Practice quiz 3

Here you go, the quiz is on Dec 1/2

Quiz: quiz3_practice

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Review of week 11/12 and preview of week 13

Hi all,

Hope you had a great Thanksgiving! In week 11, we reviewed for the test and had our test 3 on Thursday. Remember to regrade for more points!

On Tuesday, we started talking about angles, degrees and triangles, these will be the basis for trig functions, starting from next week.

For what we learned on Tuesday, make sure you understand how to compute complementary and supplementary angles, how to compute the length of edges in a right triangle. Similar triangles are triangles of the same shape, and always have the same angles. Therefore their corresponding edges have lengths of a fixed ratio.

Next week, we will introduce trig functions in right triangles and Cartesian planes. In right triangles, they are really just ratios of the length of edges.

Alright, that is it for now. Let me know if you have more questions!

Review of week 11/12, preview of week 13

Hi all,

Hope you all had a fun and full Thanksgiving! For the past one and half weeks, we have started looking at series, whether they diverge or converge, how to compute their sums and such. For the most part, we are interested in power series and when they can be treated as functions. Next week we will continue with power series in reverse, in the sense that we look at normal functions, and see how to represent them with power series.

Series’ are sums of infinitely many items. Like how we always deal with infinities, we define series to be the limit of the partial sum. If the limit exists, we say the series converges; otherwise diverges. In this regard, we can treat the partial sums as a sequence, and all the results we had for sequences apply here. This method (looking at partial sums) normally require us to know a compact form of the partial sum. An example is the geometric series, with the compact form 1+r+r^2+…+r^n=(1-r^n)/(1-r), for all |r|>=1, the series would diverge and only for |r|<1 will the series converge.

In general, it is hard to tell if the series converges without a compact form of the partial sum, but we can tell if they diverge by looking at the limit of the term: the series will diverge for sure if the terms don’t converge to zero. You can use this standard to weed out a lot of divergent series.

In power series, we are adding polynomials of the shape c_n(x-a)^n. Note here x is the variable which is independent of the index n. As long as the series converge, we can look at its sum as a function. The values of x at which the series converge is naturally the domain of the function, therefore we focus on where the series converge.

If we are dealing with a geometric power series, the series converges if |r(x)|<1 (as in the power series). If we are dealing with a more general series, we can use the root test or the ratio test. Both tests are inconclusive at the boundary, so we omit the boundary cases(see sections 11.3 and 11.5).

For next week, we will write general functions in power series. The idea behind this is super cool: functions might be of different shapes and forms, but you can “always” write them in terms of the sum of powers of x. Therefore, if you are studying a “ugly” function, your life will be easier if you instead study the power expansion.

Alright, that is it for now. There are so many cool facts about power expansions and I can’t wait to share them with you guys! See you soon!

Homework (Due Dec 3/4)

Reading: 11.8, 11.9

Problems:

11.8: 6,7,9,10,12,15,18,20,29,30(a,b)

11.9: 4,5,6,8,11,12,13,14,15,18

Remark: For interval of convergence, we only look at the open interval without the endpoints. If you are done with all the homework, you can read section 11.3 and 11.5 to check the endpoints case by case. Turn them in to me and I will give you some extra credit.

Review of week ten and preview of week eleven (Test on Thursday, Nov 19)

Hi all,

Last week we talked about quadratic functions and exponential/log functions.

The quadratic function has two forms: general form and standard form. Standard form gives more information about the function: whether it opens up or down, where the vertex is, where the x-intercepts are (if there are any). If given general form, we complete the square to get the standard form. In applications of quadratic functions for example, vertex gives information about the maximum or minimum of the function, depending on the sign of leading coefficient.

On Thursday, we did more computation on completing the square and briefly introduced exponential and log functions. For exponential and log functions, make sure you know how to evaluate the functions and have a general image of the graphs of the functions in your head (x and y intercepts, domain and range of the functions, if the function increase or decrease?).

Next week, we will review on Tuesday and have test 3 on Thursday. I will write a practice test and bring the practice test on Tuesday for you to work on. For now, review your homework problems and redo the problems we did in class (self diagnose quiz and exercises)

There is no homework due next week, so use your time to review for the test!

Let me know if you have any questions!

Review of week ten and preview of week eleven (Homework due Thursday/Friday, Nov 19/20)

Hi all,

This week we had our test 2 and started looking into sequences. Next week, we will continue with properties of sequences and start with series.

Test 2 focused on general techniques of integration, application of integrations and improper integrals. From the questions I got from you guys, the hardest step is to start a problem and choose which method to use. I suggest you do the problems in section 7.5, as those problems use all the techniques we ever discussed this semester and you need to make a judgement.

For sequences, we care about convergence and divergence as n approaches to infinity. There are some general ideas to help you determine the convergence of a sequence:

  1. Look at corresponding continuous functions. As long as the function has a horizontal asymptote as x approaches to infinity, we can conclude the convergence of sequence. Note the divergence of function *does not* conclude the divergence of the sequence! The reason of looking at continuous functions is that we can use properties of functions that discrete sequences don’t have, such as L’Hospital rule.
  2. Use the properties of limit laws. If we have two convergent sequences, we can conclude the limits of their sums, differences, products, quotients, etc. We can use these properties to chop problems into small pieces and deal with each piece individually.
  3. Compare with known sequences (Squeeze Theorem). Same for all comparison rules, try some values and get a general direction before you charge ahead to find bounding sequences.
  4. In general, if you don’t know how to deal with infinities, try to convert them into zeros.

Next week, we will start looking at “sums” of the sequences, namely, series. You will use series in all sorts of applications, and they have interesting properties themselves as well!

Homework:

Reading: 11.1, 11.2

Problems:

11.1: 5,8,9,11,13,14,17,19,20,21,26,41,54,60,62,64,70

11.2: 9,10,15,16,17,21,22,25,34

Review of week nine and preview of week ten

Hi all,

Last week, we talked about linear functions and lines and started with quadratic functions on Thursday. This week, we will finish with quadratic functions and start talking about exponential and log functions.

Linear functions represent lines on the plane, and are determined by two pieces of information: slope and an intercept, two points, slope and a point, etc. Any two points on the line can give you the slope (sometimes undefined as the line is vertical!). If the slope is positive, the linear function increases as x increases; if the slope is negative, the function decreases as x increases; if the slope is zero, the function does not increase or decrease. Setting y=0 give the x-intercept and setting x=0 gives the y-intercept. These are all information that helps you plot the graph of the line.

Quadratic functions are of the shape y=ax^2+bx+c, and setting y=0 gives the x-intercept(s) if any exists! We have seen last week, that the quadratic equation ax^2+bx+c=0 can have two distinct solutions, one solution or no solution. These three situations correspond to two x-intercepts, one x-intercept or no x-intercept in the graphs of our parabolas. In the process of looking for x-intercepts, we learned three methods: factoring, quadratic formula or completing the square. This coming Tuesday, we will keep using completing the square, as it gives more information: vertex. Vertex gives information on the maximum or minimum of the function and where the max/min is attained. We will also see some applications of quadratic functions in modeling.

Alright, that is it for now. Sorry for the late sum up, let me know if you have any questions!

Review of week nine and preview of week ten

Hi all,

Last week, we wrapped up integration with improper integrals and reviewed for the test this week on Tuesday and Wednesday. After the test, we will start with infinite sequences.

Improper integrals are all definite integrals that have singularities: one of the integration limits being infinity, the function is not defined at some point in the interval of integration or both. Improper integrals are defined as the limit of a normal definite integral, where the limit is taken for the integration interval to approach to the original interval with singularities. By definition, to evaluate an improper integral, you should follow two steps: evaluate a normal definite integral with parameters, then take the limit.

Sometimes all we care about is whether the improper integral converges and not the specific value of it, we can use the comparison rule. Given two non-negative functions f(x) and g(x), if we know f(x)>=g(x), then the improper integral of f converges implies the improper integral of g converges; the improper integral of g diverges implies that the improper integral of f diverges. A lot of the time, we use x^{-\alpha} as the comparison function,  and \int_1^{\infity} x^{-\alpha}dx converges if and only if \alpha>=1. Similarly,  \int_0^{1} x^{-\alpha}dx converges if and only if \alpha<1.

We will talk about infinite sequence on Thursday/Friday, and please review limits from Calc 1. We have already seen sequences when we studied Riemann sums, but we will study more general sequences in Chapter 11 and their properties.

Alright, that’s it for now. Sorry the sum up came later this week!