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8. (100 points) Show that if H < G,K 4 G,then H oK 4G (Hint: show that Va eG,Vx e HnK,=> axa' eHoK)...

Question

8. (100 points) Show that if H < G,K 4 G,then H oK 4G (Hint: show that Va eG,Vx e HnK,=> axa' eHoK)

8. (100 points) Show that if H < G,K 4 G,then H oK 4G (Hint: show that Va eG,Vx e HnK,=> axa' eHoK)



Answers

Show that if $h$ is near zero, $e^{h} \approx 1+h .$ Hint: Use (1 a) with $h$ replacing $u$.

Alright. The theme in this problem is just writing the chain rule in a in a a different notation. So I'm gonna go like back and forth just to show you stuff my stuff. Because when you start with H equals F. Of G. That is the same thing as saying H of X is equal to F. Of G. Of X. And so let me show you in green now where the chain was located. Well the channel says that you take the derivative of the outer function and I didn't write it out. I've been writing in other functions. Do we have? The inner function is is what's inside the parentheses? The outer function is on the outside. So the derivative of H is equal to the derivative of the outside function. You leave the inside function alone and then times the derivative of the inside function. So as you look at that closely, we have a new way of writing this. Um and I'll be H. Prime is equal to F. Prime of G. Uh That quantity is then multiplied by G crime, which is exactly how they write it. Um So hopefully you understood my step by step process to get from here to hear. Um And really I need to circle everything because they want to they want you to show the work, so it's all of this work in here. Yeah.

In this question it is given that hb the action between A. And B. Then we have to prove that H upon a plus H upon B is equal to two given pitch B. The change between okay E. And B. Therefore H. Is equal to two Into A. and two B. Yes upon a plus B. Thank you. Edge into A plus B. Is equal to to a B. Next position etched into A plus B upon. Mhm. Thank you. A. and two B is equal to two. Edge into A upon a B. Plus hatch into be upon a B. Is equal to two. Mhm. Hey edge upon B plus H upon a is equal to two, Therefore edge upon A plus etched upon B Is equal to two. Yeah. Two. Mhm. Mr. Okay. Mhm. Mhm. Yeah.

Here. We have ever backs being equal to 10 to the X. So now you want to use on a difference question here. So f off X plus h minus after vax all divided by H. Ok, and this is supposed to be equal to 10 to the ex. Um right. Wanna show that this is equal to 10 to the X times 10 to the age, minus one all over each. Okay, let's go ahead and just plug this end to a different question. So what is f of X plus h mean while f of axe is 10 to the axe, so f of X plus h would be 10 to the X plus a h. Ok, there's f of X plus age and then we have one minus aftereffects. So minus f of X is minus. Whatever backs is 10 to the X. We have minus 10 to the axe, right? I'm all divided by h. So there we have. And this is f of X plus age minus half of axe, all divided by age. Now, this is supposed be equal to this. It is not yet, but what? What What What could we do here? Well, we can go ahead and actually factor out a 10 to the X from the numerator. If we do that, we get in fact, er out of 10 to the ax. Okay, Now we'll tend to the x Times. What? Give me a 10 to the X plus h while will be 10 to the h. Why? Because we multiply right with the same base. We add the exponents. So therefore, a 10 to the axe. Times 10 to the H. Well, that's equal to 10 to the X plus a h or then right. I'm factoring out a 10 to the X. So 10 to the X Times. What gives me a negative 10 to the X. Well, that's minus one. Okay, so there's my numerator factor. Right? But then that is all over all over h. Okay, which well, that tend to the X right can just come outside and we get, um this is equal to 10 to the X times 10 to the H minus one over h and therefore right. We prove that is the expression we want, and we got it. So therefore, um, we have shown, um What? Want to show? All right, take care

In the way we have to prove If you do DeLeo inch equals f energy into age and waiting in the left hand side festival if and do you endure it de that puts the tea If the endure it in the d minus town the job everyone is equal because I didn t if I don't do integration and you know do you my last town z off y h off T minus tower minus y do you are You did so for that We have me, uh, making the change off Mary. So we're putting Why u minus so far we have if and d into gate that is even as it gets. And you're d down t if our town g off u minus town John, do you mind you? Do you dejar And that is even s todo they don't know you If the off u minus dio a job, Do you mind us? You do you So this What's to in de f in D into that is if g off you and the mask you do what? It isn't even as in de in it off Dean and that goods to outages. So finally we have Lepine science will do right hand side and the statement is fruit


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