In the previous chapters, we were introduced to differential equations. We saw that the verbal descriptions of the rate of change of a process (for example, the growth of a population
or the deca

Chapter 15
Cycles, periods, and
rates of change
15.1 Derivatives of trigonometric functions
Having acquainted ourselves with properties of the trigonometric functions and their inverses in Chapter 14, we are ready to compute their derivatives and apply ou

Chapter 14
Periodic and
trigonometric functions
Nature abounds with examples of cyclic processes. Perhaps the most familiar and earliest
one we encounter is the continually repeating heartbeat that accompanies us through life.
Powering the heart are elect

Chapter 10
Exponential functions
The mathematics of uncontrolled growth are frightening. A single cell of
the bacterium E. coli would, under ideal circumstances, divide every twenty
minutes. That is not particularly disturbing until you think about it, bu

Chapter 11
Differential equations for
exponential growth and
decay
In Section 10.2.5 we made an observation about exponential functions and a new kind of
equation - a differential equation. - that such functions satisfy. In this chapter we explore
this ob

Chapter 3
Three faces of the
derivative: geometric,
analytic, and
computational
In Chapter 2 we bridged two concepts: the average rate of change (slope of secant line)
and the instantaneous rate of change (the derivative). We arrived at a recipe for calcu

Chapter 1
Power functions as
building blocks
There is no knowledge that is not power.
Ralph Waldo Emerson, (1803-1882)
Some of the beautiful architectural marvels built by humans from ancient to modern
times though very complicated as a whole, are made of

Chapter 2
Average rates of change,
average velocity and the
secant line
In this chapter, we introduce the idea of an average rate of change. To motivate ideas, we
examine data for two common processes, changes in temperature, and motion of a falling
objec

Chapter 5
Tangent lines, linear
approximation, and
Newtons method
In a previous chapter, we defined the tangent line as the line we see when we zoom into
the graph of a (continuous) function y = f (x) at some point. In much the same sense,
the tangent lin

Chapter 4
Differentiation rules,
simple antiderivatives
and applications
In a previous chapter, we defined the derivative of a function, y = f (x) by
f (x + h) f (x)
dy
= f ! (x) = lim
.
h0
dx
h
Using this formula, we calculated derivatives of a few power

Chapter 7
Optimization
Like a bridegroom, I thought it proper to take up . mathematics . and to
investigate . the laws of measurement, so useful in housekeeping.
- Johannes Kepler, Linz, Austria, 1613
Calculus was developed to solve practical problems. In

Chapter 6
Sketching the graph of a
function using calculus
tools
The derivative of a function contains important information about the function. In this
chapter we will focus on how properties of the first and second derivative can be used to
help up refi

Chapter 8
Introducing the chain
rule
Dont be a fool, you gotta be cool; learn to use that chain rule!
(modified from) Bard Ermentrout, Pittsburgh, 1990s
So far, examples were purposefully chosen to focus on power functions, polynomial, and rational functi