Classes: Calculus 131

Class 1: Introduction to Axioms on Jan. 4

Substitute teacher E. Hunsicker covered the axioms of the number system
after some calisthenics.

Class 2: Review of Algebra, Jan 6.

Covered 1.1-1.2. After introductions, the regular teacher reviewed algebra
from pre-calculus. Examples of FOIL, factoring, fractions,
and the quadratic equation were done by students.

Class 3: Order of real numbers, Jan 8.

Covered 1.3-1.4. The real numbers have a natural order from the real number line.
For example, five is bigger than three. We write this as 5 > 3, or equivalently, 3 < 5.
The absolute value function was introduced as well. We saw how to solve
interesting algebraic problems involving order, e.g. (x+4)/(x+2) > 1.

Also, the notions of proof and implication were introduced. Axioms and already proven
statements allow us to give a rigorous proof of statements similar to the following:
|x+1| < .01 implies that |10x + 10| < .1

Class 4: Epsilon, Delta, Lines and Coordinates, Jan 11.

Covered 1.5-1.7. Proof and implications were discussed. The simplest version
of epsilon delta inequalities were solved.

By completing the square, we converted an equation for a circle into the canonical form.
The various equations for a line were reviewed.

Class 5: Functions and Graphs, Jan 13.

Covered 2.1. Graphing functions of all kinds were discussed. Lines, absolute values,
parabolas, and the all American function 1/x. The trigonometric function sin x
was graphed. The notion of function was defined as well as the domain and range.
The vertical line test is a easy tool to test for whether a graph is a function.
The bracket notation for functions was used to define some interesting functions.

Three of the functions from class had interesting limit properties. The all American
function 1/x is not bounded near zero and has no limit there. The line x+1 with the hole
at x=0 had a limit that was not the same as the value of the function defined at zero.
The function sin 1/x oscillates infinitely often near zero, and has no limit there
even though it is bounded,

By completing the square, we converted an equation for a parabola
into the canonical form. This makes it easier to graph.

Class 6: Operations on Functions and Trig, Jan 15.

After review bracket notation for functions, we saw how to perform algebra
on functions. The same rules apply to adding fractional expressions as in adding fractions.
The same thing goes with other rules of algebra such as factoring the difference of two squares.

We also saw how to compose functions to add to our list of functions.

The trigonometry was pretty light. We covered the definitions of sine, cosine and tangent
as ratios of sides of a right triangle. The mnemonic SOCOHTOA is useful
for remembering which function is which ratio. The values for the angles 0, pi/3 and pi/4
were determined. There are 2pi radians in an angle of 360 degrees.

Class 7: Limits and Epsilon-Delta, Jan. 18

For every epsilon, there is a delta, ... ...,but only when the limit exists. This is a paraphrase of the famous limit statement. Other statements are not as obscure and help us understand what a limit is. One student remembered a limit as being the value the function approaches when the x-value approaches a number.

There are three ingredients to a limit statement. It contains a real number c for the x-value to approach. There is a function f, and the limit L is the real number that the values of the function approach.

There were three important points made about limits. First, the limit does not depend on the value of the function at c. This can not be overemphasized. The function does not even need to be defined at c, which turns out to be one of its primary applications. Lastly, we noticed the obvious fact that it is local and only depends on the values near c.

Keeping things simple, we also saw three ways for a limit to not exist. The function could be unbounded near c, like the all American function 1/x is unbounded near 0. It could have some obvious break, or oscillate ina strange manner like sin 1/x.

We investigated how guaranteeing that f(x) is close to L translates into the epsilon delta statement. By picking an epsilon, you are asking for the function to be within epsilon of L near c, but not necessarily at c. If there is a delta, such that for all x-values within delta of c, the values of f(x) are withing epsilon of L, then the graph of f lies within a rectangle centered at (c, L). If these rectangles can be found statisfying these properties, no matter how close the horizontal lines are to L, then the limit statement holds.

One advantage of the rigorous statement for limits is that it can be stated succintly:

For every epsilon > 0, there is a delta > 0 such that
0 < |x - c| < delta implies that |f(x) - L| < epsilon .

Translation: whenever x is within delta of c, but not necessarily c, then f(x) is within epsilon of L. Hence f gets arbitrarily close to L, without any significant oscillation.

Class 8: Epsilon-Delta proofs, Jan 20

We saw how easy it was to do an epsilon-delta proof for a linear function.
The only technicalities are some fine points with the algebra of absolute values
and inequalities. On scratch paper, we convert the inequality
|f(x) - L| < epsilon into |x - c| < k epsilon
where k is a number which does not depend on x. The number k only depends
on the function f.

After doing the scratch work, it is easy to write the proof down.
Fix epsilon > 0. Let delta = k epsilon. Then... (the limit definition) follows
because of the scratch work.

Class 9: Limit Theorems and the Squeeze theorem, Jan 25

After seeing a more complicated epsilon-delta proof, we saw that limits of polynomials
and rational functions are made easy by the limit theorems. The book calls them
the main limit theorems. An easy corollary is that the limits of polynomials can be done
by substitution. Next time, we will call any such function a continuous function.

We began the next section by proving the basic lemma that
if f is bigger than or equal to a constant a,
then lim f is also bigger than or equal to the constant a.
We used this lemma to prove the next lemma.
If f is bigger than g, then lim f is bigger or equal than lim g.
Once we did that, it was easy to use this fact to prove the Squeeze theorem.

We finished class by using the squeeze theorem to show that lim sin x/x = 1.
Using the geometric assumption that
1 - x^2 is less than or equal to sin x/x which is less than or equal to 1,
and the fact that the limit of 1 - x^2 and the limit of 1 are the same, namely 1, at c = 0,
the hypotheses of the squeeze theorem are satisfied. The function sin x/x is squeezed
between 1 - x^2 and 1 at c = 0, and its limit has to be 1.

We also squeezed the function |x sin 1/x| between 0 and |x| near c = 0.
Both 0 and |x| have limit L = 0 at c = 0, and so must the squeezed function.

Class 10: Continuous Functions, Jan 27

A continuous function is one for which you can draw the graph without lifting your pencil.
The rigorous definition is that a continuous function is defined, has a limit,
and the limit coincides with the function. A continuous function is one for which
the limit can be calculated by substitution.

Many common functions and their combinations are continuous. This follows from
the main limit theorems discussed last time. The composite limit theorem shows that
the composition of continuous functions are continuous. The translation is that most functions
which can be written down are continuous where defined. That technicality comes up
with 1/x which is not defined at 0.

When a function is defiined and continuous on an interval, then
the Intermediate Value Theorem holds. This is analogous with the intuitive definition
of continuous. A word problem analogy is that someone driving from Chicago to Buffalo
on highway 90 has to go through Gary, Indiana.

Class 11: Derivatives, Jan 29

The derivative has many interpretations. Different applications have their own word
for it. In economics, it is marginal profit, and in mechanics it is velocity. It is the slope of
the tangent line to a graph. It gives much more accurate information about
the rate of change of a function than the average change.

The tangent line is realized as the limit of approximate tangent lines in the same way that
the instantaneous velocity is the limit of approximate average velocities. An example of
these approximations was done for the function f(x) = x^2 + 2. We also attempted
a word problem before time ran out.

Class 12: More Derivatives, Feb 1

Several examples of limits were done using the original deinition of a limit and the book's second definition of a limit. It is this second definition that Chrissy was asking about that got the teacher confused.

We also did the binomial coefficients theorem.

Class 13: Word Problems and Rules for Derivatives, Feb 3

There are three essential steps to solving a word problem. The first step is to assign
variables to the constants and functions mentioned in the problem. Later, it may be
convenient to define some new variables, but it is important to start with defining
anything mentioned in the problem.

The second step is to translate the statements and words in the problems into
mathematical expressions. A sentence like Jack has three more apples than Jane
might become a = b + 3, where a is the number of apples Jack has and b is the number of
apples that Jane has. Don't worry about solving anything until you have all of the facts.

The third step is to solve for the required variables using the appropriate tools.
Some problems involve algebraic manipulation, and are just of the form `solve for b'.
Other problems involve a rate of change or velocity and they require calculus by taking
a derivative.

During this lecture we also surveyed the basic rules for differentiation.
The product rule and quotient rule. Also, (x^n)' = n(x^(n-1)) for any integer n.
A couple of proofs were done in class. Next class we will do the chain rule.

Class 14: Chain Rule, Feb 5

Today, we covered the chain rule. Also, we introduced the Liebniz notation.

Class 15: Higher Derivatives and Review, Feb 10

Higher derivatives are very useful in applications. For instance, the second derivative of
a distance function is the acceleration. It can also be thought of as the rate of change of
the velocity.

Class 16: Implicit Differentiation, Feb 15

Today we studied implicit differentiation. We used it to prove the power rule for rational exponents. The case for real exponents was given as an extra credit problem.

Class 17: Related Rates, Feb 17

Today we did some word problems involving related rates. Some problems have formulas that are implied and not stated. Some of these we did in class. Ladder problems involve right triangles, and container problems use formulas from solid geometry.

Class 18: Approximations, Feb 19

Approximations of functions can be made by using the derivative. This can be easier and simpler than evaluating the function multiple times. Also, depending on the application, it can give all of the necessary information.

Class 19: Maximum and Minimum, Feb 22

After doing some of the old related rates, we introduced problems involving maximums and minimums. There are three possibilities, endpoints, stationary points and singular points. Hopefully, these sets will be finite and we can test them one by one to find the maximum and minimum.

Class 20: Existence of maximums, Feb 24

We saw some examples of functions without maximums, and tried to see why a continuous functoin has a maximum on a closed bounded interval.

Class 21: More maximums, Feb 26

We studied local maximums and minimums. Defined concave up and down, and did inflections points.

Class 22: More maximums, Mar 1

The first derivative test and the second derivative tests may be applied to determine whether a critical point is a local max or min.

Class 23: More maximums, Mar 3

After doing a related rates from the homework, we talked about finding a function from its derivative, qualitatively.

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Back to the winter 131 class page.

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Classes: Calculus 132

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Class 1: Overview, Mar 29

In the first class, we summarized the material we will cover this quarter. Trigonometry, and the calculus of trigonometric functions. Later, we will concentrate on integration. An application of integration is to find the area under a curve in the plane such as the area under a section of a parabola. In word problems, integration sums up the rates of change. It is the opposite of differentiation.

Class 2: Review of Pre-calculus, Mar 31

After assigning homework and making suggestions for those who feel that their math is rusty, we reviewed the basic skills of precalculus. Fractions and rational functions can be added and multiplied. Exponents can be simplified according to the basic rules.

Class 3: Review of Calculus, Apr 2

This day was for a review of calculus covered in math 131. The main tool is the concept of a limit. It is used to define the derivative and to define continuous functions. Word problems consist of rates of change. A rate of change is like the speed of a car, being the derivative of its position.

Class 4: Trigonometry, Apr 5

By this point we have covered the trigonometric functions as ratios of sides of a right triangle. Also, the functions sine and cosine give the coordinate functions for the unit circle. The trigonometric identities follow from the pythagorean theorem, and are derived by dividing both sides by either a^2, b^2, or c^2.

Today we gave a geometric proof that sine is continuous, leaving the case of cosine as extra credit. Then we showed why the limit of sin x over x as x tends to zero is 1. We used that cosine is continuous to show the other fundamental limit: (1- cos x) /x tends to 0 as x tends to zero.

Class 5: Trigonometry and Calculus, Apr 7

Today we reviewed the material from last time, and studied how the graph of sine and cosine change depending on the function in the argument. The basic example is the linear argument: Asin(mx+b). We also considered sin x^2 and sin 1/x.

Class 6: More Trigonometry and Calculus, Apr 9

Using the basic trig limits and the angle addition formulas we derived the derivatives for the functions sine and cosine. Namely, the derivative of sine is cosine and the derivative of cosine is negative sine. One way to remember the difference is for angles greater than zero and less than a right angle, these functions are both positive. For these angles, sine is increasing, going up the circle, while cosine is decreasing, moving to the left.

Class 7: Even More Trigonometry and Calculus, Apr 12

We started by showing how to draw auxilliary lines in a triangle such as an altitude to relate the sides and the trig functions when the triangle is not a right triangle. The rest of the class was spent covering the rules for differentiation like the product rule and the chain rule. From these we can derive the derivatives of the other trig functions, as well as the derivatives of trig functions with nonlinear arguments.

Class 8: Derivatives of Trig Functions, Apr 14

Taking the derivatives of the trig functions is a great way to practice using the three rules for differentiation. Product Rule, Quotient Rule, and Chain Rule. That is what we did today.

Class 9: Max and Mins and Word problems, Apr 16

First we redid the lighthouse problem using related rates. Then we reviewed how to do a maximize problems by finding the critical points. There are three kinds: end points, stationary, and singular points.

Class 10: Mean Value Theorem, Apr 19

After reviewing some basic skills in trigonometry and calculus, we studied the mean value theorem. It applies to functions differentiable on an interval. For such a function and interval, there is an average rate of change given by the toal change in the function divided by the length of the interval. The MVT says that there is a point in the interval for which the derivative, being the instantaneous rate of change, is equal to the average rate of change. An example is the position function of someone who drove a car 100 miles in two hours. Assume the function is differentiable, and the mean value theorem says that there was a time during those two hours when his speed was exactly 50 miles per hour.

Class 11: Applying the mean value theorem, Apr 21

The mean value theorem is a powerful tool for making precise the relationship between the derivative and the rate of change. We used it today to prove that a function whose derivative is zero must be constant. We also showed that a function with a positive derivative must be increasing.

Class 13: Derivatives and Anti-derivatives, Apr 26

There are a couple of basic patterns which arise when taking derivatives of functions. The power rule, the product rule, the chain rule. All of these have their own patterns. The power rule has exponents; the product rule has the sum of two terms; the chain rule has a multiplication. The anti-derivative of a function f is a function g such that the derivative of g is f. An example is that sin x is an anti-derivative of cos x, while -cos x is an anti-derivative of sin x. The trick to finding the anti-derivative of a function is to recognize the patterns that derivatives take.

Class 14: Sums and Differential Equations, Apr 28

The trick to summing up the first 11 integers is to realize that their average value is 6. Hence their sum is 6*11=66. The trick to summing the first 5 powers of two is to multiply by 2 to get the first 6 powers of 2 minus one. The difference is 2^6 -1, which is the same as the original sum. Summing changes to find the total change is the opposite process to finding the rate of changes from the total change. This is why sums are related to Calculus and we will make this idea more precise when we do the Fundamental Theorem of Calculus. In this class, we also discussed the differential equations. In an application, one can find relations between rates of change, which gives a differential equation. One example is a population model in which the rate of change is proportional to the amount of population. For example, the equation f' = 2f.

Class 15: Sums and Sequences, Apr 30

A sequence is a list of numbers. The sum of a sequence of numbers can be written in the sigma notaion. It is particularly convenient when the sum is of a long list of numbers. Another way it is convenient is that sometimes there is a formula for the sum, for instance the sum of the first n counting numbers or the first n powers of 2. Later we will take sums of arbitrarily long sequences.

Class 16: Areas, May 3

After developing an intuition about area by considering rectangles and triangles, we did Archimedes derivation of the area of a circle. It shows how an area can be computed by taking a limit of approximations. The idea of taking a limit of approximations is useful because the approximations can be calculated. In particular, we will use rectangles to approximate the areas under curves. Today we did straight lines, which correspond to triangles.

Class 17: More Area Sums, May 5

In detail, we studied the area sums for the curve f=x^2 on the interval [0,1]. The left hand sum is the sum of the areas of n rectangles where the height is given by the left hand endpoint. The base of the rectangles is taken to be the same size for each rectangle. The right hand sum is similar the only difference being that the height of one of the rectangles is the height of the graph on the right hand side. In either case, there is a general formula using the sigma notation for the sum.

Class 18: Riemann Integrals, May 7

At last, we have covered the important Riemann Integral. It is calculated the same way as the areas have been calculated. One difference is that it may not represent an area since it will be negative when a function is negative (area is never negative). Another difference is a theoretical distinction. The height of the rectangles can be f(c) for any c in the interval, including the right hand side or the left hand side. Also, the partition does not have to divide the interval into equal pieces.

These last two distinctions make the Riemann integral harder to calculate, but easier to use for theoretical purposes. In any case in which we will come across, the Riemann integral can be calculated using the right hand side formula with regular partitions that we were using before to calculate area. The reason for the greater generality of the Riemann integral for us will be seen next class in the proof of the fundamental theorem of calculus.

Class 19: Riemann Integral Close Up, May 12

A function is integrable when the limit of the Riemann sums converges. The notion of convergence is a bit more subtle than the convergence of the usual limits considered in Math 131. The idea is that the sum of the rectangles is close to the limit as long as the biggest piece of the partition is not very large --no matter what points are chosen to find the height of the rectangles.

This notion of convergence is very strong and it allows us to prove the fundamental theorem of calculus. In the proof we ssume the limit exists, and then we are free to pick the points to take the height of the rectangles to coincide with the value c from the mean value theorem.

Continuous functions are always integrable, while some noncontinuous functions are and others aren't. An unbounded noncontinuous function like 1/x or 1/x^2 is not integrable when the interval contains the unbounded part (x=0 in this case.) We also saw that the characteristic function of the rationals is not integrable.

We also saw that the linearity properties of the integral follow from the analogous properties of sums.

Class 19: More Integral Formulas, May 17

After going over the exam we reviewed some of the integral formulas. The formulas for linearity and interval additivity are important. When the Fundamental theorem of calculus holds, the linearity follows from the linearity of the derivative and interval additivity is trivial. These properties also hold when FTC does not apply.

We lloked at the comparison property and provided a short proof. It says that if g is larger than f then the integral of g is larger than the integral of f. We used this to show that for positive angles x, sin x is less than x, by expressing these functions of integrals of 1 and of cos x. Since cos x < 1, then so goes for the integral of each side. We also used the comparison property to show the boundedness property.

We ended class with the integral version of FTC and the corresponding Mean Value Theorem for integrals.

Class 20: Tricks for doing integrals, May 19

Substitution is a good way to find an integral. Typically, it makes the problem alot simpler, although it may not be clear that it will do so.

Class 21: Natural Lograithm, May 21

Then natural logarithm is defined to be the integral from 1 to x of 1/t. Using this definition, it is easy to see that it gives an anti-derivative of 1/x for x>0. It is only defined for positive values. The book has a good presentation of the properties of this function.

Class 21: Natural Lograithm and Exponential functions, May 21

The natural logarithm is increasing so has an inverse function called the exponential function. It has the opposite properties of ln. It turns addition into multiplication. Also, it cancels ln in formulas.