Hello again, Tom here, we’re gonna do a problem with linear approximation with one variable, generally in calculus 1, and let’s do it. Index 8 of course to get to my menu from, to get to the main menu, I’m already at a linear approximation with one variable here to save time on the video. Press enter, linear approximation, 1 variable, and we’re finding the equation of a tangent line, and then working finding b in that, finding, also finding m which is the slope. Usual problem in calculus, and so we have to press enter before we enter anything in here and the function is alpha x to the 5th. It’ll show you what you’ve entered, you can change it if you want, I say it’s okay, they give it as the point being alpha 3. and of course here’s the original function, x to the 5, the derivative of that is 5 x to the 4, that’ the slope of a line, remember that. We’re gonna find the slope of a tangent line, to the point 3, and we do that by taking the derivative and then substituting 3 for 4 in here, we come up with 405, the slope is equal to 405. now remember the slope, there’s a 1 over here of 405 remember the slope is rise over run, right? So you got really the rise is 405 and the run is, denominator is 1, so if you went up 405 on the x axes, and over 1 on the, on the x axes, draw a line and that’s really the origin of the graph, that’s what you’ve found here the slope, okay? Now we take the 3, we put it into the original function, we get the y value at that point, which is 243. and so here’s the point we’re at, we’re looking at 3, and y is 243. and so then we do the computations here, y=243, we do algebra stuff here, 405 and 3 for x to find b, turns out to be b is -972. so the equation of a tangent line is 405x + -972. now they give us what we’re supposed to approximate at which is alpha 3.6. so we’re gonna add 3.6 into this equation here, and we come up with y=486. So the x y coordinates 3.6 and the y is 486. pretty neat how everystepcalculus.com go to my site, buy my programs, and enjoy passing calculus. Have a good one.
Archives for September 2017
Linear Approximation, 2 var, fx = √(20 x^2 7y^2)
Hello, Tom from everystepcalculus.com and everystepphysics.com. This is a linear approximation problem in calculus 2, we’ve got two variables, x and y, and let me show you how it’s done. Index 8 to get to my menu at my programs here, I’m already at linear approximation, but you’d scroll to that if you, and we’re gonna choose two variables, here’s the formula, you have the original function plus the partial differentiation for x and partial differentiation for y, and then dx and dy, okay? So this is the formula you’re gonna use. You’re gonna enter the function, you have to press L for first before you enter anything in these entry lines, and the function is alpha. 20, I have to go slow because of the simulator it’ll mess up, minus x, squared, whoops, didn’t need second, squared, minus 7 times y squared, minus 7 times y squared. Oh, that’s a square root so we’ve gotta, we’ll put the parentheses in here, and then I’ll go all the way back to the front, put the square root sign in, the alpha x, see square root of, yeah, it’s correct, it’ll give you a chance to change it in case it’s incorrect, and they’re gonna give us the they want you to approximate the x variables and the y variables at, here’s your alpha first, the x is 1.95, 1.95, and the y value is 1.08, so you alpha 1.08, and again that’ll show you what you’ve entered, you can change it, say it’s okay, and then I choose the points [inaudible] you’re supposed to choose the points closest to it, and equals a and b, okay? So dx is equal to x-a is equal to the x value minus a, -.05, and dy is equal to y -b, 1.08-1 = .08. now we do the derivatives [inaudible] to x, with respect to x, turns out to be this here, with respect to y turns out to be here, and that a and b 2 and 1 you add that into the, you know, the derivative with respect to x, and the function here, and you come up with -.6667. here the same thing for y, for 2 and 1 comes up to -2.3330 or 333. and then we do the original function with 2 and 1, comes up with, here’s the visual function, comes out to 3. so, here’s the formula again, f + fx dx + fy times dy, you add these variables, here’s the answer 2.8467, that’s the answer linear approximation of that, of the original function. Pretty neat how every step calculus got [inaudible] buy my programs and help you do all this stuff for your homework and your tests. Calculus is nonsense so treat it that way, try not to learn too much about it, just pass the tests and get out of there. you have a good one.
Linear Approximation, 1 variable, f(x) = 3*√(x), x=27
Hello again, Tom from everystepcalculus.com everystepphysics.com this is a video on linear approximation for calculus. It’s a one variable situation, so probably calculus 1, and I’m gonna read it to you. Use linear approximation, in other words the tangent line to approximate 3 times square root of 27 as follows. A and f(x) equals 3 square root of x the equation the tangent line of f(x) at x=27. If you’ve written in the form y=mx+b, find a and b. then get m and b. so, let’s do it. Index 8 to get to my menu, and we’re gonna scroll down here to linear approximation in the L section, and we’ve got to choose one variable because there’s only x given, okay? 1 variable. The point slope form is y=mx+b, it’s a standard tangent line equation, and we’re gonna enter what’s given. Function is given, and that would be if you press L [inaudible] if there’s any lines here and they give it as alpha 3 times e square root of x. and I’ll show you what you’ve got here, you can change it if you made a mistake. Now I’ve gotta enter the point that we’re given, which is alpha 27. I say that’s okay also. So the first thing you do, you write the function here, you take the derivative of a function equals three, I’m gonna find the slope. Okay, so here’s the derivative of 3 times the square root of x, and they, they’re supposed to decide the closest square root to that number, I do that for you which is 25. and so we enter 25 into the derivative, and so the m is the slope = 3/10. now we take 25 and put it into the original function here, comes up with y=15. So here’s the point, 25 for x, 15 for y. at y=15 then we solve for b using this calculations here, just straight ahead algebra, and equals this right down here, mx+b okay? Now we plug in the point to evaluate, 27, and with that we come up with these calculations here which is 78/5, so, the answer’s 27 and 78/5. so m is 3/10 and b is 15/2, that would be the answer to this problem, okay? Go to my site if you like these programs, buy them, and enjoy passing calculus, okay? Have a good one.
Mean Value Theorem, y=x^3+x 1, [0,2]
Transcript
Hello! Tom from everystepcalculus.com and everystepphysics.com. Mean value theorem problem for calculus II. Index 8 to get to my menu. I’m going to scroll down here to mean value theorem and we are going to enter our function. You have to press alpha first. The function is alpha x^3 +x-1, [0,2] is the given limit here so alpha(0), alpha(2). It will show you what you’ve entered, you can change it if you want, else say it’s okay. It’ll show you the formula. I know we have f, Notice we have f(b)/2 – f(a) and (b-a), and we have f’(c). So, f(a) is equal to this. We put zero I for all x’s in the function, come up with -1. Put 2, the upper limit into the function, come up with 9. We subtract b-a, here is b-a, was 2, and we put it back into the formula: [9-(-1)]/2. Turn out to be 5. Now we are going to take our function, take the first derivative of it, 3x^2+1, change it to c, x’s to c: 3c^2+1=5. We are going to solve for c which equals this here. And the answer is two [2(3)^(1/2)]/3, the approximation is 1.1547. Go to my site, buy my programs and have a great time passing calculus otherwise suffer because it’s such a worthless course, lot of study but no reward except to get through calculus and move on with your life. So worthless math concept or subject and it goes along with crossword puzzles or Sudoku, [unclear 03:06] maybe. Same thing. After 25 years of studying this this is what I believe. So anyways, have a good one!
Mean Value, 400x+3x^2, [0,3]
Transcript
Hello! Tom from everystepcalculus.com and everystepphysics.com. This is a mean value problem in calculus II not a mean value theorem problem, mean value. It finds the average value or the mean value under a curve using an integral. Integrals are areas under the curves, and this finds, actually finds nonsense in calculus, [unclear 0:31] calculus. Sorry. 25 years of studying this I find no value in it, practical value. So, index 8 to get to my menu. We have to pass our calculus class though. Okay. I’m already at mean value so I’m going to choose that. These programs are in your titanium calculator that you have in your purse or your schoolbag, and you’ll be able to solve all these problems in my menu forever. Compare to your calculus book, you can throw away your calculus books or your study materials from your professors, absolutely worthless but this calculator has all your calculus computations forever as long as you have the calculator for your children, your grandchildren, your partners or your spouses or sisters or brothers, whoever in the future. And not for a very much price either, let me tell you. Finding a mean value, let’s do it. Here’s the formula here. Compare to the root mean square formula which has the square root, all these are no square root signs. I’m going to enter the function, you have to press alpha first. The function is 400x+3x^2, the lower limit is alpha(0), the upper value is alpha(3). And it will show you which values you’ve entered, you can change in case you made a mistake, else say it’s okay. And we start the process of computations of these. Okay, here’s the formula. I put the integral sign down here, this is the mid of the [02:49] because it will off the screen if I did it any other way. So, remember that. No big deal. It still has a dx with respect to x here (3*x^2+400*x)*dx. We just subtracted the b-a here and get 3, and we still have the integral over the limits of 0 and 3 right here. We are going to do the integration now. Here’s the integration x^3+200x^2, and we are going to compute that, these x values with the upper limit and lower limit. It actually equals 3. You write these on your paper, you look like you are a genius, right? 1827, okay, now equals to 0. You write this on your paper equals 0. And the upper minus the lower always in physics, in calculus, upper minus lower. And the answer is, after we’ve put it back into the formula, the answer is 609. Hey, have a good one! Go to my site buy my programs and enjoy passing calculus.
Have a good one! Bye, bye.
U substitution, cos(ln(x))/x and cos(ln(x)), integration by parts
Transcript
Hello! Tom from everystepcalculus.com and everystephysics.com. We are going to do a problem in calculus U substitution, and we are going to do a function with cosine in it. Index 8 to get to my menu. Press enter. I already had U substitution but you’ll scroll with this cursor here down or up or whatever to get to the [unclear 00:33] for the problem that you have at hand. So, U substitution is this problem and we are going to enter the function, we have to press alpha first before you enter anything in these entry lines here. So, we are going to press alpha, and then we want cosine, secant z here which is cosine, and then secant x which is log of x, close up the parentheses, and divided by x, is the function. So, it will show you what you have enter. You can change it if you want, else say it’s okay. Notice that the trick to use substitution is that whatever inside parenthesis here, you take the derivative of that and it’s got to be able to be matched to the outside somehow, okay? So, we are going to rewrite this because here we have 1/x here dx, so we are going to kind of isolate that so it’s already cleared for you. Cos(ln(x))*(1/x)dx, and then u = ln(x) du = 1/x
You have to memorize that. The integral of 1/x is ln(x) etc. so the opposite derivative of ln(x) is 1/x. So, now we are going to do the integral of cos u, du equal sin(u) + c. When you do the integral of cos u it’s sin u. so, now we have the answer of sin[ln(x)] + c, as the answer. Now, we are going to do another problem here, and we are going to press alpha, we are going to have secant. Cosine of second log of x, and without the x, divided by that. Okay? It shows you what you have entered, say it’s okay. And notice that this is not a U substitution problem but an integration by parts problem because notice that the derivatives of the parenthesis here, ln(x) is equal to 1/x but there’s nothing on the outside that equals 1/x. So, if that’s the case then you cannot integrate it by U substitution, you have to go to integration by parts. And that’s just pathetic here with the long, long here of getting the answer, more of Sudoku of math. So we have,
u = cos[ln(x)],
du = -sin[ln(x)],
Mean Value Theorem, 2√(x), [4,9]
Transcript
Hello! Tom from everystepcalculus.com and everystepphysics.com. This is a mean value problem in calculus II not a mean value theorem problem, mean value. It finds the average value or the mean value under a curve using an integral. Integrals are areas under the curves, and this finds, actually finds nonsense in calculus, [unclear 0:31] calculus. Sorry. 25 years of studying this I find no value in it, practical value. So, index 8 to get to my menu. We have to pass our calculus class though. Okay. I’m already at mean value so I’m going to choose that. These programs are in your titanium calculator that you have in your purse or your schoolbag, and you’ll be able to solve all these problems in my menu forever. Compare to your calculus book, you can throw away your calculus books or your study materials from your professors, absolutely worthless but this calculator has all your calculus computations forever as long as you have the calculator for your children, your grandchildren, your partners or your spouses or sisters or brothers, whoever in the future. And not for a very much price either, let me tell you. Finding a mean value, let’s do it. Here’s the formula here. Compare to the root mean square formula which has the square root, all these are no square root signs. I’m going to enter the function, you have to press alpha first. The function is 400x+3x^2, the lower limit is alpha(0), the upper value is alpha(3). And it will show you which values you’ve entered, you can change in case you made a mistake, else say it’s okay. And we start the process of computations of these. Okay, here’s the formula. I put the integral sign down here, this is the mid of the [02:49] because it will off the screen if I did it any other way. So, remember that. No big deal. It still has a dx with respect to x here (3*x^2+400*x)*dx. We just subtracted the b-a here and get 3, and we still have the integral over the limits of 0 and 3 right here. We are going to do the integration now. Here’s the integration x^3+200x^2, and we are going to compute that, these x values with the upper limit and lower limit. It actually equals 3. You write these on your paper, you look like you are a genius, right? 1827, okay, now equals to 0. You write this on your paper equals 0. And the upper minus the lower always in physics, in calculus, upper minus lower. And the answer is, after we’ve put it back into the formula, the answer is 609. Hey, have a good one! Go to my site buy my programs and enjoy passing calculus. Have a good one! Bye, bye.
Long Division (x^4-9*x^3+25*x^2-8*x-2)/(x^2-2)
Transcript
Hello, Tom from everystepcalculus.com and everystepphysics.com. Long division problem regarding functions you know in calculus if it comes up in calculus whatever, index 8 to get to my menu your going to change long division and your going to enter your function, you have to press alpha before you enter anything in these functions here I’ve already entered the problem so I’m going to press cause it’s a long problem and it takes time save time and you notice this is called the dividend (x^4-9*x^3+25*x^2-8*x-2) and this is called the divisor (x^2-2) you set it up just like a regular numerical long division problem, this off to the left if the screen was wider I could do that for you but I can’t so I have to do it this way and remember your dividing x ^2 into x ^4 first getting the answer of x ^2 and your multiplying x ^2*x^2 which is x^4 and then your going to multiple x^2*-2 which gets -2x^2and your going to change the signs because your subtracting from this so I’ll show you how to do that but remember that this is a divisor we’re operating dividing this into several things that come along here okay so it always give you a chance to change incase you’ve made a mistake but I’ll say it’s okay now here we’ve divided the divisor x^2 into x^4 and you get x^2 and you multiple that times it and then change the sign -x^4 we multiple x^2*-2 and we get -2x^2 and we change the sign to make it +2x^2 okay and then we add them, you notice now that we have 25x^2 up here we’ll if the screen was wider we would have -8x over here and -2 over here but I have to bring it down to show you that it’s still there okay but we are going to deal with this one and this one and this plus this is 27 and this of course becomes 0 so we have 0+(27*x^2) and we bring down the (-26*x)+(-2) okay we are going to divide x^2 into 27*x^2 and we are going to get 27 and we change the sign, write this all on your paper exactly as you see it and you have to keep the coulomb’s of x^2 and x and x^3 all together okay now 27*-2 because a minus -18*x okay -18x-8x is 26x-26x and we still have to bring down the -2 okay and we of course multiple that to come up with a minus which negates these two here and then when we multiple the minus 2 times the minus 27 we get -54 and that change its sign and become 54 okay and 54-2 here becomes 52 so we get 52 we got -2x now once in long division the divisor cannot be divided into this anymore this all becomes as one then you divide that by the divisor so we have [(-26*x)+(52)]/(x^2-2) here is the answer (x^2)+(-9*x)+(27)+[(-26*x)+(52)]/(x^2-2) complicated you have to do it on paper and you will see where it’s much clearer I have trouble explaining it here because of such a small screen and it’s all crammed into one but all the answer is correct and you will get all the problems are correct. Pretty neat hah everystepcalculus.com go to my site buy my programs and pass your calculus, have a good one.