The document explains how to find the equation of tangent... Show more
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
29
0
Diana
05/09/2022
Maths
Higher Maths - Equation Of A Tangent
1,045
•
5 Sept 2022
•
Easy Guide: Find the Tangent to a Curve or Circle
Diana
@ddovgy
More Tangent Examples
This page expands on the concept of finding tangent equations with more complex examples. It demonstrates how to find the equation of a tangent to a curve without a given point, specifically when only the x-coordinate is provided.
The first example deals with the curve y = x² - 2x at the point where x = -1. This problem requires an additional step of calculating the y-coordinate before proceeding with the tangent equation.
Example: For y = x² - 2x at x = -1, the process involves:
- Finding y when x = -1
- Differentiating to get dy/dx = 2x - 2
- Calculating the gradient at x = -1
- Using the point-slope form to derive the tangent equation: y = x + 2
The page also introduces a more challenging example involving a rational function, y = / , where the gradient of the tangent is given as 11.
Highlight: When the gradient is given, the process involves solving an equation to find the x-coordinate of the tangent point.
Tangents with Specified Gradients
This page focuses on finding tangent equations when the gradient is specified, particularly for the curve y = x² - 4x. It demonstrates how to find the equation of a tangent to a curve without a given point, using only the gradient information.
The problem involves finding two tangent equations where the gradient is 8. The process includes:
- Differentiating the curve equation: dy/dx = 2x - 4
- Setting the derivative equal to the given gradient: 2x - 4 = 8
- Solving for x to find the points of tangency
- Calculating the y-coordinates for these x-values
- Using the point-slope form to derive the tangent equations
Example: For y = x² - 4x with gradient 8, two tangent points are found: and (2, 0).
Highlight: This example showcases that a single curve can have multiple tangents with the same gradient, emphasizing the importance of considering all possible solutions.
The resulting tangent equations are y = 8x - 16 and y = 8x + 16, demonstrating how two different tangents can have the same gradient but different y-intercepts.
Equation of a Tangent
This page introduces the concept of finding the equation of tangent to the curve at a point. It explains that to determine the tangent equation, two pieces of information are crucial: the point on the curve and the gradient at that point. The gradient is calculated by differentiating the curve's equation and substituting the x-value of the given point.
Definition: The equation of a tangent is a linear equation that represents the line touching a curve at a single point.
The page provides a detailed example of finding the tangent equation for the curve y = x² - 3 at the point . It demonstrates the step-by-step process:
- Differentiate the curve equation to find dy/dx
- Calculate the gradient by substituting the x-value
- Use the point-slope form y - y₁ = m to derive the tangent equation
Example: For y = x² - 3 at , the tangent equation is derived as y = 4x - 7.
Highlight: The point-slope form of a line, y - b = m, is crucial for determining the tangent equation once the gradient and point are known.
We thought you’d never ask...
What is the Knowunity AI companion?
Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
Where can I download the Knowunity app?
You can download the app from Google Play Store and 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
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
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
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
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 apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
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
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
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
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
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 apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
Easy Guide: Find the Tangent to a Curve or Circle
Diana
@ddovgy
The document explains how to find the equation of tangent to the curve at a pointusing differentiation. It covers the process of calculating the gradient and using the point-slope form of a line to determine the tangent equation. The... Show more
More Tangent Examples
This page expands on the concept of finding tangent equations with more complex examples. It demonstrates how to find the equation of a tangent to a curve without a given point, specifically when only the x-coordinate is provided.
The first example deals with the curve y = x² - 2x at the point where x = -1. This problem requires an additional step of calculating the y-coordinate before proceeding with the tangent equation.
Example: For y = x² - 2x at x = -1, the process involves:
- Finding y when x = -1
- Differentiating to get dy/dx = 2x - 2
- Calculating the gradient at x = -1
- Using the point-slope form to derive the tangent equation: y = x + 2
The page also introduces a more challenging example involving a rational function, y = / , where the gradient of the tangent is given as 11.
Highlight: When the gradient is given, the process involves solving an equation to find the x-coordinate of the tangent point.
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
Tangents with Specified Gradients
This page focuses on finding tangent equations when the gradient is specified, particularly for the curve y = x² - 4x. It demonstrates how to find the equation of a tangent to a curve without a given point, using only the gradient information.
The problem involves finding two tangent equations where the gradient is 8. The process includes:
- Differentiating the curve equation: dy/dx = 2x - 4
- Setting the derivative equal to the given gradient: 2x - 4 = 8
- Solving for x to find the points of tangency
- Calculating the y-coordinates for these x-values
- Using the point-slope form to derive the tangent equations
Example: For y = x² - 4x with gradient 8, two tangent points are found: and (2, 0).
Highlight: This example showcases that a single curve can have multiple tangents with the same gradient, emphasizing the importance of considering all possible solutions.
The resulting tangent equations are y = 8x - 16 and y = 8x + 16, demonstrating how two different tangents can have the same gradient but different y-intercepts.
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
Equation of a Tangent
This page introduces the concept of finding the equation of tangent to the curve at a point. It explains that to determine the tangent equation, two pieces of information are crucial: the point on the curve and the gradient at that point. The gradient is calculated by differentiating the curve's equation and substituting the x-value of the given point.
Definition: The equation of a tangent is a linear equation that represents the line touching a curve at a single point.
The page provides a detailed example of finding the tangent equation for the curve y = x² - 3 at the point . It demonstrates the step-by-step process:
- Differentiate the curve equation to find dy/dx
- Calculate the gradient by substituting the x-value
- Use the point-slope form y - y₁ = m to derive the tangent equation
Example: For y = x² - 3 at , the tangent equation is derived as y = 4x - 7.
Highlight: The point-slope form of a line, y - b = m, is crucial for determining the tangent equation once the gradient and point are known.
We thought you’d never ask...
What is the Knowunity AI companion?
Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
Where can I download the Knowunity app?
You can download the app from Google Play Store and 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
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
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
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
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 apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
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
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
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
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
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 apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user