Archives for September 2014
Calculus Q & A
Question:
An email from a Calculus student:
Hi Tom,
First, I want to sincerely thank you for the support of your wonderful programs. You’ve inspired a knowledge of calculus that my prof cannot.
I’ve got a problem to find the extremas of an exponential function over a given closed interval but don’t know which program to use. Here’s the problems:
f(x) = (3x-1)e^(-x), on the interval [0, 2]
and
f(x) = (ln (x+1))/(x+1), on the interval [0, 2]
any advice would be greatly appreciated!
Thanks!
Answer:
In short you enter the function into my “graph by hand” menu program.
Calculus to me is like teaching you how to multiply through the 9’s and then make a student take three semesters siting a million areas of the usage of multiplication.
Calculus finds two things, the derivative finds the “slope of a line”, that’s it!!! and the integral finds the “area” under a smooth curve (x^2,x^3,x^4,x^5 etc and sin(x) and cos(x) (called a sine wave) as you graph those on an x y graph, and only finds the area, if you give that integral a range, called a “definite integral” an indefinite integral finds nothing.
To me calculus is the Sudoku of math, the study of cross word problems. Something to do while waiting for a plane to Phoenix. In calculus they don’t say “rose” (ya know the flower) they say “hibiscus mutabilis” they constantly make easy things into extremely hard things. Linear approximation is a prime example, as well as related rates.
That said, “extrema” “max and mins” of a function is the absolute or maximum high points or low points as you look at a graph of a function. If you graph -x^2 (minus) this is a smooth mountain, extrema is standing at the top. The opposite is true of x^2 (positive) this is a smooth valley, if you graph it, and you are under that valley, touching it with your finger at the lowest point.
That said, it just so happens that when the slope of line is “horizontal” it has a slope of zero, and if you set that horizontal line on top of the mountain it will touch at only one point (tangent) and that point will be the highest possible point on that mountain. So calculus take the first derivative of a function (slope of a line), sets it equal to zero and then solves for the x values. It then puts those x values back into the origininal function, and when solved, finds the “y” value. That point (x,y) is the maximum or extrema of that graphed function. Those x values are called “critical numbers” because they lie on the x axis. They become “critical points” when you plug them into the original function and solve for “y”.
Now to me in your first example, given what I’ve just taught you, They say “over an interval” and then give you the inteval [0,2] (Notice the brackets which indicate an interval where parenthesis would indicate a point (x,y) in math — There is only one critical point (no interval) where the hoizontal line would touch the graph (tangent), so I guess this “over the interval” is to throw you off, or check whether your understanding is as good as mine.
My graph program will do all this for you, but in the first example if written properly (get used to doing this) would look like this: (3*x-1)*e^(-x). Notice the times sign in front of the “e” that tells you product rule to find the derivative. In the second example quotient rule would be used to find the derivative.
Thanks, for the kudos, Tom
Partial Fraction Example 6
Raw Transcript
Hello Everyone, Tom from everystepcalculus.com and everystepphysics.com. I’m going to do a test problem on Partial Fraction Decomposition, again.Let me show you how my programs work on this. Index 8 to get to my main menu. And we scroll down the Partial Fractions which is in the menu. You can scroll up or down here with the cursor. You have to press Alpha before we enter anything in these entry lines here.Press Alpha and the problem is one divided by, I’m going to use parentheses for the denominator And numerator. This one is x squared minus nine.I always show you what you’ve entered. Now notice that x squared minus nine is a difference of squares so when you factor that it’s always X minus three times x plus three. And that’s a big deal in calculus so always always be on the lookout for these type of things. These difference of squares. Sometimes I’ll have X minus 3 up in the numerator and it will cancel and they use all kinds tricks to throw you off. If you’re hip and I’m gonna try to make you hip, you’re always are looking for the difference in squares. Okay. We’re doing Partial Fractions so I say it’s okay. I’ll always give you a change it if it’s not. And I factor it for you. Here’s x minus three times x plus three. If you can’t factor the denominator, you can’t do Partial Fractions. So keep that in mind, too. And the problems have to be relatively simple in tests because not that many people, including myself, are not very good factoring. In our heads, you know. They get more complicated than this, so. So, you start out by putting one,numerator, divided by x squared minus 9 times x squared minus nine. What you’re doing is multiplying both sides the denominator. This is factored but this is not. And of course, when you multiply the numerator by this, you’re going to eliminate. So there’s going to b one up there. And of course these switch around as we do times this portion here. A divided by x minus three plus B divided by x plus three. I switched it around now so it’s one equals A times x plus three plus b times x minus three. Now the first thing you want to do is eliminate some of these. So you’re going to put a minus three here to make this zero. I’ll do that for you. I have x equals minus three. I put, anytime you see quotation marks, you put parentheses. Parentheses minus three plus three which is zero. B times parentheses minus three parentheses minus three which is minus 6. Zero minus six so one equals a B minus 6. One equals B minus 6 to B is equal to minus one sixth. And X equals three parentheses three in here. open and close plus three And this becomes six and this becomes zero. Which a is equal to one sixth. Partial Fractions, enter one sixth divided by by X minus three plus minus one sixth divided by X plus three. Those are the Partial Fractions. And when you integrate these, one sixth logged absolute value X minus three plus minus one sixth times log absolute value X plus 3. So add all the stuff on your paper and homework, or whatever. Get the problem perfect. Pretty neat, huh? everystepcalculus.com. Go to my site and buy my programs.
Thank you
Limit at Infinity Example 1
Limit to Infinity Example 2-Solved by TI-89 Video
Lim (7x^3+9(x^2+3)
–> Infinity
Raw Transcript
Hello Everyone, Tom from everystepcalculus.com and everystepphysics.com. I’m going to do another Limit to Infinity problem. Index 88 to my menu. Choose Limits. Scroll to compute limit. Enter the function by pressing alpha. This test problem is seven times x cubed plus nine. Uh oh, I forgot the left parentheses. Press second, go there quick. Divided by x squared plus 3. Add the Infinity indication. Choose the other button, and the yellow register is underneath. See the yellow letters, that’s what that does for us to that register when you press the yellow button there. I always show you what you’ve entered, you can change it if you want. which you better you can change it if want. You’re dividing everything, every term in that problem by the highest order term in the denominator. This here equals this and then we add infinity to these, it’s going to come seven because anytime you take a number. So when you multiply times Infinity it becomes Infinity. So here’s Infinity and second one is zero.Numerator Limit is Infinity. Denominator divided by the highest order term of denominator is this. That equals. When you’re add Infinity, that equals one zero. The Denominator limit is one. The Problem limit is Infinity then. Infinity over one. Pretty neat, huh? everystepcalculus.com, visit my site and buy my programs. Thank you
Limits to Infinity on TI-89 Q4
Lim (7x^3+9)/(x^2+3)
Raw Transcript
Hello Everyone, Tom from everystepcalculus.com and everystepphysics.com. I’m going to show you how to do a Limit at Infinity, on this video. I’m doing several of them so you can get a great idea of what happens.Index 8 to get to my menu. Limits, Choose Compute Limit. Enter the function. Alpha Seven times x cubed plus nine. Oops forgot the other parentheses,
So we press second and the arrow here to get quickly over to the beginning. Add our parentheses and press second and the right arrow to get back where we were. precedent for a second and the right. Divided by parentheses x squared plus three. Need to add the Infinity. Yellow button here.I show you what you’ve entered. You can change it if you want. I say it’s okay. Again, we divide all the terms by the lowest ordered by the lowest quartered term in the denominator. And that equals this for the numerator. And when you do Infinity, the Numerator Limit is Infinity.Denominator…you take those terms and divided by x squared also. Equals one. Infinity divided by one is Infinity.That’s the answer. Problem Limit.
Pretty neat, huh? Visit my site and buy my programs. Thank you
Limits at Infinity on TI-89 Video
Raw Transcript
Hello again from everystepcalculus.com and everystepphysics.com. Another Limit to Infinity problem. Let’s do it. Index 8 to get to my menu. Choose Limits. Choose Compute Limit. Alpha before you enter anything in here. Alpha, parentheses, three times X plus two divided by x.
Add the infinity sign. Alpha yellow button here in the catalog which is the register for infinity. Show you what you’ve entered. I say it’s okay. We take all the terms and divided by the lowest order. X in the denominator, 3 0, the numerator limit is three. Denominator divided by Xis one. Answers is limit is three.
Calculus Q & A-Limits
Question:
Evaluate the limits:
lim(theta symbol–>0-) csc theta symbol
lim(theta–>pi/2) 1/2 tan theta
Calculate this limit at infinity (+/- infinity) for the following function:
f(x) = e^x+e^-x/2
Answer:
When you say f(x), calculus and programmable calculators are saying “y”. When they ask you for a limit, it is a trick question meaning find y. They then give you an x value and say as x approaches 5 (for instance) what is the limit? You then would plug 5 in for x ( which my programs do) and the answer is the limit is the value of “y”. Most of the time they trick you again do to the function they give you which is always a fraction such as (3x+7)/(x^2-4), (always a division sign). Then they ask you as x–>2 (as x approaches 2) what’s the limit. You’ll notice that when you plug in 2 for x you get a zero in the denominator which becomes an “undefined” answer.
If you then (like my program does) add some very small number to 2 such as .001 to get 2 + .001 = 2.001 or take a small number away from 2 to get 2-.001 = 1.999. Setting that equal to x then you actually get a number and that is the limit. I would actually graph the original function quickly on the calculator so I could see the picture of the function and then use the cursor to scroll back and forth at the x value they give you to do things like what happens when x approaches this from the right or left, to give me any clue of what their talking about.
Same thing with infinity. Graph the function first and then logically think about what they are asking. Also start to make things perfectly clear in your calculation examples so there is no question of what you’re asking.
For instance, you have:
f(x) = e^x+e^-x/2 which to a calculator, and me, means
f(x) = e^(x)+e^(-x)/2
and when you mean to say:
f(x) = e^(x)+e^(-x/2)
The problem says calculate the limit at infinity. I would graph the function on the calculator. You can see it’s just a parabola (valley looking, because no minus sign before it).
Here’s what they might be looking for:
f(x) = e^(x)+e^(-x/2)
Convert to:
= x*ln(e)+(-x/2)*ln(e)
factor out the ln(e)
= ln(e)[x+(-x/2)]
ln(e) = 1 So:
(1)[x+(-x/2)]
The limit as x-> ∞
= [ ∞ + (-∞/2)]
= ∞
Partial Fraction Decomposition Ex 11
Partial Fractions Example 3
Limits: Negative Infinity on TI-89 Video
Lim (7x^3+4x-17)/(4x^3-x^2)
Raw Transcript
Hello Everyone, Tom from everystepcalculus.com and everystepphysics.com. A problem regarding limits at Infinity. Index 8 to get to my menu. Scroll down to Limits. You can actually use second to go quicker on the menu, in other words, I want to go up a screen second, and up, second and up,or second and down. And here’s Limits. Click on that. And my other videos that I did previously I didn’t put Limit to Infinity here. You would access it by going to Compute Limit but when you want to put minus Infinity in there. It’s a little bit tricky do that. Well, it’s a little tricky on the Titanium to do that so I decided to put it in a actual menu. So here’s Limit to Infinity. We’re going to put the function in. Press Alpha first. Alpha left parentheses. This is a test problem. 7 times X cubed plus 4 times X minus 17, right parentheses, divided by, left parentheses, 4 times X cubed minus x squared, right parentheses. And let’s go negative infinity. Number two. And there it is there. This is the limit of this with X going to minus infinity. And I always show you if that’s correct. I always give you a change it in my programs. I say it’s okay. Then we’re going to divide all terms by the highest order term in the denominator. Which will be x cubed. Here’s the terms numerator. All divided by x cubed. Here’s the results. Now we put Infinity in for every X. Or minus Infinity, sorry. And that turns out Numerator Limit of 7. These turn out to be 0. Denominator
4 divided by the x cubed. That equals 4. And x squared divided by x cubed equals one over x. We add the minus Infinity to every x.
It turns out to be 40 Denominator Limit. The Problem Limit therefore is seven divided by four or seven fourths. Pretty neat, huh? everystepcalculus.com Go to my site and buy my programs. If you want to pass calculus and/or do your homework. And remember you’ll have these programs for life if you do that.
Limits Infinity Solved TI-89 Video
Raw Transcript
Hello Everyone. Tom from everystepcalculus.com and everystepphysics.com. Limits to Infinity. Let’s get started. Index 8 to get to my menu. Scroll to limits and choose number 2, compute the limit.And the function, you have to press alpha before you enter anything in here. Alpha left parentheses five plus two times X. This appeared on a test. Divided by,left parentheses, 3 minus X right parentheses.
Then press alpha and we press this yellow button here which goes to the letters with the yellow letters about the keys. The yellow register, I call it. And this is infinity right here. Alpha. There’s Infinity. I always show you what you’ve entered. The limit of two times x plus five over 3 minus X, as X approaches infinity. That’s the way you state it. I will show you what you’ve entered, you can
change it if you want. I say it’s okay. You divide all terms by the highest order term in the denominator. And that becomes a minus X
And you take the limited as that approaches infinity and the numerator limit is minus 2.Denominator, you have three and minus x in the denominator and that’s divided by minus x.Three minus x and you take the limit, you put Infinity in here. Anytime you take a number and divide it by Infinity, you get a very large number. It gets smaller and gets closer to zero. So that equals zero. And the other one equals one. That’s the denominator limit. You divide that numerator by the denominator, of course and you get minus two for the problem. Pretty neat, huh? everystepcalculus.com Go to my site and buy my programs.Thank you
Partial Fractions of Integral-Video
Raw Transcript
Hello Everyone, Tom for everystepcalculus.com and everystepphysics.com. I’m going to do partial fraction problem. This person said that it’s a hard partial fraction problem. Let’s see about that. It is hard if you ask me, without my programs. But let’s do it. Index 8 to get to my menu. Scroll down to partial fractions. We’re going to enter the integral. You have to press alpha before you enter anything in these entry lines. We’re going to go alpha and then going to do left parentheses, X squared plus 3 times X plus 1 close off the parentheses divided by, open up the parentheses, X to the fourth power plus 5 times X squared plus 4, close off the parentheses.
I always show you what you’ve entered. Now maybe you’re better than I am but you have to factor this denominator in partial fractions every time. So this it makes it hard for a class in it for a test problem is way too hard for a test probably for homework because I don’t know who would be able to write off from memory to factor that. I say it’s okay. We factor the denominator; here it is right here. And we start doing our partials. The idea is that you are going to deliminate the denominator here by multiplying at times the same thing with the numerator this. This here is the the factored part ion is here and this is the original denominator. And so then we have
the numerator is equal to Ax plus B times x square plus 4 plus Cx plus D times x squared plus 1.You multiply that out using the foil method. Remember the foil method? First outside inside and last. First would be Ax times x squared,outside would be Ax times four, et cetera. I multiply it out here. The calculator uses small letters rather than capitals. No problem. They combine those terms and come up with this. A plus C is x squared, x squared, 4a plus c, et cetera. Now we have to figure out the coefficients for this numerator. And since there’s no X cubed, we have to put one in there, which is 0 times x cubed. And then we have one coefficient here, three coefficient here, and one. So here we have one, three, and one. So then therefore AC is equal to 0, BD equal to one. 4a plus c is equal to three and 4b plus d is equal to one. And we work out, we use subtraction. We’re subtracting like terms. So AC is subtracted. 4a plus c and that equals three. A equals one. Do the same for each one of them. D equals one, C equals a minus one. Notice here we’re replacing A with what we found which is one. Which I do that for you here. When they’re in quotation marks, I replaced it in there. Partial fractions are this. Right here. And I also do the integrals for you. Log of x squared plus one divided by two, et cetera, et cetera. Pretty neat, huh?
everystepcalculus.com. Go to my site and buy my programs.
Thank you