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Exercises 23 and 24 refer to a difference equation of the form $y_{k+1}-a y_{k}=b,$ for suitable constants $a$ and $b$ A loan of $\$ 10,000$ has an interest rate of 1$\%$ per month and a monthly payment of $\$ 450 .$ The loan is made at month $k=0$ , and the first payment is made one month later, at $k=1 .$ For $k=0,1,2, \ldots,$ let $y_{k}$ be the unpaid balance of the loan just after the $k$ th monthly payment. Thus a. Write a difference equation satisfied by $\left\{y_{k}\right\}$b. $[\mathbf{M}]$ Create a table showing $k$ and the balance $y_{k}$ at month $k .$ List the program or the keystrokes you used to create the table.c. $[\mathbf{M}]$ What will $k$ be when the last payment is made? How much will the last payment be? How much money did the borrower pay in total?

a. $y _ { k + 1 } = 1.01 y _ { k } - 450 , y _ { 0 } = 10,000$b. $\mathrm { m } =$25$\mathrm { y } =$114.8802c. $\$ 11,364.88$

Calculus 3

Chapter 4

Vector Spaces

Section 8

Applications to Difference Equations

Vectors

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a ton going on in this problem. So let's get right right to it. Um, some parts easier than others. For instance, What's the loan amount immediately? The loan amount is right there in 35,000 so try to keep track of these part by part, so it's really easy for everybody to fall along. 35,000. The balance before anything's been paid off. OK, Part B. What is the length of alone? Which isn't? Is easy to see. Does it cause it doesn't say is year 0123 But here's the thing. 2027 minus 2010 is 17. It's actually 18 years. If you include this is your zero and then 18 years from here to here. If this is your one, this will be your 18. Feel free to write all the numbers in between. The Czech totally works. So 18 years. All right, Part C of leaves, a monthly payment, right monthly payment. It might be tempting to just say I need to add the principal and the interest, and we actually do definitely have to do that. So if we add those two numbers, we get 4000 $65.22 but that's a yearly amount paid. So I gotta take this and divide it by 12. And that's gonna give me the approximately month approximate monthly payment, which is 300 $38. And if I'm rounding to the nearest sent 77 cents Very nice. Okay, Party, party, party. What is the total interest? So you would just take every bit of interest at all of them up. I kind of manually put this into Google sheets and said, Hey, find me the sum of all these things and it did. $38,174 and nine cents. Kind of interesting to know this is the money on top of that. So you pay more in interest than you do. Then the actual loan amount is like this is only $35,000 alone, and by the end, you would pay more than double that just to pay this whole thing off. Interesting note. But I think next is the scatter plots. Let's do it. I included the linear aggression with the scatter plot. I didn't have to, but I went on to Dismas ideas. Does most for all the regression. So let me just write it out. If you've watched any of my videos, you know, I'm very dependent on it. You it helps me out a ton. I input all the numbers and you notice I you can't see all of them. It's 18 years worth of numbers, and I made this scatter plot and the linear regressions. Well, you kind of see the X values are from zero toe, 50,000 more or less. Why values? Excuse me in the exits from zero to se 21 So feel free to check that out. And so here's how does most spits out a linear, quadratic and cubic um, regression. You basically type in. Hey, take these CS his wife for next four. That's because those were the kind of sub values there. Take those and give me an approximate M and B values so I can have an equation in the same for a B and C value in a, B, c and D for the cubic. So let me write it up, and I believe around part f, right, So it's a part F is a linear aggression. Here we go. Okay, linear aggression is is why equals. I was looking for the n value negative 100 or 1870 0.6 to multiply by X and then plus a B value of 38,000 814.6. I recognize it says around the nearest 100th. I'm so, so sorry. But that is one limitation that I couldn't find a way around. So this value here here it looks like there's just a couple that will be rounded to the nearest 10th. So I apologize for that. But if we're looking at the quadratic quadratic, we're got our A B in vite in C values that we gotta plug in to the equation. Why equals C that a value of negative 86 pulling 5 to 700 X rayed B is negative screen or 13 points 31 rounded to the nearest hundreds as well. Multiply by next. Okay, see, value is 34,000 400 and to again it's rounded to the nearest 10th. I apologize. I could not find a way around that, but I feel like it's such a small little air that hopefully the remainder of this video makes up for and let's go ahead and do the cubic alright, ABC and devalues. So are a values negative. 2.69 right? It's the nearest hundreds x 1/3 is what that psycho official to minus 13.77 x squared. See that devalue got right there. See, Negative 820 three cowfish X and D is a positive 35,000 61 again rounded to the nearest tent. So I just got these three constant terms that around into the 10th. Otherwise, they're all run into the hundreds, so there's every single part to this quite large problem.

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