Feedback on Assessment quiz for Math 231Complex numbers:If you didn’t know this stuff: It’s an easy fix. You should remember complex numbersbeing introduced in the context of quadratic equations. i is a quantity whose square is −1.It can obviously not be a real number. So we are extending the number system of realnumbers, by includ ing i, and all other quantities entailed by it, like, for ins tance 3 + 5i.This larger number system is called the set of complex numbers.For instance, when you looked at a quadratic equation like x2+ 4x + 13 = 0, the quadraticformula would give you the solutions−4±√42−4·132=−4±√−362=−4±6i2= −2 ± 3i.If you think introducing ‘crazy numbers’ as solutions to equations that otherwise wouldn’thave solutions is a weird idea, you are perfectly right. The quadratic equation context inwhich you first encounted complex numbers like 2 + 3i is a poor reason for encounteringthem; but they don’t have a better motivation at this level. A better reason will be thatcomplex numbers actually “come out of the elctric AC outlet”, so they are very practicaland genuine! We’ll see how and why this is the case, when we comne to chapter 4; and we’llembrace complex numbers in this class rather than giving them a brief guest appearanceand then snubbing them again, as the book seems to do.If you had p roblems with this part, ou need to review the algebra (adding, multiplying ofcomplex numbers) by chapter 4. As for dividin g them, the trick is similar as when youmade the denominator rational in expressions like1+√24+√2: Compare:1 +√24 +√2=(1 +√2)(4 −√2)(4 +√2)(4 −√2)=4 − 2 + 4√2 −√216 − 2=2 + 3√214with2 + 3i4 + 2i=(2 + 3i)(4 − 2i)(4 + 2i)(4 − 2i)=8 − 6i2+ 12i − 4i16 − 4i2=14 + 8i20=7 + 4i10Everything else you need to know about complex numbers in this course will be taught aspart of th e course.Integrals:If your integral skills are a bit rusty now, you should begin revamping them by reviewingthe key techniques, and where they apply. Refer to the appendix of the book, and to notesI make available for this purpose.If I believe, based on the quiz, that you’ll need a lot of revisiting, I’ll tell you so individually.Exponentials and Logs:If you have difficulty with this pre-cal material, this may already have shown up duringyour calculus class. And then it is a predictor for trouble, and you may expect seriousdifficulties in 231.If you recognize yourself in this pattern: “never really understood exponentials, logs (andpossibly trigs) thoroughly, got through Calc 1 ok, but with an uneasy feeling, alwaysalgebra trouble in Calc 2”, then your chances for 231 are precarious and you need tutoring1asap. And you may need to bud get the equivalent of extra credit hours’ worth of time todeal with thes issues along with the new material.Trigs:I’m not concerned on whether you have memorized the addition theorems for trigs atthis p oint. What you should have memorized is the graphs of sine and cosine (and notconfusing them), and the fact that sin2x + cos2x = 1. And that sin(−x) = −sin x andcos(−x) = cos x, which info is contained in the symmetry of the graphs.Now if you didn’t get the quiz questions, then what you really want is a look at is thefollowing bit of wisdom:“I memorize thattrig(sum or difference) = trig(one) trig(other) plus or minus trig(one) trig(other),where trig is sin or cos, and all I have to worry is which goes where”. Can you memorizethat much? Try to.Now look at sin(x + y) Clearly, exchanging x and y does not change the expression. Thisrules out that the expression could be T1. Also, if I take x = 0, the result must be sin y.This rules out T1, T2, T3, T4. Or, it has to work with y = −x, when the formula has toproduce 0.Look at cos(x + y) now. Again, the result cannot change under swapping of x and y. Ifx = 0, the result should be cos y. Also, if y = −x, we want the result to be cos 0 = 1.Which options survive?If you only memorize the general pattern as outlined above, and then rule out the unfeasiblechoices by the above type of reasoning, you will be able to reconstruct all the additiontheorems. And once you have done this reconstruction work several times (looking up theformula afterward s, just to ch eck), you’ll have a more thorough understanding of the trigsand will see that you suddenly have memorized these addition theorems almost painlessly,as a side effect.Question 5:The vast majority had difficulty with this, and it is core matter. I’ll address it in class;but I am not sure how much I can predict success or trouble fr om this one problem.Series:You should commit the series of ex, cos x and sin x to memory, as we’ll rely on them tounderstand why “complex numbers come out of the AC wall outlet”. If you have neverbeen easy with the discussion about power series in the first place, refer to calculus notesI am providing as a review. They are actually in a different pedagogical design that theusual Stewart Calculus textbook approach .Catching up here is not such a daunting task and you want to do this before chapter