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Car tires need to be inflated properly because overinflation or underinflation can cause premature treadware. The data in the table show tire life $L($ in thousands of miles) for a certain type of tire at various pressures $P\left(in 1 \mathrm{b} / \mathrm{in}^{2}\right)$ $$ \begin{array}{|c|c|c|c|c|c|c|}\hline P & {26} & {28} & {31} & {35} & {38} & {42} & {45} \\ \hline L & {50} & {66} & {78} & {81} & {74} & {70} & {59} \\ \hline\end{array} $$ (a) Use a graphing calculator or computer to model tire life with a quadratic function of the pressure. (b) Use the model to estimate $d L / d P$ when $P=30$ and when $P=40 .$ What is the meaning of the derivative? What are the units? What is the significance of the signs of the derivatives?

That's a problem about a car that talks about the entire life and the tire pressure cars. Let's say we have this chart of data and l is the entire life in thousands of miles and T is the tire pressure and pounds for inches square. And let's say we wanted to try to use a calculator to calculate a model Teoh to look at this information. So here I have the data plotted and I use Dez knows to practice. And then I did a quadratic regression because it looked quite are because, you know, the data points, they they went up to a certain bond. Then they started to come back down. So I did a quadratic regression for that to get the quadratic function right there. So that will be my model. They want to use the model to do some estimating we wanna estimate d L D P, which is the change in the life of the tire, according to the change in the pressure of the tire. All right, so let's say that we wanted to estimate this when P equals 30 so we're gonna have to look at our table values and notice we don't have a 30 in p ro, but we do have 30 would be in between 28 31. So let's say we use our slope formula then on this. So I'm going to use my l my change in why per se ah, 78 minus 66 over my pressure, which is X said change in X 31 minus 28. That would be 12/3 or four. Now, the units of measure since we had our r l on the numerator would be thousands of miles her and then the change in pressure per pounds per square inch. Notice that our answer was positive. So what that means is, if I find 30 down here on our on our graph, so 30 is about right here. So what we did was we just estimated that slow using these two points did that slope and use it to estimate this slope right there. And what that means is that the life of the tire is increasing as the pressure is increasing. Since it was a positive four for the slip, there's gonna come a point in time where, as you continue, Teoh, increase the pressure that entire life is gonna go down cause you're the tires. Overinflated. So let's take a look at maybe when that would be So let's say we also want to estimate the pressure when of the life of the tire here when the pressure was 40. So 40 is about right here, So I'm gonna use changing Why 70 minus 74 changing X 40 to minus 38 because it's negative for four or negative one slow. Well, let's look at our graph there. So 40 is, like, right here. So that means we used this point at this point. This slope estimate this slope right there and you can see that the slope is negative were calculated a negative one. So that means, you know, as we increase the tire pressure, the life of our tire has now decreased because now the tire isn't over inflated

It's clear someone named Reed here. So we're finding the tire tire life and thousands so we'll get and I could have 275 0.4 to 77 three piece square plus 19,748 0.5 to 7. Five p minus 2.7 35 for part B. We're gonna use our equation for l that we call it and differentiate it. So I get d l over d p this sequel to 0.55 through 896 p plus 19 0.748 5 to 8 to differentiate. I didn't use the equation above. I used the equation known in thousands. So when P is equal to 30 yet approximately 3.22 and when P is equal to 40 get approximately negative 2.286 and the unit are inches cubes per pound, and the derivative is the rate of change of life, of tyre and respect to the pressure. And when the derivative is positive, it means that, um, there's a further increase in pressure will increase the life of the tire and for negative means that a further increase in pressure will decrease the life of the tire