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6. (10 points) A researcher studying housing in Austin has leamed that the number N ofhomes in particular East Austin neighborhood that have been renovated and reso...

Question

6. (10 points) A researcher studying housing in Austin has leamed that the number N ofhomes in particular East Austin neighborhood that have been renovated and resold has been growing according 5 the function N = f()= ~0.2t where is the number of years since 2000. Find and interpret the 1+9e following (decimal approximations OK for b and c); be sure to include units in your interpretations_f(0)() f (10)f' (10)

6. (10 points) A researcher studying housing in Austin has leamed that the number N ofhomes in particular East Austin neighborhood that have been renovated and resold has been growing according 5 the function N = f()= ~0.2t where is the number of years since 2000. Find and interpret the 1+9e following (decimal approximations OK for b and c); be sure to include units in your interpretations_ f(0) () f (10) f' (10)



Answers

Approximating $f(x)=e^{x} \quad$ Refer to Problem 89 (a) Approximate $f(-2.4)$ with $n=3$ (b) Approximate $f(-2.4)$ with $n=6$ (c) Use a calculator to approximate $f(-2.4)$ (d) Using trial and error along with a graphing utility's SEQuence mode, determine the value of $n$ required to approximate $f(-2.4)$ correct to eight decimal places.

For this problem we are asked to approximate or eat to the negative 2.4 years. Using this, um, some nation rule right here. So the first one we're asked to um, Carre is asking us to approximate what f of Make it a 2.4 years Within equals 2. 3. Okay, So, so therefore this is going to be equal to. So using this, uh, this summation rule, so K equals zero. We're in is three X. Is gonna be negative 2.4 race K over K. Factorial. So now I'm going to expand this summation. So this is equal to negative 2.4 K. Starts at 0/0 factorial Plus -2.42. The first over, well, first factorial plus negative 2.4 to the second, Power over two factorial plus negative 2.4 to the third fact oil over three facts. I mean negative 2.4 to the third power over three factorial. And if you use your calculator to simplify this, you will get the sum of approximately negative point A 24 So now the next thing is asking us to approximate f of -2.4 within Equals to six. So therefore my summation rule is going to be okay Equals to 0-6 of -2.4 is my ex to the K power over K. Factorial. So basically I'm adding when K go zero and K goes one, K goes to and keiko's three, which I have already done from part A. So I'm gonna separate this summation. So this is this is equivalent to The sum of K to the 0-3, which which we we we did from part a class. The summation of K equals 4-6. Okay, so we want to end at six at the end. So negative two point forward to the K over K Factorial. So we already know what the answer for this one is. It's negative point point You know, 8-4. So now I just need to figure out what the some of this part is which is going to be right here. So this sound is gonna be negative 2.4 where K starts at floor Over four factorial plus -2.4 To the 5th power over five factorial plus negative 2.4 to the six Power over six factorial. And if you use your calculator to simplify this, you will get approximately .984- 6, 88. So therefore my final answer, I'm going to add negative negative point a 24 plus 0.984 to 688 So my final answer equals two for part B. Yes .160 2688. So, and parsi is asking us to approximate f of negative for using our calculator. So f of negative 2.4. So if we use our calculator, our calculator has the E. So I can just type in what E. Two negative 2.4 years. So the exact answer for this one should be So find the E. In your calculator raised that too, Parentheses -2.4. Close the parentheses. So the answer should be approximately .09071 795. Okay, so you can see that when we did into the sixth. Our previous previous answer is still not very close to the exact answer and that's what D. Is asking us to do and carry D. Is asking us to figure out, you know, using our calculator in the sequence mode. Figure out, you know, the value of N. Required to obtain Up to eight correct small places. So to do that, the first thing you're gonna do is turn your calculator a into a sequence mode. So click on mode and go down to function and go over to sequence mode. And now I'm gonna tap in our sequence rule for this. Um ah For this summation. So our sequence rule is access negative 2.4 raised to K over K. factorial. So you're going to go to second, start And go to hopes and then choose five. Which that sequence. So you're going to type in your sequence rule, your summation rule which is negative 2.4. Grace too. And okay, because we're in the sequence mode and then divided by and factorial. So math and go to prob and for so that's how single sequence rule and are variable here is in Based on our calculator, we started at zero And you can see that, you know, and you go six is still very far away from the correct answer. So we need to really, you know, go up a lot higher. So let me double the six, you know, try 12 and see if that's anywhere close. And now I'm going to paste that sequence rule in my homepage. So there's a sequence. And I'm interested in the summer because the sound will will help us approximate, you know, We'll eat to the negative 2.4 years. So you're going to do second in the elite, which means insert. And now you're going to do a second, start again And go over to math and then choose five. So I'm adding the some of the sequence. So and I said earlier that we're letting in At end at 12 C if that's anywhere close. Okay, so I should get you know 0.071795. Okay so that's what I need to get so And I forgot to write the last 133. So And we have 0.07 299 instead 179. So I still need to go harder than 12. So let me let's say in is 15 then. Okay I'm getting closer uh 7179 but I need a five not a zero at the end so second enter again And let me change to 16. I'm still now their second enter by 17 And there we go. So the answer here is in equals 2 17.

Okay, so it says, Explain how you could use the falling table by used in order to find the given quantity. So we're looking to find the log of 1000. So we're gonna do is treat that so it's equal. Why? And we know that any lock without a base is a base of 10. So we're gonna rewrite this so that it is in the form A to the y equals X, so that's tend to the y equals 1000. And then what that means is I can go over here to the table. And since it's 10 to the X equals 1000 that equals 1000 when X is three. So that means, in terms of this problem that my answer is, why equals three So 10 cubed makes sense is 1000.

Hello were given an equation. FX is he quoted 10 very super Rx beginning a table, but the values of X and F X corresponding right and then were given long screw off. 10. He has to find out the value of this without using a calculator. He is in the table, right? So first of all, let's rewrite lock scored of 10. Using their different definition off log, they should be equal to square root of 10 is equal to 10. Raised to Power X, that's a seem. This is X here, but this is similar to dis equation, right? So to find the body of eggs, we just have to find the course for England. You off FX, where FX is equal to root temp and using the table. We can see that, Ben, if X is equal to root, 10 x is equal to 0.5. So lock 10. I mean, square root of 10 is equal to 0.5. All of this is done using the table and not a calculator

Okay, So our goal is to find the log based 10 of 100.5 equals X. So what we're gonna do is change this. So it's as an exponential. So what that means is, this is going to become 10 to the X, equals my five, and then we can go over to the table given to us, and it says 1 10 to the exes. 100.5 x must be negative point 3010 So, using the table and our formula to change it to an exponential week, it X equals negative 0.3010


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