Solve 5 Calculus Questions in JPG File

[hannibalisfun]

problem in JPG file in order

1. I really don’t know how to do this with the integration methods I can remember right now and I don’t know how to get the greater than and less than numbers either.
2.Evaluate . This one I used L'Hopitals Rule once. Then I think I can use it one more time to get the answer but I need to look through old calc notes.
3.If what is f’(x). I just pulled the x squared out front and then used chain rule to find the answer.
4.On this one I’m not sure how to handle the integral in side the integral. I think you replace t with x after that I’m not sure how to handle the stuff in side the second integral.
5.All I can think of is making this a integral but I need to pull out my calc book because I don’t remember exactly what else I need to put it in integral form.

 

[HallsofIvy]

problem in JPG file in order

1. I really don’t know how to do this with the integration methods I can remember right now and I don’t know how to get the greater than and less than numbers either.

What is the largest possible value of $\frac{1}{1+x^4}$ between x=1 and x= 2? What is the smallest possible value? (Hint: x is clearly a decreasing function.) If $u\le f(x)\le v$, then
$$\int_a^b u dx\le \int_a^b f(x)dx\le \int_a^b v dx$$

2.Evaluate . This one I used L'Hopitals Rule once. Then I think I can use it one more time to get the answer but I need to look through old calc notes.

Don't integrate! Just notice that it is of the form
$$\lim_{x\rightarrow0}\frac{F(x)}{x}$$
Doesn't that look a lot like the definition of F'(0) to you? What is the derivative of an integral?

3.If what is f’(x). I just pulled the x squared out front and then used chain rule to find the answer.

Or the "fundamental theorem of calculus" together with the product rule.

4.On this one I’m not sure how to handle the integral in side the integral. I think you replace t with x after that I’m not sure how to handle the stuff in side the second integral.

Again, "fundamental theorem of calculus", together with the chain rule.. The second derivative is, of course, the "derivative of the derivative". After taking the first derivative you have left
$$\frac{d}{dx}\int_1^{sin x}\sqrt{1+ u^4}du$$
You will need the chain rule to handle that "sin(x)".

5.All I can think of is making this a integral but I need to pull out my calc book because I don’t remember exactly what else I need to put it in integral form.

I agree. This looks a lot like a "Riemann sum"! In creating a Riemann sum to integrate f(x) from, say, 0 to 1, if we divide it into n equal intervals, then each interval would have length 1/n and we would multiply that by f(x_i): $\frac{1}{n}f(x_i)= \frac{1}{\sqrt{n+i}{n}}$ so that $f(x_i)= \frac{n}{\sqrt{n+i}\sqrt{n}}= \frac{\sqrt{n}}{\sqrt{n+i}}$.
Dividing both numerator and denominator by $\sqrt{n}$, we have
[tex]\frac{1}{\sqrt{1+ \frac{i}{n}}}[/itex]
If we are dividing the interval from 0 to 1 into n equal intervals, and then take x_i to be the left endpoint, we would have x_i= i/n. Okay, this is a Riemann sum for the integral, from 0 to 1, of what function?
(You should notice that it could also be the integral from 1 to 2 of a slightly simpler function.)