# Solve 5 Calculus Questions in JPG File

• hannibalisfun
In summary, the conversation discusses various integration methods and techniques, such as L'Hopital's Rule, the fundamental theorem of calculus, and the chain rule. The focus is on solving integrals and determining derivatives, with the use of Riemann sums also mentioned. The conversation highlights the importance of using proper techniques and tools to solve integration problems.
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.

#### Attachments

• math problems.JPG
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hannibalisfun said:
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(xi): $\frac{1}{n}f(x_i)= \frac{1}{\sqrt{n+i}{n}}$ 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 xi to be the left endpoint, we would have xi= 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.)

## 1. How do I solve calculus questions in a JPG file?

To solve calculus questions in a JPG file, you will need to first convert the image into a digital format that can be edited, such as a PDF or a Word document. There are various online tools and software programs that can help you do this. Once the image is converted, you can then use a calculator or a math software program to solve the equations and find the solutions.

## 2. Can I use a calculator to solve calculus questions in a JPG file?

Yes, you can use a calculator to solve calculus questions in a JPG file. However, it is important to note that some calculators may not have the necessary functions or capabilities to solve complex calculus equations. In such cases, it may be more efficient to use a math software program specifically designed for calculus.

## 3. Are there any tips for solving calculus questions in a JPG file?

One helpful tip for solving calculus questions in a JPG file is to first identify the type of problem you are dealing with, such as differentiation or integration. This will help you determine the appropriate method or formula to use. It is also important to carefully read and understand the given question, and to double-check your calculations to avoid errors.

## 4. What are some common mistakes to avoid when solving calculus questions in a JPG file?

One common mistake to avoid when solving calculus questions in a JPG file is not using the correct units or not including units in your final answer. It is also important to pay attention to signs and make sure to include them in your calculations. Another mistake to avoid is not simplifying your final answer, which may result in an incorrect solution.

## 5. Can I use a math software program to solve calculus questions in a JPG file?

Yes, using a math software program can be a helpful tool for solving calculus questions in a JPG file. These programs often have advanced functions and features that can make solving complex equations easier and more accurate. It is important to choose a reputable and reliable program to ensure accurate results.

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