Today we went over the slides of algebra review in the folder of Week 01, covering the following topics in class:

• Multiplying polynomials. 
• Binomial theorem. 
• Difference of squares and cubes. 
• Factorizing quadratic polynomials.
• Fractions.

We did not finish everything but stopped at the formula for compound fraction. Please go over the rest of the slides, including the examples of compound fractions and rationalizations. You may also use the video (either on YouTube or in the folder).

On Friday, Collier will conduct a review of trigonometry. The quiz on next Monday will be on the algebra we reviewed today together with the trigonometry. You may find some exercise problems in 0.1-1.1-HW.pdf and 0.2-Trigonometry.pdf.

In the Week 00 folder you may also find the PDF file containing some selected sections of the precalculus textbook by Stewart. You may also use that PDF file to review precalculus. 

Please also note that we do not have any lecture time for Chapter 1 of the textbook that covers the basics of functions. Although we will not quiz those topics next Monday, please do spare some time and go over the materials in the Week 00 folder. 

Additional Announcements: 

1. The location of Friday Recitation is Boettcher Center Auditorium 102. It should appear also on your schedule by now. 
2. Collier will also hold office hours on Thursdays 9-10AM. 
3. The information regarding Math Help Center is supplemented to the syllabus. 

We covered the Stewart Textbook Section 2.2. Here are some key facts you should keep in mind. 

1. Numerical experiments help to guess the limit. But it is not reliable. 
2. The limit of a function f(x) at a number x = a depends only on the behavior NEAR x = a, while independent of the function behavior AT x = a. 
3. . 
4. The limit exists if and only if both one-sided limits exist and are equal to each other. 
5. If the left-hand limit is not equal to the right-hand limit, then the limit does not exist. 
6. Basic principles in deciding infinite limits: 
1. A positive number dividing a small positive number is positively large. So the limit is . 
2. A positive number dividing a small negative number is negatively large. So the limit is . 
3. A negative number dividing a small positive number is negatively large. So the limit is . 
4. A negative number dividing a small negative number is positively large. So the limit is . 

Please keep in mind  that essential materials will be re-written on the board. If I am reading the slides without writing on the board, I do not expect you to master the knowledge. For example, I introduced the intuitive definition of limit in class. Having some ideas of this definition would be far more than enough. To seriously handle this definition requires much more sophisticated training in first-order logic, which is way beyond the level of first-year Calculus.

Another example is the numerical experiment. I went over the topic very fast simply to assist the understanding of limits and, more importantly, show the existence of pitfalls. I will not expect the students to carry out any numerical experiments. 

As a result, please do not waste your energy on taking notes of the slides. All slides are released in the course folder. You may revisit it by yourself and at your own pace any time at your convenience. Please keep you energy only for the materials I write on the board. 

Homework Problems today will be 2.2 exercises. You are now ready for almost every problem except for 8(c) and 44(a). Problem 12 requires certain knowledge of trigonometry functions and piecewise functions. We will not have more lecture time to review these concepts. Please go over either the videos or the slides for 1.1, 1.2, 1.3 (in Week 00 folder). 

We finished Section 2.2 and the first half of 2.3. Here are the things to keep in mind: 

1. 
2. Algebraic laws for limits concern the following operations: Sum, Difference, Constant Multiple, Product, Quotient, Power, and Root.
   These laws are intuitive. But please make sure you use them correctly. The laws work only when the conditions are satisfied. Anything can happen if the conditions fail (cf. Example 1b, Example 1').
3. Another set of laws concerns the limit of x, the limit of constant, and the limit of powers of x.
   3.1. Please make sure you can justify the steps as I did in Example 2. Please practice with Problem 6 in the 2.3-HW.
   3.2. Using these laws together with algebraic laws, we can prove the direct substitution property for polynomial and rational functions. Please find the details in the homework solution. They will not be required in quizzes or exams.
4. The direct substitution property fails when you get 0/0, which means you are dealing with a type of indefinite limit. You should use algebra (factorization, expansion, rationalization) and transform it to a definite limit. 

For HW-2.3, you are now ready for everything in the first three pages and the first three on the fourth page. These problems, together with the ones in 2.2, may appear in the quiz next Monday. 

Note: Generally the homework problems are more difficult than the example problems I went over in class. So it is completely natural to struggle. Please do not hesitate to ask for help from me and from Collier. 

Today we finished Section 2.3 and started the discussion of Section 2.5. Here is the knowledge you need to keep in mind. 

1. For the limit of a piecewise function at a critical number (where the function has two different definitions to the left and the right), you should use one-sided limits to take care of the ambiguity. 

2. The limit comparison theorem is very important theoretically, but we will not be using it a lot.

3. For the squeeze theorem, please make sure you are comfortable with the example x^2 sin(1/x), and Problem 38, 40 in the Homework. For Problem 40, you should use the fact that the function e^x is increasing to obtain upper and lower bound functions. 

4. Example 10 and Example 11' will not be required, and I skipped them in class. You are welcome to work on them by yourself. Feel free to ask for assistance for sure. 

5. A function is continuous at a number a if

• is defined.
• exists.
• coincides with the function value .

    These three conditions are compactly incorporated into the equation

6. Types of discontinuities:

• Removable: exists but not equal to or does not exist. We call it removable because it can be removed by redefining (or defining) .
• Infinite discontinuity: One of the one-sided limits is infinity (positive or negative). Equivalently, when is a vertical asymptote.
• Jump discontinuity: Both the left-hand limit and the right-hand limit exist, but they are not equal.
• Other types, e.g., .

7. We use left-hand-limits and right-hand-limits to define continuity from the left and continuity from the right.

8. A function is continuous on an interval if the function is continuous at every number in the interval.
8.1. If the number happens to be the left endpoint of the interval, we require the function is continuous at the number from the right.
8.2. If the number happens to be the right endpoint of the interval, we require the function is continuous at the number from the left.
Make sure you know how to appropriately justify that a function is continuous on an interval.

Today we covered the four important trig limits, namely, 

These limits are obtained using geometry. More details are included in the videos and slides. You may go over them at your own pace. 

Basically after today's lecture, you should know that most elementary functions are continuous on its domain. The direct substitution property works not only for polynomial and rational functions, but also works for trig, exp and log functions. Next lecture we will finish 2.5 with compositions of continuous functions and intermediate value theorem. You are now ready for all homework problems in 2.5 except for 36, 38, 53, 55, 62, 52. Please attempt them to prepare for the quiz on Wednesday. 

Since we will have two quizzes on Monday and Wednesday, plus that we have not finished 2.7 and 2.8, I decided to push the quiz back for another week. That shall leave you enough time preparing for it. 

We finished the discussion of continuous functions. Here are things you need to keep in mind: 

1. If while is continuous at , then the limit of the composite function as x approaches is simply . You may also interpret it as that the limit operation commutes with , i.e.,
   
   This fact is useful in the computation of limits of complicated functions.
2. Compositions of continuous functions are continuous. This fact is useful in deciding where a complicated function is continuous.
3. Elementary functions (functions obtained by sum, difference, product, quotient, root, composition of polynomial polynomial functions, rational functions, trig functions, inverse trig functions, exp functions, log functions) are continuous everywhere on their domain.
4. The absolute value function and piecewise functions are not considered as elementary functions. You need to apply the standard techniques of piecewise functions to determine the continuity.
5. Intermediate value theorem: For a function that is continuous on , if is a number such that (or ), then there exists such that . 
   This theorem is especially useful for getting numerical solutions to an equation.

You are now ready for all the exercises in 2.5. Please go over them and make sure you are comfortable with all of them. 

Today we almost finished the 2.6. 

1. Limits at infinity and asymptotes. The definitions are too long to fit in here. Please find them in the slides.
2. To compute the limits of fractions at infinity, the standard technique is to divide both the numerator and the denominator by the highest power in the denominator. This technique is especially helpful in handling a -type indefinite limit. 
3. Be careful when radicals are involved. When you are computing the limit at negative infinity and want to put something inside a radical, please make sure you follow the rules carefully:

• • 
  • for every
  • Be aware that . Saying that they are equal constitutes the crime of abusing algebra, punishable by the banishment from science and engineering.

The only left-over is the indefinite limits . Next lecture we shall cover that and start derivatives. You are now ready for all the homework exercises in Section 2.6 except for 28 and 30. 

The midterm next week will be covering everything up to the lecture today. Topics discussed next week will not be included. I will release more information regarding the midterm once I finish the first draft. 

We finished the leftovers in 2.6 and all the discussions in 2.7. Homework for today will be the exercises in those two sections. 

Here are things you need to keep in mind: 

1. Currently we have seen three types of indefinite limits: . For such indefinite limits, you need to use algebra to transform them into definite limits. It requires experience and practice. 

2. Generally speaking, for any quantity that changes over time, the average rate of change from to is given by the difference quotient 

2.1. Geometrically, the difference quotient is the slope of the secant line passing through and . 
2.2. In case is the position function of an object moving along the number line, the difference quotient stands for the average velocity from to . 

3. The instantaneous rate of change is given by the limit of the difference quotient 

3.1. Geometrically, the limit is the slope of the tangent line passing through . 
3.2. In case is the position function of an object moving along the number line, the limit stands for the instantaneous velocity from to . 

4. The derivative of a function at a number a is defined as the following limit 

Equivalently, 

The computation of derivatives by its definition is an important skill. Please make sure you fully understand all the details. Next lecture, we will see some more examples. 

Today we finished most of Stewart 2.8. Please pay special attention to the piecewise function problems. 

Here are what you need to keep in mind: 

1. The derivative , as the number a varies, defines a function, which will also be called the derivative of , denoted . The derivative might have its own derivative, called the second derivative, denoted . We can repeat the process and define the n-th derivative for any positive integer. 

2. If is the position function of an object moving along a straight line, then the first derivative is the velocity, the second derivative is the acceleration, and the third derivative is the jerk. 

3. If a function has a horizontal tangent at a number , the . We have seen in class that if a function is increasing (resp. decreasing) on an interval I, then the derivative (resp. ) for every in the interval I. Using this observation, we can roughly sketch the graph of based on the graph of . 

4. The domain of can be smaller than the domain of . In fact (not in the slides or videos), the domain of is ALWAYS an open interval. 

5. A function differentiable at a number must be continuous at that number. But the converse does not hold: is continuous at 0 but not differentiable at 0. 

6. If is differentiable at a number , then as we zoom in on the graph of a function near the point , the graph of should be closer and closer to a straight line (with a finite slope). 

7. If is a corner point, a point of discontinuity, or a point with a vertical tangent, then is not differentiable at . 

Today we finished 2.8 and almost everything in 3.1. Here is the knowledge you should keep in mind. 

Here is what you should keep in mind: 

1. The derivative , as the number a varies, defines a function, which will also be called the derivative of , denoted . The derivative might have its own derivative, called the second derivative, denoted . We can repeat the process and define the n-th derivative for any positive integer .

2. If is the position function of an object moving along a straight line, then the first derivative is the velocity, the second derivative is the acceleration, and the third derivative is the jerk.

3. Differentiation rules: For any function differentiable functions f, g, and any constant c,

4. The slope of the normal line of y = f(x) at x = a is simply -1/f‘(a).

5. Power rules . Please make sure you can comfortably rewrite a radical expression into power functions and vice versa.

6. . Essentially, these two formulas are justified with the fact that and . Justification of these two limits is beyond the reach of this course.
   (Warning: L'Hospital's rule does not provide a justification. Using L'Hospital's rule to compute these limits is essentially committing the crime of circular reasoning, punishable by banishment from mathematical research)

Please ensure you finish the review of trigonometric, exponential, and logarithmic functions as soon as possible. We didn't have to worry about them. We will have to very soon. 

You are now ready for all homework problems in 2.8 and 3.1. The only leftover in 3.1 are the normal lines. We will go over a few examples on Wednesday before getting into the product rule and the quotient rule. 

We finished the justification of the product rule, the quotient rule. Using the derivatives of sine and cosine function, we deduced the derivatives of all other trigonometric functions using quotient rules.

Here is what you should keep in mind: 

1. The normal line of passing through is a line perpendicular to the tangent line. 
   1.1. As a result, the slope of normal is . 
   1.2. The equation of tangent is
   1.3. The equation of normal is
2. Product rule: . Quotient rule: . 
   1.1. Professor Rob Borgersen at the University of Manitoba invented this way of memorizing the quotient rule:
          square low, then low D high, minus high D low. 
   1.2. You may also find many crazy ways on YouTube that help you to memorize the quotient rule. Just pick the one you like.
3. The formulas and are obtained using the following facts:
   2.1.
   2.2.
         We will provide a full argument next lecture. We will also discuss other applications of these limits. 
4. The derivatives of are obtained from the formulas in 2 together with the quotient rule.
   4.1. Whenever there is a co-prefix in the trig function, the derivative contains a negative sign. 

The examples we did in class are less technical than the course requirement. Please make sure you work out ALL homework problems. 

You are now ready for the homework problems from 3.1, 3.2, and almost all problems in 3.3 (excluding the last six: 40, 42, 44, 46, 48, 50). All these problems are straightforward (unlike those in Chapter 2). Please make sure you go over every one of them. 

We finished the leftovers in the section on derivatives of trigonometric functions and half of the chain rule. You are now ready for all the leftover problems (40 - 50) in the Homework for 3.3. Please review trigonometry and make sure you are comfortable with everything in the notes.  

Here is what you need to keep in mind. 

1. Many trigonometric limits can be computed by the knowledge of. To figure out when to apply this formula, you must be skillful in trig identities. You should do the trig review now if you haven't done it yet. 
2. Chain rule can be applied either using the formula in terms of outer and inner functions, i.e., 

or using the formula with Leibniz notation, i.e., 

The former is faster and recommended by the textbook, but sometimes may cause confusion when the function is complicated. The latter is slower but will usually clear all possible confusion. I would suggest using the latter for complex problems. 

We finished the tricky chain rule problems and the implicit differentiation. Note that the video and the slides does not contain any discussion of inverse trigonometric functions because Canadian universities normally leave them to Calculus II. In case you did not make it to the lecture, please make sure to get the notes from your peers. We discussed the inverse function and the formula for the derivative. As an application, we have obtained the derivatives of ln x, arcsin(x), and arctan(x). 

Homework problems this week will be the leftover trig limits in 3.3, together with the problems in 3.4 and 3.5. Please go over them and get fully prepared for the quiz and midterm next week. 

Here is what you need to keep in mind: 

1. The implicit differentiation is convenient in obtaining from a relation of x an y.

1.1. When you take the derivatives of an expression of y, remember to use the chain rule that . 

1.2. Your final answer of should be in terms of both x and y. 

1.3. To find the second derivative, simply apply to the result of obtained from the previous steps. Make sure you replace any occurrence of y' or by the result from the previous steps, and you final expression is in terms of x and y only without y' or . 

2. A function is one-to-one if Equivalently,

2.1. The square function is not one-to-one, while the cube function is one-to-one. 
2.2. The graph of a one-to-one function should pass the horizontal line test: any horizontal line intersects the graph of the function at most once. 

3. A one-to-one function with domain A and range B admits an inverse with domain B and range A, satisfying

3.1. The inversion law holds: for every x in B; for every x in A. 
3.2. The graph of can be obtained by reflecting the graph of f about the diagonal . 
3.3. We may apply on the inversion law to find the following formula for the derivative of the inverse function, namely, 

4. The inverse of the natural is . The derivative of is

5. The sine function does not have an inverse in general. But if we restrict its domain to , then the graph passes the horizontal line test. 
5.1. The inverse of the function is commonly denoted by . 
5.2. Since , we know that
5.3. Using 5.2 and 3.3, we can obtain the formula that

6. The tangent function does not have an inverse in general. But if we restrict its domain to , then the graph passes the horizontal line test.
6.1. The inverse of the function is commonly denoted by . 
6.2. Using the formula for , we have
6.3. Since , we have

Today we talked about derivatives involving logarithmic functions, together with logarithmic differentiation. I will not require logarithmic differentiation in the quizzes and the exams, though it is a good technique to know. 

We also discussed related rates. The mathematics of related rates is easy: once you get the relation, simply apply d/dt on both sides. The difficult part is to mathematicalize the problem. We will not be very demanding on the kind of models 

Here is the knowledge you need to keep in mind: 

1. for every .

1.1. These formula can be extended as for every . I did skip the details in class. But these extensions will be important in Calculus 2. Rigorously this extension is also necessary for logarithmic differentiation. You do not need to worry about it for now. 

1.2. More generally, we have for every differentiable function .
2. Logarithmic differentiation can be applied to differentiate very complicated functions involving complicated products, quotients and powers. 
2.1. Please keep the following logarithmic laws in mind: , as well as the change of basis formula . In case you are not familiar with these rules, please go over Eddie Woo's video on logarithmic review. 
2.2. To perform logarithmic differentiation on , we first take the logarithm on both sides, getting , then use logarithmic laws to organize the right-hand-side, then take the derivative to get , finally multiply both sides by y. 
2.3. For functions of the form , logarithmic differentiation is the only possible way. You should make a clear distinction between the derivatives of which are obtained respectively by the derivative of constant, power rule + chain rule, exponential function + chain rule, and logarithmic differentiation. 

3. If two quantities are related, then their rates of change are related. 

3.1. The relation of the rates of change are obtained by implicit differentiation. Note that you are supposed to view all the variables as functions of the time, and use chain rule accordingly. 

3.2. Please make sure you do not plug in any numbers before taking d/dt. Only plug in the numbers after taking d/dt. 

3.3. We will only be requiring similar models in Example 1-4. Today we finished Example 1-3. I will still go over Example 4 together with Example 5 on Wednesday. 

We finished the discussion of related rates in Section 3.9 and started the discussion of maximum and minimum values in Section 4.1.

In the final exam, only the models of Example 1 - 4 in 3.9 will appear. Please make sure you understand those models and the related homework problems. The model in Example 5 will appear in many engineering and science courses. If you do have time and energy, please go over it. I will not require that model in the final exam. 

I have posted two video explaining all the 3.9 HW problems on Canvas. Note that the first 26:30 of the video "3.6-3.9-HW.mp4" covers logarithmic differentiation, which will not be required in the final. 3.9 homework starts from 26:30. There is a supplementary video "3.9-HW-Last-Three.mp4" covering the last three problems in 3.9-HW. 

Regarding 4.1, here is the knowledge you should keep in mind. 

1. What is the absolute maximum, the absolute minimum, a local maximum, a local minimum of a function. 
2. Extreme Value Theorem: A continuous function on a closed interval always attains absolute maximum and absolute minimum. 
3. Critical number c: either f'(c) = 0 or f'(c) DNE. 
4. Modified Fermat's theorem: If f takes a local maximum or local minimum at x = c, then c is a critical number of f. 
5. Closed interval method: To find the absolute maximum and minimum of a continuous function, we first find all the critical numbers, then compare the function values at the critical numbers and endpoints. The largest number will be the absolute maximum; the smallest will be the absolute minimum. 

We have only done one example today in class. We shall see more examples next Monday before going into 4.2 and 4.3. 

We did another example from Section 4.1, then briefly discussed Mean Value Theorem and discussed almost everything in 4.3 (only leaving Second Derivative Test). You will need to use inequalities extensively in this section to handle the related problems. If you don't feel confident, please go over the techniques in second chance materials for midterm 1. 

Here is what you should keep in mind: 

1. Mean value theorem: If a function f is continuous on [a, b], differentiable at (a,b), then there exists c \in (a,b), such that
2. If a function has zero derivative on an interval, then the function is constant on that interval.
3. If two functions have the same derivatives on an interval, then they differ by a constant.
4. If a function has positive derivative on an interval, then the function is increasing on that interval.
5. If a function has negative derivative on an interval, then the function is decreasing on that interval.
   Remark: All these conclusion may fail if we are not discussing on an interval.
   Remark: 4 and 5 are commonly referred as the increasing / decreasing test, short for I/D test.
   Remark: When you present your result on increasing / decreasing intervals, do not use the union notation.
   Remark: All these results are proved by the Mean Value Theorem.
6. Knowing where a function is increasing and decreasing, we can decide if a function takes local max or local min at a critical number. This method is called first derivative test.
   Remark: First derivative test is convenient for single-variable calculus, but usually not possible to carry out for multi-variable calculus. Next lecture we will discuss second derivative test, which is not convenient for single-variable calculus and contains some caveats, but might be the only tool you have for multivariable calculus.
7. A function f(x) is concave upward on an interval if, at each number c of the interval, the graph y=f(x) near c is above the tangent line at c.
   A function f(x) is concave downward on an interval if, at each number c of the interval, the graph y=f(x) near c is below the tangent line at c.
8. Concavity test: 
   f(x) is concave upward on an interval if f''(x) is positive over I. 
   f(x) is concave downward on an interval if f''(x) is negative over I. 
9. Inflection point: (c, f(c)) is an inflection point if the concavity changes at x=c. 

We certainly covered a lot of materials today during the online lecture. Please make sure you get enough practice from the Homework Exercises of 4.1 and 4.3. Homework Problems in 4.2 will not be required. You may safely skip those problems. 

We summarized the guideline of graph sketching in class. In order to convey the ideas for the initial examples, I deliberately ignored the subtle issues involving critical numbers where the derivative does not exist.

Here is the complete version of the guideline, which has some discrepancy to the version in the video and slides. 

• Domain: Find the domain of the function.
• Intercepts: Find the y-intercept by setting x = 0 (which is usually very easy). Find the x-intercept by setting y = 0 (if the equation is too complicated then skip this step).
• Symmetry: Decide if the function is odd, even or periodic.
  • Recall: f(x) is an even function if In this case, the graph of f is symmetric about the y-axis.
  • Recall: f(x) is an odd function if In this case, the graph of f is symmetric about the origin.
  • Recall: f(x) is T-periodic if In this case, we may skech the graph of f over one period, then translate it to get the full graph. 
•  Asymptotes:
  • Compute the limits at positive infinity and negative infinity and decide if there exists any horizontal asymptotes.
  • For the vertical asymptote, find out if there exists numbers where the function value explodes to positive infinity and negative infinity. Usual candidates for such numbers are at the boundary of the domain.
  • (not required) For slant asymptotes, find if there exists m, b making . For rational functions, slant asymptote exists only when the degree of numerator = 1 + degree of denominator. The slant asymptote can be found by long-division. 
• Compute the derivative. Find the critical numbers by figuring out where f'(x) = 0 and where f'(x) does not exist.
  • If f'(c) = 0, then f has a horizontal tangent at c.
  • (Did not mention in class) If f'(c) does not exist and c is in the domain, find out the left-hand limit and right-hand limit of f'(x) at c and decide if we have a vertical tangent or a cusp point. In greater detail (did not emphasize in the video):
    • If , then f is increasing near c and has a vertical tangent at c. 
    • If , then f is increasing near c and has a vertical tangent at c. 
    • If , then f is decreasing to the left of c and increasing to the right of c and has a cusp point of shape. 
    • If , then f is increasing to the left of c and decreasing to the right of c and has a cusp point of  shape. 
• I/D test: Critical numbers separate the real line into intervals. For each interval, figure out the sign of f'(x) to decide if the function is increasing or decreasing. 
• Local maximum and minimum: Use the results from I/D test to figure out if the function takes a local max or local min at each critical number. 
• Compute the second derivative. Find the critical numbers (of the first derivative) that separate the real line in different intervals. Use the concavity test to decide if the function is concave upward or downward on each interval. Find out the coordinates of each inflection point. 
• Mark the points of interest on the coordinate plane. Draw the vertical and horizontal asymptotes in dash lines. Analyze the behavior near these points and asymptotes. Then connect these bits of information by curve using the information we have obtained above.  

You should now be ready for all homework problems in 4.1 and 4.3. 4.2 will not be required this quarter. You may also try some problems in 4.5 but note that we haven't finished that section. We shall finish it next Monday, then turn to applied optimization in 4.7. 

We finished Example 4.5.1 and went over Example 4.3.7, 4.3.8 in the textbook. So far, we have discussed the graph sketching process for

• Polynomial function (Example 4.3.6)
• Rational function (Example 4.5.1)
• Algebraic function (Example 4.3.7)
• Exponential function (Example 4.3.8)

We haven't been able to discuss the process for 

• Trigonometric function (Example 4.5.4)
• Logarithmic function (Example 4.5.5)

Unfortunately we are left with only two lectures. The plan is to use the Wednesday lecture for optimization, and use the next Monday lecture for a review. I'll try to cover Example 4.5.4 and 4.5.5 after conducting a review. Although I will not ask you to graph such functions, seeing the process may help you to review the necessary knowledge regarding trigonometric functions and logarithmic functions in other sections. 

There are two more examples in Section 4.5, namely, an algebraic function in Example 4.5.2, and an exponential function in Example 4.5.3. I have created the videos discussing all these examples, together with all the exercises in 4.5. You may find them in the OneDrive folder.