Math Related Web Sites
Web Site
www.wyzant.com
www.khanacademy.org
www.math.com
Summary
This site covers everything from elementary math through
calculus, with detailed, image-rich explanations that go
beyond static content with check-able quizzes, a Question
and Answer forum and interaction with tutors.
The premier online education site on the internet.
The Math.com site is a good reference for help with prealgebra, algebra, trig., and geometry.
coolmath.com
Coolmath.com is designed for people from ages 13 to 100!
Their main goal is to provide resources for teaching math
and making it fun for all. Check out the sections on fractions
and geometry.
www.netcomuk.co.uk/
~jenolive/trig.html
This web site provides web pages that describe some
properties and physical applications of vectors. Each section
builds on the previous ones to make a logical sequence and
there are hot links within sections so that it is easy to refer
back if you want to. This particular link will connect you to a
page that has information on basic trig.
www.quickmath.com
The QuickMath site provides a means for getting help with
common math problems over the Internet. Think of it as an
online calculator that solves equations and does all sorts of
algebra and calculus problems. It contains sections on
algebra, equations, inequalities, calculus, and matrices.
www.mathleague.com
The Math League site offers a 'Help Facility' that covers
decimals, fractions, geometry, introductory algebra, and
much more. While the site targets students in grades 4-8, the
information is useful to those who need a refresher on basic
math. This particular link will bring you to a page on ratio
and proportions."
algebrahelp.com/
AlgebraHelp.com uses some of the latest technology to help
users learn and understand algebra. The site features lessons,
calculators that show how to solve problems step-by-step,
and interactive worksheets to test skills.
www.sosmath.com
S.O.S. MATHematics is a free resource for math review
material on subjects such as algebra, trigonometry, calculus,
and differential equations. A good site for high school,
college students and adult learners.
www.math2.org
This web site provides math reference tables for General
Math, Algebra, Geometry, Trig., Stat., Calc., and more. It
also contains a math message board that allows users to have
web-based discussion about math.
aleph0.clarku.edu/
~djoyce/java/trig/index.html
Dave's Short Trig Course is useful for people who would
like to learn or brush up on trigonometry. The notes are more
of an introduction and guide than a full course.
www.collegeonline.org
This math resource is buried in the library section of the
collegeonline.org website under "Education Articles." The
resource provides a listing of math reference tables and other
useful information.
Calculus Help and Problems
This section contains in depth discussions and explanations on key topics that appear throughout
Calculus 1 and 2 up through Vector Calculus. The topics are arranged in a natural progression catering
typically to late highschool and early college students, covering the foundations of calculus, limits,
derivatives, integrals, and vectors.
Still need help after using our calculus resources? Use our service to find a calculus tutor.
Introduction to Calculus
Main Lesson: Introduction to Calculus
An introduction to Calculus and its applications.
Limits
Main Lesson: Limits
An overview of limits as it applies to differential and integral calculus.
Differential Calculus
Main Lesson: Differentiation - Taking the Derivative
An overview of Differential Calculus.
Integral Calculus
Main Lesson: Integration - Taking the Integral
An overview of Integral Calculus.
Multivariable / Vector Calculus
Main Lesson: Multivariable / Vector Calculus
Introduction to Calculus
Calculus is the study of change and motion, in the same way that geometry is the study of shape and
algebra is the study of rules of operations and relations. It is the culmination of algebra, geometry, and
trigonometry, which makes it the next step in a logical progression of mathematics.
Calculus defines and deals with limits, derivatives, and integrals of functions. The key ingredient in
calculus is the notion of infinity. The essential link to completing calculus and satisfying concerns about
infinite behavior is the concept of the limit, which lays the foundation for both derivatives and integrals.
Calculus is often divided into two sections: Differential Calculus (dealing with derivatives, e.g. rates of
change and tangents), and Integral Calculus (dealing with integrals, e.g. areas and volumes). Differential
Calculus and Integral Calculus are closely related as we will see in subsequent pages. It is important to
have a conceptual idea of what calculus is and why it is important in order to understand how calculus
works.
History of Calculus
Main Lesson: Brief History of Calculus
A brief history of the invention of calculus and its development.
Difference Between Calculus and Other Mathematics
Main Lesson: Difference Between Calculus and Other Mathematics
A comparison of calculus against other mathematical disciplines.
Calculus Applications in Algebra and Geometry
Main Lesson: Calculus Applications in Algebra and Geometry
A Brief History of Calculus
Calculus was created by Isaac Newton, a British scientist, as well as Gottfried Leibniz, a self-taught
German mathematician, in the 17th century. It has been long disputed who should take credit for
inventing calculus first, but both independently made discoveries that led to what we know now as
calculus. Newton discovered the inverse relationship between the derivative (slope of a curve) and the
integral (the area beneath it), which deemed him as the creator of calculus. Thereafter, calculus was
actively used to solve the major scientific dilemmas of the time, such as:


calculating the slope of the tangent line to a curve at any point along its length
determining the velocity and acceleration of an object given a function
describing its position, and designing such a position function given the object's
velocity or acceleration


calculating arc lengths and the volume and surface area of solids
calculating the relative and absolute extrema of objects, especially projectiles
For Newton, the applications for calculus were geometrical and related to the physical world - such as
describing the orbit of the planets around the sun. For Leibniz, calculus was more about analysis of
change in graphs. Leibniz's work was just as important as Newton's, and many of his notations are used
today, such as the notations for taking the derivative and the integral.
Difference Between Calculus and Other
Math Subjects
On the left, a man is pushing a crate up a straight incline. On the right, a man is pushing the same crate
up a curving incline. The problem in both cases is to determine the amount of energy required to push
the crate to the top. For the problem on the left, you can use algebra and trigonometry to solve the
problem. For the problem on the right, you need calculus. Why do you need calculus with the problem on
the right and not the left?
This is because with the straight incline, the man pushes with an unchanging force and the crate goes up
the incline at an unchanging speed. With the curved incline on the right, things are constantly changing.
Since the steepness of the incline is constantly changing, the amount of energy expended is also
changing. This is why calculus is described as "the mathematics of change". Calculus takes regular rules
of math and applies them to evolving problems.
With the curving incline problem, the algebra and trigonometry that you use is the same, the difference is
that you have to break up the curving incline problem into smaller chunks and do each chunk separately.
When zooming in on a small portion of the curving incline, it looks as if it is a straight line:
Then, because it is straight, you can solve the small chunk just like the straight incline problem. When all
of the small chunks are solved, you can just add them up.
This is basically the way calculus works - it takes problems that cannot be done with regular math
because things are constantly changing, zooms in on the changing curve until it becomes straight, and
then it lets regular math finish off the problem. What makes calculus such a brilliant achievement is that
it actually zooms in infinitely. In fact, everything you do in calculus involves infinity in one way or
another, because if something is constantly changing, it is changing infinitely from each infinitesimal
moment to the next. All of calculus relies on the fundamental principle that you can always use
approximations of increasing accuracy to find the exact answer. Just like you can approximate a curve by
a series of straight lines, you can also approximate a spherical solid by a series of cubes that fit inside the
sphere.
Algebra and Geometry with Calculus
One of the earliest algebra topics learned is how to find the slope of a line--a numerical value that
describes just how slanted that line is. Calculus gives us a much more generalized method of finding
slopes. With it, we can find not only how steeply a line slopes, but indeed, how steeply any curve slopes
at any given point.
Without calculus, it is difficult to find areas of shapes other than those whose formulas you learned in
geometry. You may be able to find the area of commons shapes such as a triangle, square, rectangle,
circle, and even a trapezoid; but how could you find the area of the shape like the one shown below?
With calculus, you can calculate complicated x-intercepts. Without a graphing calculator, it is pretty
difficult to calculate an irrational root. However, a simple process called Newton's Method (named Isaac
Newton) allows you to calculate an irrational root to whatever accuracy you want.
Calculus makes it much easier to visualize graphs. You may already have a good grasp of linear functions
and how to visualize their graphs easily, but what about the graph of something like y= x^3 + 2x^2 - x
+ 1? Elementary calculus tells you exactly where that graph will be increasing, decreasing, and twisting.
You can even find the highest and lowest points on the graph without plotting a single point.
One of the most useful applications of calculus is the optimization of functions. In a small number of
steps, you can answer questions such as:
If I have 500 feet of fence, what is the largest rectangular yard I can make? or Given a rectangular sheet
of paper which measures 8.5 inches by 11 inches, what are the dimensions of the box I can make
containing the greatest volume?
The traditional way to create an open box from a rectangular surface is to cut congruent squares from
the corners of the rectangle and then to fold the resulting sides up as shown:
Calculus develops concepts in other mathematics that lets us discover more about them and enables us
to achieve greater feats than the mathematics that it is built on. It is vital to understanding and making
sense of the world we live in.
Applications of Calculus
With calculus, we have the ability to find the effects of changing conditions on a system. By studying
these, you can learn how to control a system to make it do what you want it to do. Because of the ability
to model and control systems, calculus gives us extraordinary power over the material world.
Calculus is the language of engineers, scientists, and economists. The work of these professionals has a
huge impact on our daily life - from your microwaves, cell phones, TV, and car to medicine, economy,
and national defense.
Credit card companies use calculus to set the minimum payments due on credit card statements at the
exact time the statement is processed by considering multiple variables such as changing interest rates
and a fluctuating available balance.
Biologists use differential calculus to determine the exact rate of growth in a bacterial culture when
different variables such as temperature and food source are changed. This research can help increase the
rate of growth of necessary bacteria, or decrease the rate of growth for harmful and potentially
threatening bacteria.
An electrical engineer uses integration to determine the exact length of power cable needed to
connect two substations that are miles apart. Because the cable is hung from poles, it is constantly
curving. Calculus allows a precise figure to be determined.
An architect will use integration to determine the amount of materials necessary to construct a curved
dome over a new sports arena, as well as calculate the weight of that dome and determine the type of
support structure required.
Space flight engineers frequently use calculus when planning lengthy missions. To launch an
exploratory probe, they must consider the different orbiting velocities of the Earth and the planet the
probe is targeted for, as well as other gravitational influences like the sun and the moon. Calculus allows
each of those variables to be accurately taken into account.
Statisticians will use calculus to evaluate survey data to help develop business plans for different
companies. Because a survey involves many different questions with a range of possible answers,
calculus allows a more accurate prediction for appropriate action.
A physicist uses calculus to find the center of mass of a sports utility vehicle to design appropriate
safety features that must adhere to federal specifications on different road surfaces and at different
speeds.
An operations research analyst will use calculus when observing different processes at a
manufacturing corporation. By considering the value of different variables, they can help a company
improve operating efficiency, increase production, and raise profits.
A graphics artist uses calculus to determine how different three-dimensional models will behave when
subjected to rapidly changing conditions. This can create a realistic environment for movies or video
games.
Obviously, a wide variety of careers regularly use calculus. Universities, the military, government
agencies, airlines, entertainment studios, software companies, and construction companies are only a few
employers who seek individuals with a solid knowledge of calculus. Even doctors and lawyers use calculus
to help build the discipline necessary for solving complex problems, such as diagnosing patients or
planning a prosecution case. Despite its mystique as a more complex branch of mathematics, calculus
touches our lives each day, in ways too numerous to calculate.
Help with Limits in Calculus
All of calculus relies on the principle that we can always use approximations of increasing accuracy to find
the exact answer, such as approximating a curve by a series of straight lines in differential calculus (the
shorter the lines and as the distance between points approaches 0, the closer they are to resembling the
curve) or approximating a spherical solid by a series of cubes in integral calculus (as the size of the cubes
gets smaller and the number of cubes approaches infinity inside the sphere, the end result becomes
closer to the actual area of the sphere).
With the help of modern technology, graphs of functions are often easy to produce. The main focus is
between the geometric and analytic information and on the use of calculus both to predict and to explain
the observed local and long term behavior of a function. In Calculus classes, limits are usually the first
topic introduced.
In order to understand the workings of differential and integral calculus, we need to understand the
concept of a limit. Limits are used in differentiation when finding the approximation for the slope of a line
at a particular point, as well as integration when finding the area under a curve. In calculus, limits
introduce the component of infinity. We can ask ourselves, what happens to the value of a function as
the independent variable gets infinitely close to a particular value?
The graph illustrates finding the limit of the dependent variable f(x) as x approaches c. A way to find
this is to plug in values that gets close to c from the left and values close to c from the right.
To further illustrate the concept of a limit, consider the sequence of numbers of x:
These values are getting closer and closer to 2 (i.e. they are approaching 2 as their limit). We can can
say that no matter what value we consider, 2 is the smallest value that is greater than every output f(x)
in the sequence. As we take the differences of these numbers, they will get smaller and smaller. In
calculus, the difference between the terms of the sequence and their limit can be made infinitesimally
small.
Sometimes, finding the limiting value of an expression means simply substituting a number.
(1) Find the limit as t approaches 10 of the expression
We write this using limit notation as
In this example, we simply substitute and write
There is no complication because M = 3t + 7 is a continuous function, but there are cases where we
cannot simply substitute like this.
(2) Find the limit as x approaches 0 of
Continuity and Limits
Many theorems in calculus require that functions be continuous on intervals of real numbers. To
successfully carry out differentiation and integration over an interval, it is important to make sure the
function is continuous.
Definition
A function f is continuous at a point (c, f(c)) if all three conditions are satisfied:
1) An output of c exists:
2) The limit exists for c and
3) The limit equals the output of c
This definition basically means that there is no missing point, gap, or split for f(x) at c. In other words,
you can move your pencil along the image of the function and you would not have to lift up the pencil.
These functions are called smooth functions.
Continuous function on [a,b]
To see if the three conditions of the definition are satisfied is a simple process.
1) Plug in the value assigned to c into the function and see if f(c) exists.
2) Use the limit definition to see if the limit exists as x approaches c.
The limit is the same coming from the left and from the right of f(c)
3) If the limit exists, see if it is the same as f(c). If it is all of the above, it is continuous.
We can see that functions need to be continuous in order to be differentiable. Are all continuous
functions differentiable?
The answer is no. In taking the derivative we did an example of a continuous function that was not
differentiable at x = 0.
f(x) = |x| is a continuous function but it is not differentiable at x = 0. Even though it is continuous and
we can draw the graph without lifting our pencil, it is not differentiable. Conversely, all differentiable
functions are continuous.
Discontinuous Graphs
There are three types of discontinuities - infinite discontinuities, jump discontinuities, and point
discontinuities.
Infinite Discontinuity
Infinite discontinuities break the 1st condition: They have an asymptote instead of a specific f(c) value.
Jump Discontinuity
Jump discontinuities break the 2nd condition: The limit approaching from a specific c from the left is not
the same as the limit approaching c from the right.
Point Discontinuity
Point discontinuities break the 3rd condition: The limit of c is not the same as (c).
These graphs are discontinuous because they cannot be drawn without lifting up the pencil.
Discontinuous graphs can be differentiated and integrated, but only over a continuous interval of the
graph.
Intermediate Value Theorem
The Intermediate value theorem states that if we have a continuous function f(x) on the interval [a,b]
with M being any number between f(a) and f(b), there exists a number c such that:
1) a < c < b
2) f(c) = M.
The Intermediate Value Theorem is a geometrical application illustrating that continuous functions will
take on all values between f(a) and f(b). We can see if we draw a horizontal line from M, it will hit the
graph at least once. If the function is not continuous on the interval, this theorem would not hold.
It is important to note that this theorem does not tell us the value of M, but only that it exists. For
example, we can use this theorem to see if a function will have any x intercepts.
(1) Use the Intermediate Value theorem to determine if f(x) = 2x3 - 5x<SUP2< sup> - 10x + 5 has
a root somewhere in the interval [-1,2].
In other words, we are asking if f(x) = 0 in the interval [-1,2]. Using the theorem, we can say that we
want to show that there is a number c where -1 < c < 2 such that M = 0 in between f(-1) and f(2).
We see that p(-1) = 8 and p(2) = -19. Therefore, 8 > 0 > -19, and at least one root exists for f(x).
Similarly to the concept of a limit, it is important to develop an intuitive understanding of continuity and
what it means in terms of limits. By taking infinitesimally close values of x (the domain), we can make
each f(x) as close as we want. We should also have a geometric understanding of continuous functions
(Intermediate Value Theorem).
Using L'Hopital to Evaluate Limits
L'Hopital's Rule is a method of differentiation to solve indeterminant limits. Indeterminant limits are limits
of functions where both the function in the numerator and the function in the denominator are
approaching 0 or positive or negative infinity. It is not clear what the limit of indeterminant forms are, but
when applying L'Hopital's Rule, indeterminant limits can be made easier to evaluate.
Evaluate the following limits:
(1)
(2)
These limits are indeterminant because the quotient on the left is 0⁄0 when x = 3, and the limit on the
right is ∞⁄-∞ when x approaches infinity. We cannot simply plug in the approaching value for x to find the
limit. Luckily, there are different methods we can use.
(1) For the first limit, we could factor out an (x-3)
It is easy to see that when x is 3, the limit is 6.
(2) For the second limit, we can factor out an x2
Knowing that the limit of any number over infinity is 0, we can plug 0 into the limit and simplify to 6⁄-5.
(3) But what about this limit?
We cannot factor anything out, so how to we evaluate it?
We can see that when x approaches 0, both the numerator and denominator approach 0. Because the
quotient will be 0⁄0, it is not clear what the limit will be. In the limits page, we evaluated this limit by
looking at the graph of the function's behavior as it approached 0 from the left and the right. Using
L'Hopital's rule, we can now evaluate the limit in a determinant form. Since both the numerator and the
denominator go to 0 and both functions have a deritive, we can apply L'Hopital's rule.
L'Hopital's Rule states that for functions f(x) and g(x):
L'Hopital's Rule
Let's use L'Hopital's rule for our limit.
Differentiating both the numerator and the denominator will simplify the quotient and make evaluating
the limit easier.
Taking the derivative of the numerator and denominator, the limit is easier to see. We know that the
cos(0) is 1, so the limit as x approaches 0 is 1. To check, we can graph both functions and see that they
both converge to y = 1 as x approaches 0.
Let's use L'Hopital's rule on our first two limits to see if it works.
(1) and (2) Evaluate the following limits:
(1)
We take the derivative and plug in 3 for x to get our limit.
(2)
We take the derivative twice and simplify. After the first derivative, the quotient is still ∞⁄-∞, so we can
apply L'Hopital's rule again and take the derivative. Simplifying, we get 6⁄-5.
Differentiation - Taking the Derivative
Differentiation is the algebraic method of finding the derivative for a function at any point. The derivative
is a concept that is at the root of calculus. There are two ways of introducing this concept, the
geometrical way (as the slope of a curve), and the physical way (as a rate of change). The slope of a
curve translates to the rate of change when looking at real life applications. Either way, both the slope
and the instantaneous rate of change are equivalent, and the function to find both of these at any point
is called the derivative.
The Geometrical Concept of the Derivative
If you have ever found the slope of a line on a graph, that is the derivative. When we are looking at
curves instead of linear graphs, it gets difficult to find the slope at every point, because the slope is
constantly changing. A way to find the slope is to zoom in on the graph at a point and find the slope at
that point.
A way to find the slope is using the rise over run method, or the formula for slope:
The way to get a better approximated slope, or derivative, is to make the two x values as close as
possible. This is a tedious process when you want to find the slope for many points on the graph. This is
where differentiation comes in. The definition of a derivative comes from taking the limit of the slope
formula as the two points on a function get closer and closer together.
For instance, say we have a point P(x, f(x)) on a curve and we want to find the slope (or derivative) at
that point. We can take a point somewhere near to P on the curve, say Q(x+h, f(x+h)), where h is a
small value. Now we can plug these values into the slope formula:
Solving for this will get us an approximation of the slope, but it still will not get us an exact value. We
want h to be as small as possible so we can get the slope at P, so we let h approach 0.
Limit Definition for the Derivative
This is the slope of the tangent line, or derivative at point P. This gives us the instantaneous rate of
change of y with respect to x.
Let's do an example. Consider the function:
Then we substitute x+h in for x
Taking the limit, we would get
Now we simplify
Factor out an h
We can see as h goes to 0, we are left with 6x+2.
This linear expression 6x+2 is the derivative for the function, and we can find the slope of the tangent at
any point on the curve by plugging in the x value of the coordinate.
In the graph below, the original function is red and the derivative is green.
Notice that when the slope of the parabola is negative, the function of the derivative is below zero, and
when the slope of the parabola is positive, so is the function of the derivative. When the parabola dips
and the slope changes from negative to positive, the function of the derivative goes from negative to
positive. We can see that at f(-1), f'(-1) = -4, so the slope at -1 is -4. Similarly, at f(0), f'(0) = 2, so the
slope at 0 is 2.
Though we have seen the form of the derivative using the limit, it can also be notated as dy/dx, f'(x), or
y'
Different notations for the derivative
d/dx means that we are taking the derivative with respect to x.
f'(x) denotes the derivative of f(x), and y' denotes the derivative of y.
Taking the Derivative of Polynomials
Finding the derivative for some functions is harder than others, and can be a tedious process when using
the slope formula. Luckily, there is an easier way of obtaining the derivative of polynomials without using
limits. Newton and Leibniz discovered an easy way to find the derivative of harder functions that only
takes a few steps. Let's look at an example:
The first step to finding the derivative is to take any exponent in the function and bring it down,
multiplying it times the coefficient.
We bring the 2 down from the top and multiply it by the 2 in front of the x. Then, we reduce the
exponent by 1. The final derivative of that term is 2*(2)x1, or 4x.
For the second term, the exponent is assumed to be 1, so we bring it down and multiply it by the
coefficient in front of the x. Then, we reduce the exponent by 1, making it 0. The final derivative of this
term is 1*(-5)x0. Note that any number raised to the 0th power is 1, so our simplified answer is 1*(5)*1, or -5.
The third term is eliminated because it does not have an x, which means it is a constant. The reason for
this is because the number 3 can be written as 3x0, and when the 0 comes down the whole term
becomes 0. Now we are left with our simplified derivative:
Notice that the derivative is linear and the original function is quadratic. The derivative will always be one
degree less than the original function. Here is a general rule for taking the derivative of all terms of a
polynomial where c is a constant:
This is commonly called the Power Rule (see proof of power rule).
Let's do another graphical example
Differentiable and Non Differentiable
Now, you must be careful when finding the derivative, because not every function has one. Most
functions are differentiable, which means that a derivative exists at every point on the function. Some
functions, however, are not completely differentiable.
Let's find the derivative of the following function at x = 0.
The limit as h approaches 0 from the left is different than when h approaches 0 from the right. This is
equivalent to saying the derivative (or slope) on the left is -1, whereas the derivative of the right side is
1. What is the slope where they meet at the origin?
Looking at the graph, we can see that at the origin there is not a definite slope because there are
multiple tangents, so there is not a derivative at that point. Therefore, the function does not have a
derivative at x=0, so it is differentiable everywhere except for x = 0.
We must note that in order for a function to be differentiable, it must be continuous.
Finding the Tangent Line
Earlier, we found the slope of the tangent line at a point using the limit definition of a derivative. Let's do
an example finding the tangent line at a given point using the power rule for polynomials.
Find the equation to the tangent line to the graph of f(x) = x 2 + 3x at (1,4).
We find the derivative using the power rule for differentiation
Plug in our x coordinate into the derivative to get our slope
Now we can use point slope form to find the equation of the tangent line. (1,4) is our point and 5 is our
slope
The Physical Concept of the Derivative
Isaac Newton focused on the physical concept of differentiation as it applied to mechanics and
instantaneous rate of change. As it relates to mechanics, the rate of change is defined as velocity, or
speed, when we are talking about distance over a period of time. Just like the geometrical approach,
visualize that you are traveling from point A to point B. We use the formula for the slope to find the
average velocity:
Now, if we want to find the instantaneous velocity, we want the change in time to get smaller and
smaller. We introduce the concept of a limit as the change in time approaches 0. We end up with
Notice that this is the exact same as the geometric definition of the derivative, but with different
variables. The physical definition is based off of the geometric definition, and all of the rules of
derivatives apply to both. While you can find velocity by taking the derivative, you can also find the
acceleration by taking the second derivative, i.e. taking the derivative of the derivative.
Let's do an example.
Find the velocity and acceleration of a particle with the given position of s(t) = t3 - 2t2 - 4t + 5 at t = 2
where t is measured in seconds and s is measured in feet.
Velocity is found by taking the derivative of the position.
At 2 seconds, the velocity is 0 feet per second.
The acceleration is found by taking the derivative of the velocity function, or the second derivative of the
position.
At 2 seconds, the acceleration is 8 feet per second squared.
Let's analyze the graph from a physical perspective. The black curve is the object's position. Notice that
when the curve has a hump, the velocity function hits 0. Picture an object going a certain distance in a
straight line and then coming back -- the object cannot turn around without the velocity resting at 0. This
is the same for the acceleration as it relates to the velocity function. Also, when the acceleration is 0, the
graph of the position function looks like a straight line around that point. This is because when the
acceleration is 0, the velocity of the object is staying the same, therefore the slope will be constant.
Differentiation Summary
We should understand




the definition of a derivate as a limit as two points of a function get
infinitesimally close
the relationship between differentiability and continuity
how derivatives are presented graphically, numerically, and analytically
how they are interpreted as an instantaneous rate of change.
In summary, the derivative is basically the slope, or instantaneous rate of change, of the tangent line at
any point on the curve. When you take the derivative of a function, you end up with another function
that provides the slope of the original function. The derivative of a function must have the same limit
from left to right for it to be differentiable at that point. The derivative can also tell us the rate of change
from one quantity compared to another when looking at real world situations. If we know how much
distance a car has traveled over time, the derivative can tell us it's velocity and acceleration at any point
in time.
Rules for Differentiation
Here is a list of general rules that can be applied when finding the derivative of a function. These
properties are mostly derived from the limit definition of the derivative.
Linearity
Product Rule
see product rule
Quotient Rule
see quotient rule
Reciprocal Rule
Chain Rule
see chain rule
List of Derivatives
Simple Functions
Proof
Exponential and Logarithmic Functions
Proof
Proof
Proof
Trigonometric Functions
Proof
Proof
Proof
Proof
Proof
Proof
Inverse Trigonometric Functions
Proof
Proof
Proof
Proof
Proof
Proof
Product Rule Explanation
It is not always necessary to compute derivatives directly from the definition. Several rules have been
developed for finding the derivatives without having to use the definition directly. These rules simplify the
process of differentiation. The Product Rule is a formula developed by Leibniz used to find the derivatives
of products of functions.
The Product Rule is defined as the product of the first function and the derivative of the second
function plus the product of the derivative of the first function and the second function:
The Formula for the Product Rule
Product Rule Example
Find f'(x) of
We can see that there is a product, so we can apply the product rule. First, we take the product of the
first term and the derivative of the second term.
Second, we take the product of the derivative of the first term and the second term.
Then, we add them together to get our derivative.
Notice that if we multiplied them together at the start, the product would be 21x 5. Taking the derivative
after we multiplied it out would give us the same answer - 105x4. The product rule helps take the
derivative of harder products of functions that require you use the rule instead of multiplying them
together beforehand.
Let's look at a harder example:
Differentiate:
We can see that we cannot multiply first and then take the derivative. We must use the product rule.
Quotient Rule Explanation
Similar to product rule, the quotient rule is a way of differentiating the quotient, or division of functions.
The quotient rule is defined as the quantity of the denominator times the derivative of the
numerator minus the numerator times the derivative of the denominator all over the
denominator squared.
The Formula for the Quotient Rule
The quotient rule can be more difficult to remember because the order of functions matters. An easier
way to remember it is saying "Low D High take High D Low - Cross the line and square the Low"
Quotient Rule Examples
(1) Differentiate the quotient
We take the denominator times the derivative of the numerator (low d-high). . .
Then subtract the numerator times the derivative of the denominator ( take high d-low). . .
Divide it by the square of the denominator (cross the line and square the low)
Finally, we simplify
(2) Let's do another example. Find the derivative of
A quick glance at this may fool us into thinking it requires quotient rule because of the clear numerator
and denominator. If we look closely, we can see that we can rearrange this function.
The 1⁄7 is a constant, so we can pull it in front as a coefficient and use the good old power rule to
differentiate.
Chain Rule Help
The chain rule is similar to the product rule and the quotient rule, but it deals with differentiating
compositions of functions. If we recall, a composite function is a function that contains another function:
The Formula for the Chain Rule
The capital F means the same thing as lower case f, it just encompasses the composition of functions. As
a motivation for the chain rule, let's look at the following example:
(1)
This function would take a long time to factor out and find the derivative of each term, so we can
consider this a composite function. The two functions would look like this:
Notice that substituting g(x) for g in f(x) would yeild the original function. We will see that after
differentiating, we will then substitute g(x) back in for g.
So the composite function would be
Now, we can use the chain rule, which is defined by taking the derivative of outside function times
the inside function, and multiplying it by the derivative of the inside function:
Using this rule, we have:
Let's do another example.
(2) Differentiate the following function:
We define the inside and outside function to be
Then, the derivative of the composition will be as follows:
Think of the chain rule as a process. The derivative of the composite function is the derivative of the
outside function times the derivative of the inside function.
Mean Value Theorem Explanation
The Mean Value Theorem states that, given a curve on the interval [a,b], the derivative at some point
f(c) where a < c < b must be the same as the slope from f(a) to f(b).
In the graph, the tangent line at c (derivative at c) is equal to the slope of [a,b] where a < b.
The Mean Value Theorem is an extension of the Intermediate Value Theorem, stating that between the
continuous interval [a,b], there must exist a point c where the tangent at f(c) is equal to the slope of the
interval.
This theorem is beneficial for finding the average of change over a given interval. For instance, if a
person runs 6 miles in an hour, their average speed is 6 miles per hour. This means that they could have
kept that speed the whole time, or they could have slowed down and then sped up (or vice versa) to get
that average speed. This theorem tells us that the person was running at 6 miles per hour at least once
during the run.
If we want to find the value of c, we

i) Find (a,f(a)) and (b,f(b))



ii) Use the Mean Value Theorem
iii) Find f'(c) of the original function
iv) Set it equal to the Mean Value Theorem and solve for c.
Mean Value Theorem Examples
Let's do the example from above.
(1) Consider the function f(x) = (x-4)2-1 from [3,6].
First, let's find our y values for A and B.
Now let's use the Mean Value Theorem to find our derivative at some point c.
This tells us that the derivative at c is 1. This is also the average slope from a to b. Now that we know
f'(c) and the slope, we can find the coordinates for c. Let's plug c into the derivative of the original
equation and set it equal to the result of the Mean Value Theorem.
We have our x value for c, now let's plug it into the original equation.
Let's do another example.
(2) Consider the function f(x) = 1⁄x from [-1,1]
Using the Mean Value Theorem, we get
We also have the derivative of the original function of c
Setting it equal to our Mean Value result and solving for c, we get
c is imaginary! What does this mean? The function f(x) is not continuous over the interval [-1,1], and
therefore it is not differentiable over the interval. For the Mean Value Theorem to work, the function
must be continous.
Rolle's Theorem
Rolle's Theorem is a special case of the Mean Value Theorem. It is stating the same thing, but with the
condition that f(a) = f(b). If this is the case, there is a point c in the interval [a,b] where f'(c) = 0.
(3) How many roots does f(x) = x5 +12x -6 have?
We can use Rolle's Theorem to find out. First we need to see if the function crosses the x axis, i.e. if at
some point it switches from negative to positive or vice versa.
We can see that as x gets really big, the function approaces infinity, and as x approaches negative
infinity, the function also approaches negative infinity.
This means that the function must cross the x axis at least once.
If the function has more than one root, we know by Rolle's Theorem that the derivative of the function
between the two roots must be 0.
This is not true. The only way for f'(c) to equal 0 is if c is imaginary. f'(c) is always positive, which means
it only has one root.
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Derivative Proofs
Though there are many different ways to prove the rules for finding a derivative, the most common way
to set up a proof of these rules is to go back to the limit definition. This way, we can see how the limit
definition works for various functions.
We must remember that mathematics is a succession. It builds on itself, so many proofs rely on results of
other proofs - more specifically, complex proofs of derivatives rely on knowing basic derivatives. We can
also use derivative rules to prove derivatives, but even those are build off of basic principles in Calculus.
For the sake of brevity, we won't go through every proof, but it is important to know how many of these
derivatives were obtained.
It is important to understand that we are not simply "proving a derivative," but seeing how various rules
work for computing the derivative.
Derivative proof of Power Rule
Derivative proofs of ex
Derivative proof of ax
Derivative proof of lnx
Derivative proof of sin(x)
Derivative proof of cos(x)
Derivative proof of tanx
Derivative proofs of cotx, secx, and cscx
Derivative proofs of Inverse Trig Functions
Derivative Proof of Power Rule
This proof requires a lot of work if you are not familiar with implicit differentiation, which is basically
differentiating a variable in terms of x. Some may try to prove the power rule by repeatedly using product
rule. Though it is not a "proper proof," it can still be good practice using mathematical induction. A
common proof that is used is using the Binomial Theorem:
The limit definition for xn would be as follows
Using the Binomial Theorem, we get
Subtract the xn
Factor out an h
All of the terms with an h will go to 0, and then we are left with
Implicit Differentiation Proof of Power Rule
If we don't want to get messy with the Binomial Theorem, we can simply use implicit differentiation,
which is basically treating y as f(x) and using Chain rule.
Let
Take the natural log of both sides
Take the derivative with respect to x
Notice that we took the derivative of lny and used chain rule as well to take the derivative of the inside
function y.
Multiply both sides by y
Substitute xc back in for y
Integration - Taking the Integral
Integration is the algebraic method of finding the integral for a function at any point on the graph.
Finding the integral of a function with respect to x means finding the area to the x axis from the curve.
The integral is usually called the anti-derivative, because integrating is the reverse process of
differentiating. The fundamental theorem of calculus shows that antidifferentiation is the same as
integration.
The physical concept of the integral is similar to the derivative. For the derivative, the motivation was to
find the velocity at any point in time given the position of an object. If we know the velocity of an object
at a particular time, the integral will give us the object's position at that time. Just as the derivative gave
the instantaneous rate of change, the integral will give the total distance at any given time.
The integral comes from not only trying to find the inverse process of taking the derivative, but trying to
solve the area problem as well. Just as the process of differentiation is used to find the slope at any point
on the graph, the process of integration finds the area of the curve up to any point on the graph.
Riemann Integration
Before integration was developed, we could only really approximate the area of functions by dividing the
space into rectangles and adding the areas.
We can approximate the area to the x axis by increasing the number of rectangles under the curve. The
area of these rectangles was calculated by multiplying length times width, or y times the change in x.
After the area was calculated, the summation of the rectangles would approximate the area. As the
number of rectangles gets larger, the better the approximation will be. This is formula for the Riemann
Summation, where i is any starting x value and n is the number of rectangles:
This was a tedious process and never gave the exact area for the curve. Luckily, Newton and Leibniz
developed the method of integration that enabled them to find the exact area of the curve at any point.
Similar to the way the process of differentiation finds the function of the slope as the distance between
two points get infinitesimally small, the process of integration finds the area under the curve as the
number of partitions of rectangles under the curve gets infinitely large.
The Definition for the Integral of f(x) from [a,b]
The integral of the function of x from a to b is the sum of the rectangles to the curve at each interval of
change in x as the number of rectangles goes to infinity.
The notation on the left side denotes the definite integral of f(x) from a to b. When we calculate the
integral from an interval [a,b], we plug a in the integral function and subtract it from b in the integral
function:
where F denotes the integrated function. This accurately calculates the area under any continuous
function.
The General Power Rule for Integration
To carry out integration, it is important to know the general power rule. It is the exact opposite of the
power rule for differentiation.
Let's look at a general function
When we take the integral of the function, we first add 1 to the exponent, and then divide the term by
the sum of the exponent and 1.
After we have done this to each term, we add a constant at the end. Recall that taking the derivative of a
constant makes it go away, so taking the integral of a function will give us a constant. We label it C
because the constant is unknown - it could be any number! Because we can have infinitely as many
possible functions for the integral, we call it the indefinite integral.
Let's do an example.
Find the integral of
We start with the first term. We look at the exponent of 2 and increase it by 1, then we divide the term
by the resulting exponent of 3.
Then we look at the next term and do the same thing. Since it has an exponent of 1, the resulting
exponent will be 2, so we divide the whole term by 2.
The last term has an x value but we just don't see it. We can imagine the last term as -3x0. This is the
equivalent to -3(1). If we use the power rule of integration, we add 1 to the exponent to raise it to the
first power, and then we divide the term by 1.
All we need to do is add a constant at the end, and we are done.
This power formula for integration works for all values of n except for n = -1 (because we cannot
divide by 0). We can take the opposite of the derivative of the logarithmic function to solve these cases.
In general,
Integration Summary
We should understand




the Definite Integral as a limit of Riemann sums
the Definite Integral as a change of quantity over an integral
how integrals are presented graphically, numerically, and analytically
how they are interpreted as the position of an object at a given velocity.
To recap, the integral is the function that defines the area under a curve for any given interval. Taking
the integral of the derivative of the function will yield the original function. The integral can also tell us
the position of an object at any point in time given at least two points of velocity of an object
Integration by Parts
Integration by Parts is a method of integration that transforms products of functions in the integrand into
other easily evaluated integrals. The rule is derivated from the product rule method of differentiation.
Recalling the product rule, we start with
We then integrate both sides
We then solve for the integral of f(x)g'(x)
Integration by Parts
This is the formula for integration by parts. It allows us to compute difficult integrals by computing a less
complex integral. Usually, to make notation easier, the following subsitutions will be made.
Let
Then
Making our substitutions, we obtain the formula
The trick to integrating by parts is strategically picking what function is u and dv:
1. The function for u should be easy to differentiate
2. The function for dv should be easy to integrate.
3. Finally, the integral of vdu needs to be easier to compute than the integral of udv.
Keep in mind that some integrals may require integration by parts more than once. Let's do a couple of
examples
(1) Integrate
We can see that the integrand is a product of two functions, x and ex
Let
Then
Substituting into our formula, we would obtain the equation
Simplifying, we get
Integration by parts works with definite integration as well.
(2) Evaluate
Let
Then
Using the formula, we get
Then we solve for our bounds of integration : [0,3]
Let's do an example where we must integrate by parts more than once.
(3) Evaluate
Let
Then
Our formula would be
It looks like the integral on the right side isn't much of a help. Let's try integrating by parts and see if we
can make it easier.
Let
Then
Our second formula would be
Substituting into our original formula, we would have
Notice that the integral on the left hand side of the equation appears on the right hand side as well, so
we can solve for it.
Simplified, we get