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"Mark's Song" System Check www.reverbnation.com/systemcheck 2017

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"Mark's Song" System Check www.reverbnation.com/systemcheck 2017

For more infomation >> Mark's songMusic Video 2 - Duration: 1:57.

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Video: Mayor ticks off priorities of violence-reduction plan - Duration: 1:53.

For more infomation >> Video: Mayor ticks off priorities of violence-reduction plan - Duration: 1:53.

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Video Solution #1 - Duration: 3:46.

So for my video presentation, I'll be doing Question 42 from Chapter 3.1, which is a part

of Unit 4.

So, in this question, they ask us to use the limits definition of a derivative to determine

these derivatives.

And so in part (a), the first step is to write out the limits definition of this derivative.

So we take the limit as h goes to 0 of the square root of a(x+h)+b subtract the square

root of ax+b, and we take that all over h.

So as we learned in the limits section, when we have a sum or difference of square roots

in a fraction that we're trying to find the limit of, it often helps to multiply the entire

fraction by the conjugate.

So that's what we're going to do here.

We're going to multiply the numerator by the conjugate, which is the square root of a(x+h)+b

plus the square root of ax+b.

And of course, whatever we do to the numerator, we also have to do to the denominator, in

order to maintain the equality of the fraction.

We'll then combine the two terms of the product.

In the numerator, we see that we just get a difference of squares; so we just square

the first square root, to give us a(x+h)+b, and we subtract the square of the second square

root, which is just ax+b.

And we do that all over the denominator, which is h times the conjugate that we got above.

We then expand the numerator to see if we're able to simplify it.

So the numerator becomes ax + ah + b - ax - b and the denominator remains the same.

As we see in the numerator, ax minus ax is 0, b minus b is 0, so we're just left with

one term, ah, in the numerator.

But we see that we can factor out an "h" from both the numerator and the denominator, so

that cancels out.

And we're left with the limit as h goes to 0 of a over the square root of a(x+h) + b

plus the square root of ax + b.

So, in this case, as h goes to 0, this term goes to 0 as well.

And so what we get is a over the square root of ax+b plus the square root of ax+b, and

this gives us the final answer of a over 2 times the square root of ax+b.

For part (b), they want us to use our answer from part (a) to determine the derivative

of the square root of 5x+9.

We quickly see that these two expressions are essentially identical.

They simply replaced the a with a 5, and they've replaced the b with a 9.

So to find the derivative of the square root of 5x+9, all we have to do is substitute these

values of a and b into our solution from part a.

So we get 5 over 2 times the square root of 5x+9, and that's our solution for part b.

And of course, we could have rationalized the denominator by multiplying the top and

bottom by the square root terms in both of the solutions for part (a) and (b)...as so...although

these answers are essentially equivalent.

So those are the answers for question 42, and this is my video presentation.

For more infomation >> Video Solution #1 - Duration: 3:46.

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All my animations in one video (Updated) October 2012 - July 2017 :) - Duration: 3:46.

presents...

All my animations in one video (updated version from the summer, 2017) :)

Thanks for watching :)