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Finding the Gradient of a Curve and Understanding Stationary Points

4

0

user profile picture

Millie Roberts

9/21/2022

Maths

Differentiation Q and A

78

•

Sep 21, 2022

•

3 pages

Finding the Gradient of a Curve and Understanding Stationary Points

user profile picture

Millie Roberts

@millierobertsz

Learn how to find the gradient of a curve using dy/dx and calculate it at points like x=2. Discover how to find the gradient of a function f(x, y, z) and explore tangents. Understand stationary points and how to determine if they're maximum or minimum using the second derivative. Get examples and use calculators for finding stationary points and gradients. Perfect for kids curious about calculus and PDF resources!

1. y=2x²³ x^ + 5x² - 7x + 10 a) Find dy doc dy = 6x² + 10x -7. dx. b). Find gradient when x = 2 dy = 6x² +10x-7 dx cunen x=2 dy = 6(2)² + 10

Finding Stationary Points

This page focuses on finding the coordinates of stationary points and determining their nature. It continues the example from the previous page and introduces a new, more complex problem.

For the function y = 3x + 1/x², the page demonstrates how to find stationary points by setting dy/dx = 0 and solving for x.

Example: For y = 3x + 1/x², the stationary points occur at x = ±1/√3.

The page then introduces a new problem: y = 4x³ + 15x² - 18x + 5. It guides the reader through the process of finding the stationary points for this function.

Highlight: Stationary points occur where dy/dx = 0, which is crucial for finding the gradient of a tangent at these points.

Vocabulary: Stationary points are points on a curve where the gradient is zero or undefined.

This section provides valuable insight into how to determine nature of stationary points using second derivative, which is explored further on the next page.

1. y=2x²³ x^ + 5x² - 7x + 10 a) Find dy doc dy = 6x² + 10x -7. dx. b). Find gradient when x = 2 dy = 6x² +10x-7 dx cunen x=2 dy = 6(2)² + 10

Analyzing Stationary Points

The final page delves deeper into the analysis of stationary points, focusing on determining their nature using the second derivative test.

For the function y = 4x³ + 15x² - 18x + 5, the page shows how to calculate the second derivative (d²y/dx²) and use it to classify the stationary points.

Example: For the stationary point at x = 1/2, d²y/dx² > 0, indicating a minimum point. For x = -3, d²y/dx² < 0, indicating a maximum point.

Definition: The second derivative test states that if d²y/dx² > 0 at a stationary point, it's a minimum; if d²y/dx² < 0, it's a maximum.

This page effectively demonstrates how to determine nature of stationary points using second derivative, providing a comprehensive understanding of curve analysis.

Highlight: Understanding the nature of stationary points is crucial for graphing functions and solving optimization problems in calculus.

The document concludes by emphasizing the importance of these techniques in analyzing functions and their behavior, which is essential for advanced mathematical studies and applications.

1. y=2x²³ x^ + 5x² - 7x + 10 a) Find dy doc dy = 6x² + 10x -7. dx. b). Find gradient when x = 2 dy = 6x² +10x-7 dx cunen x=2 dy = 6(2)² + 10

Finding the Gradient of a Curve

This page introduces methods for finding the gradient of a curve using dy/dx. It presents two main examples to illustrate the process.

The first example deals with the function y = 2x³ + 5x² - 7x + 10. The page demonstrates how to find dy/dx and then calculate the gradient at a specific point.

Example: For y = 2x³ + 5x² - 7x + 10, dy/dx = 6x² + 10x - 7. When x = 2, the gradient is 37.

The second example involves a more complex function, y = 3x + 1/x². The page shows how to differentiate this function and find its gradient.

Highlight: The differentiation rule for x^n is nx^n−1n-1, which is crucial for finding dy/dx.

Vocabulary: dy/dx represents the derivative of y with respect to x, which gives the gradient function of the curve.

This page provides a solid foundation for understanding how to find the gradient of a curve at a point where x=2 or any other given x-value.



We thought you’d never ask...

What is the Knowunity AI companion?

Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.

Where can I download the Knowunity app?

You can download the app in the Google Play Store and in the Apple App Store.

Is Knowunity really free of charge?

That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.

Can't find what you're looking for? Explore other subjects.

Students love us — and so will you.

4.9/5

App Store

4.8/5

Google Play

The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan S

iOS user

This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha Klich

Android user

Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

Anna

iOS user

I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️

Thomas R

iOS user

Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

iOS user

This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!

Paul T

iOS user

The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan S

iOS user

This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha Klich

Android user

Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

Anna

iOS user

I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️

Thomas R

iOS user

Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

iOS user

This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!

Paul T

iOS user

 

Maths

•

78

•

Sep 21, 2022

•

3 pages

Finding the Gradient of a Curve and Understanding Stationary Points

user profile picture

Millie Roberts

@millierobertsz

Learn how to find the gradient of a curve using dy/dx and calculate it at points like x=2. Discover how to find the gradient of a function f(x, y, z) and explore tangents. Understand stationary points and how to determine if they're maximum or minimum using the second derivative. Get examples and use calculators for finding stationary points and gradients. Perfect for kids curious about calculus and PDF resources!

1. y=2x²³ x^ + 5x² - 7x + 10 a) Find dy doc dy = 6x² + 10x -7. dx. b). Find gradient when x = 2 dy = 6x² +10x-7 dx cunen x=2 dy = 6(2)² + 10

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Finding Stationary Points

This page focuses on finding the coordinates of stationary points and determining their nature. It continues the example from the previous page and introduces a new, more complex problem.

For the function y = 3x + 1/x², the page demonstrates how to find stationary points by setting dy/dx = 0 and solving for x.

Example: For y = 3x + 1/x², the stationary points occur at x = ±1/√3.

The page then introduces a new problem: y = 4x³ + 15x² - 18x + 5. It guides the reader through the process of finding the stationary points for this function.

Highlight: Stationary points occur where dy/dx = 0, which is crucial for finding the gradient of a tangent at these points.

Vocabulary: Stationary points are points on a curve where the gradient is zero or undefined.

This section provides valuable insight into how to determine nature of stationary points using second derivative, which is explored further on the next page.

1. y=2x²³ x^ + 5x² - 7x + 10 a) Find dy doc dy = 6x² + 10x -7. dx. b). Find gradient when x = 2 dy = 6x² +10x-7 dx cunen x=2 dy = 6(2)² + 10

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Analyzing Stationary Points

The final page delves deeper into the analysis of stationary points, focusing on determining their nature using the second derivative test.

For the function y = 4x³ + 15x² - 18x + 5, the page shows how to calculate the second derivative (d²y/dx²) and use it to classify the stationary points.

Example: For the stationary point at x = 1/2, d²y/dx² > 0, indicating a minimum point. For x = -3, d²y/dx² < 0, indicating a maximum point.

Definition: The second derivative test states that if d²y/dx² > 0 at a stationary point, it's a minimum; if d²y/dx² < 0, it's a maximum.

This page effectively demonstrates how to determine nature of stationary points using second derivative, providing a comprehensive understanding of curve analysis.

Highlight: Understanding the nature of stationary points is crucial for graphing functions and solving optimization problems in calculus.

The document concludes by emphasizing the importance of these techniques in analyzing functions and their behavior, which is essential for advanced mathematical studies and applications.

1. y=2x²³ x^ + 5x² - 7x + 10 a) Find dy doc dy = 6x² + 10x -7. dx. b). Find gradient when x = 2 dy = 6x² +10x-7 dx cunen x=2 dy = 6(2)² + 10

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Finding the Gradient of a Curve

This page introduces methods for finding the gradient of a curve using dy/dx. It presents two main examples to illustrate the process.

The first example deals with the function y = 2x³ + 5x² - 7x + 10. The page demonstrates how to find dy/dx and then calculate the gradient at a specific point.

Example: For y = 2x³ + 5x² - 7x + 10, dy/dx = 6x² + 10x - 7. When x = 2, the gradient is 37.

The second example involves a more complex function, y = 3x + 1/x². The page shows how to differentiate this function and find its gradient.

Highlight: The differentiation rule for x^n is nx^n−1n-1, which is crucial for finding dy/dx.

Vocabulary: dy/dx represents the derivative of y with respect to x, which gives the gradient function of the curve.

This page provides a solid foundation for understanding how to find the gradient of a curve at a point where x=2 or any other given x-value.

We thought you’d never ask...

What is the Knowunity AI companion?

Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.

Where can I download the Knowunity app?

You can download the app in the Google Play Store and in the Apple App Store.

Is Knowunity really free of charge?

That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.

Can't find what you're looking for? Explore other subjects.

Students love us — and so will you.

4.9/5

App Store

4.8/5

Google Play

The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan S

iOS user

This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha Klich

Android user

Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

Anna

iOS user

I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️

Thomas R

iOS user

Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

iOS user

This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!

Paul T

iOS user

The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan S

iOS user

This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha Klich

Android user

Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

Anna

iOS user

I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️

Thomas R

iOS user

Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

iOS user

This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!

Paul T

iOS user