How do we find the derivative of 1/(1+x) ?

I would recommend rewriting the expression first.

Sometimes if you change how something looks, it becomes easier to work with.

Now you can do the power rule and the chain rule.

Tutoring Calculus, Series, Sequences, Improper Integrals and Limits

We focused on series and sequences and looked a bit at improper integrals.

All of these are connected to limits. If something diverges, which is easier to determine than convergence, you simply say it diverges. If a test is inconclusive, you try another test.

Geometric series might be the easiest to deal with. Sometimes a series doesn’t exactly look like a typical geometric series and needs to be manipulated a bit algebraically. If you have the ratio and the first term, there is a simple formula to determine the sum.

Many times for series, the variable with the highest exponent becomes more important.

An alternating series just needs to be decreasing to be convergent. To be absolutely convergent requires more tests.

The algebra before doing the calculus can be very important. Changing the look of an expression can make it easier to work with.

Knowing the trig functions is necessary for some of these problems.

U-substitution was a useful technique for some of the integration. Before calculus 3, it’s probably a good idea to review integration by parts.

e ~ 2.7

Remember the chain rule when taking the derivative of e^u (most of the time) unless taking the derivative with respect to u.

If you make a substitution in integration, you must change the bounds or label the bounds more carefully.

Reviewing Some Ideas From Calculus I and leading up to it

Tutored calculus for a couple of hours the other day, just before the student started Calculus II. Here are some things we went over.

We talked about the purpose of a derivative and how you can optimize things early on and how integrals can find the areas under various curves and on into three dimensions, etc.

Went over how logarithms can be useful, what the natural number e is, and the behavior of the natural logarithm.

How algebra can simplify calculus before doing derivatives or integrals.

How to deal with fractional exponents.

How the order of approach for fractional exponents can simplify things.

Thinking about the behavior of functions for limits.

How exponents like 1/4 and 1/6 work.

How the ln function increases and decreases very slowly.

How to divide by fractions by multiplying by the reciprocal.

Using common denominators.

Going over integration by parts.

Doing some derivatives and keeping track of the sign.

How squared trig functions look.

Factoring out algebraic terms.

Use the product rule when differentiating a product.

Picking u and v for integration by parts won’t always work out, if it seems to not be working, you may want to choose something else.

Doing an integral is taking an anti-derivative and finding something that when you take the derivative of it will give you what you started with between the integral sign and the differential.

Always take the chain rule for derivatives

Sometimes, if you take the derivative with respect to the variable in the expression (and there is only one variable), you can ‘not’ use it since, for example, dx/dx = 1. But in general, you use the chain rule for derivatives.

Reviewing for the final, we started with the earlier version of a derivative which involved limits.

The letter ‘a’ is used sometimes to denote a constant.

With multiple variables, the chain rules becomes even more important.

The general approach for a problem involving changing dimensions of a rectangle was to find an expression for a particular characteristic and then take the derivative. Certain rates were known in the problem.

How small mistakes propagate and multiply in Calculus problems

We looked some at the previous test. But focused on material for the upcoming test.

One thing was the product rule, he had made a mistake before with it on another test.

Often there were small algebraic errors. The new material seems fairly comfortable though.

If you make a mistake early on it can continue to be in your work for the entirety of the problem and depending on the direction of where you go with the problem, it can get bigger. Let’s say you have four instead of two, then you square that number, suddenly the mistake got larger.

Calculus involves a lot of algebra, geometry, etc. Sometimes there can be many steps and mistakes add up and get bigger. Often times it is better to slow down a bit and be careful rather than hunt for mistakes later on.

“What are some tips on how to solve Integration problems using U-Substitution?”

Question from Quora

u-sub-integration-1

There are basically two terms here, the 5x and the 1- x^2 in parentheses.

The general idea for u-substitution is to make u something that when you take the derivative of it can be substituted for what is left over (with a little manipulation).

So if you chose 5x as your u, then du would be 5dx which you could not easily substitute.

If what you choose seems to not be working, either you made a mistake or you should try a different option.

For this problem choosing u = 1 -x^2 seems to work better. You can probably even take the derivative of that in your head to figure out in advance if it will work well.

u-sub-integration-2

At that point it becomes much easier to solve.

How to do a phasor calculation with radians using a principle of Eulers Formula

phasors problem

“How has this been calculated? I try to convert the 7.11 to radians and sum it up, then converting it back to degrees. However I do not get 1.61.”

-Quora question

My response:

The numbers are already in radians.

Dividing the coefficients does not seem to be the problem you had.

If you divide the e^x terms you would normally subtract the jπ/4 from the -j7.11.

That would get you

~-j7.8954

It’s best to save this in memory to not lose accuracy.

From there we can think about Euler’s formula. Shown on the right side in green. Much like the trigonometric functions we learn much before doing anything with phasors, you can add or subtract 2π to the input and get the same result for periodic functions. You can also do that with e^ix.

Eulers Formula imaginary calculation

For this problem we add j2π to the -j7.8954 to get -j1.61. The answer is an approximation.

“Should I skip Pre-Calculus?”

Trigonometry, if it’s included, is very important for several things and could very well be included in precalculus. Check out the syllabus and book for what would be included.

I would also think about who teaches the class. If it’s a challenging class with a good teacher, you could learn a lot. If it’s easy, maybe you could learn it on your own.

I skipped Spanish 2 in high school. That would have been no problem if the Spanish 3 class was easy, but it wasn’t. So I worked very hard and got a lot of help to catch up.

What is the rush to take calculus?

Seems like you might want to take precalculus in the summer since it would likely be at an accelerated pace given the constrained time frame. Or you could maybe just go through it on your own. But just reading the book probably wouldn’t be enough. You would want to go through a lot of problems.

“why does $x^{1/2} = +\sqrt{x}$ not $±\sqrt{x}$?”

Seems you are asking about, in the context of differentiation, you’re not asking about how to actually differentiate.

I think you’re asking why it is not true that x^{1/2} = ±\sqrt{x}?

Since it seems logical that it would be similar to this, \sqrt{x^{2}} = ±x

Basically, I would say it’s because you can write -x^{1/2}

If that’s what you want to say, you put the negative sign in front of it. If you want it to be positive, you don’t write the negative sign.

If you square ±\sqrt{x}

In either case, you will get x.

“What are the practical uses of calculus, other than calculating the area under curves?”

Among other things, one thing you learn early on in calculus that can be very powerful is Optimization.

If you know how to take a derivative, you can find where the first derivative is equal to zero which can often be a maximum or minimum.

Knowing exactly how to get the maximum or minimum of something can be very useful.

Let’s say you have a certain amount of material to build something and you want the maximum volume. Use optimization.

Let’s say you want to minimize the cost of doing something or making something. Use optimization.