# Equations of Parallel Tangent Lines to f(x)=3x(5x^2+1)

• Physics345
In summary: Yes, I do have a step-by-step process that I follow when I am struggling to write clearly. The first step is to take some time to relax and focus on what you are trying to say. Once you have a solid understanding of the main points you are trying to make, then it is important to organize your thoughts in a way that is easy to read. Lastly, make sure that your sentences are neither too long nor too short, and that they flow smoothly from one to the next.The first step is to take some time to relax and focus on what you are trying to say. Once you have a
Physics345

## Homework Statement

Determine the equations of the tangent lines to the graph of f(x)=3x(5x^2+1) that are parallel to the line y=8x+9

y=m(x-x_1 )+y_1

## The Attempt at a Solution

11.f(x)=3x(5x^2+1)
The slope of the tangent line y=8x+9
f^' (x)=3(5x^2+1)+3x(10x)
f^' (x)=15x^2+30x^2+3
f^' (x)=45x^2+3
Finding the point of tangency:
m=8
8=45x^2+3
5=45x^2
√(x^2 )=√(45/5)
x=±1/3
f(1/3)=(3(1/3))(5(1/3)^2+1)
f(1/3)=1(5(1/9)+1)
f(1/3)=5/9+9/9=14/9
f(1/3)=14/9

f(-1/3)=(3(-1/3))(5(-1/3)^2+1)
f(-1/3)=-1(5(1/9)+9/9)
f(-1/3)=-1(5/9+9/9)
f(-1/3)=-14/9

y=m(x-x_1 )+y_1
y_1=14/9 x_1=1/3 m=8
y=8(x-1/3)+14/9
y=8x-8/3+14/9
y=8x-24/9+14/9
y=8x-10/9

y=m(x-x_1 )+y_1
y_1=-14/9 x_1=-1/3 m=8
y=8(x-(-1/3))-14/9
y=8x+8/3-14/9
y=8x+24/9-14/9
y=8x+10/9
Therefore, the equations of the tangent lines are y=8x+10/9 and y=8x-10/9 I'm pretty sure I did this correctly, but I'm not confident in my final statement.

You seem correct.

Physics345
It looks good to me. After all those calculations, it's good to plug numbers back in and make sure it works as advertised: At x=1/3, does f' = 8 and f(x) = 3x(5x^2+1) = 8x+10/9? Do the same type of checks for x=-1/3.

Physics345
FactChecker said:
It looks good to me. After all those calculations, it's good to plug numbers back in and make sure it works as advertised: At x=1/3, does f' = 8 and f(x) = 3x(5x^2+1) = 8x+10/9? Do the same type of checks for x=-1/3.
Of course, I did that on paper, but I thought it would be pointless, to show to the teacher when I handed in my work. Especially since my teacher understands my enjoyment and capabilities when it comes to math. Thanks guys, I appreciate your advice.

lekh2003
You said that you were not confident in your answers. Plugging the numbers back into verify that the answers are correct should have given you confidence.

The one exception is in the slope of 8, which depends on whether you calculated the derivative correctly. That is hard to verify by plugging numbers directly into f(x) except by plugging in x=1/3 and something like x+Δx = 1/3+0.001.

Physics345
Plugging in numbers back again is a very fast way to figure out whether your answer is right, especially under time-intensive conditions or when the actual working is so long that you need an alternative method rather than skimming the working.

I just today used the technique of plugging the numbers back into the start in a long time-intensive financial math test, where most of the working is hidden in the calculator.

FactChecker said:
You said that you were not confident in your answers. Plugging the numbers back into verify that the answers are correct should have given you confidence.

The one exception is in the slope of 8, which depends on whether you calculated the derivative correctly. That is hard to verify by plugging numbers directly into f(x) except by plugging in x=1/3 and something like x+Δx = 1/3+0.001.
Oh I was referring to my therefore statement when I said that sir, and when I said "I'm pretty sure I did this correctly" I was referring to the math and using "pretty sure" was my scapegoat in case my math was somehow wrong in a way beyond, my spectrum of understanding. Sorry for the confusion! I have a problem conveying my thoughts through writing, it's something I've been working on a lot lately.

Last edited:
lekh2003 said:
Plugging in numbers back again is a very fast way to figure out whether your answer is right, especially under time-intensive conditions or when the actual working is so long that you need an alternative method rather than skimming the working.

I just today used the technique of plugging the numbers back into the start in a long time-intensive financial math test, where most of the working is hidden in the calculator.
Yes of course, I completely agree. The number one rule of doing math is checking your work, since there is always a way to do it in algebraically when it comes to functions, and by "always" I mean in relation to my current level of math.

Physics345 said:
Sorry for the confusion! I have a problem conveying my thoughts through writing, it's something I've been working on a lot lately.
Ha! I can really understand that! Being able to write clearly is something I have struggled with all my life. It's very good that you recognize how important that is.

Physics345
FactChecker said:
Ha! I can really understand that! Being able to write clearly is something I have struggled with all my life. It's very good that you recognize how important that is.
Oh, I've always had this problem as well, unless I spend endless hours reviewing/refining what I write. I wish there was a easier solution, but I tend to rush into things without thinking, which I have found extremely hard to stop. Do you have any techniques or advice that could possibly help me out? I honestly thought I was crazy and this wasn't normal, I'm glad I found someone that can relate to my problem.

Physics345 said:
Oh, I've always had this problem as well, unless I spend endless hours reviewing/refining what I write. I wish there was a easier solution, but I tend to rush into things without thinking, which I have found extremely hard to stop. Do you have any techniques or advice that could possibly help me out? I honestly thought I was crazy and this wasn't normal, I'm glad I found someone that can relate to my problem.
Just keep word-smithing your work to make it clearer. Eventually I learned what bad habits I have and look to correct those. I usually have to break up very long sentences and replace ambiguous pronouns with the specific nouns.

Physics345
FactChecker said:
Just keep word-smithing your work to make it clearer. Eventually I learned what bad habits I have and look to correct those. I usually have to break up very long sentences and replace ambiguous pronouns with the specific nouns.
That's what I've been doing for a while now, it just takes a very long time. I guess eventually I'll be able to speed up the process with practice and time.

## 1. What are parallel tangent lines?

Parallel tangent lines are two lines that never intersect and have the same slope at any given point on a curve. They are always equidistant from each other and are found on the same side of the curve.

## 2. How are parallel tangent lines used in mathematics?

Parallel tangent lines are used to approximate curves and calculate the slope of a curve at a specific point. They are also important in geometry and trigonometry, as well as in calculus for finding the derivative of a function.

## 3. Can parallel tangent lines exist on any type of curve?

No, parallel tangent lines can only exist on curves that have a constant slope at every point. This includes straight lines, circles, and parabolas. Curves with changing slopes, such as sine and cosine waves, do not have parallel tangent lines.

## 4. How do you find the equation of parallel tangent lines?

To find the equation of parallel tangent lines, you need to know the slope of the curve at a specific point. Then, you can use the point-slope form of a line to create the equation. The slope of the parallel tangent line will be the same as the slope of the curve at that point.

## 5. What is the significance of parallel tangent lines?

Parallel tangent lines are important in calculus for finding the derivative of a function and determining the rate of change at a specific point. They also have applications in physics and engineering, where they are used to approximate curves and calculate the slope of a curve at a given point.

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