In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve y = f (x) at a point x = c on the curve if the line passes through the point (c, f (c)) on the curve and has slope f'(c) where f' is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space.
As it passes through the point where the tangent line and the curve meet, called the point of tangency, the tangent line is "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point.
Similarly, the tangent plane to a surface at a given point is the plane that "just touches" the surface at that point. The concept of a tangent is one of the most fundamental notions in differential geometry and has been extensively generalized; see Tangent space.
Finding The Tangent Line Equation With Derivatives - Calculus Problems
This calculus video tutorial shows you how to find the equation of a tangent line with derivatives. Techniques include the power rule, product rule, and implicit differentiation. This video shows you how to write the equation of the tangent line in point slope form and slope intercept form. This video contains plenty of examples and practice problems.
Application of Derivatives - Free Formula Sheet:
https://bit.ly/4eV6r1b
_____________________________
Derivatives - Fast Review:
https://www.youtube.com/watch?v=5yfh5cf4-0w&t=1s
Equation of the Tangent Line:
https://www.youtube.com/watch?v=5NyeGzbBJQM
Derivatives - Horizontal Tangent Line:
https://www.youtube.com/watch?v=aNfoxbMUOHk
The Equation of The Normal Line:
https://www.y...
published: 21 Feb 2016
The Tangent Line and the Derivative (Calculus)
In calculus, you’ll often hear “The derivative is the slope of the tangent line.” But what is a tangent line? The definition is trickier than you might think.
Tangent lines are important because they are the best way to approximate a curve using a line. We can then use the slope of the line as a way to measure the “slope” of the curve.
Be sure to subscribe so you don't miss videos from Socratica!
http://bit.ly/1ixuu9W
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Ways to support our channel:
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Connect with us!
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published: 06 Nov 2016
Understanding Differentiation Part 1: The Slope of a Tangent Line
The first operation in calculus that we have to understand is differentiation. So what is it, exactly? Well there are a couple of ways of looking at it. The first one involves finding the equation of a tangent line for some point on a curve. We can't do this with algebra, because for that we would need two points on the line, and we only have one. Enter calculus!
Watch the whole Calculus playlist: http://bit.ly/ProfDaveCalculus
Watch the whole Mathematics playlist: http://bit.ly/ProfDaveMath
Classical Physics Tutorials: http://bit.ly/ProfDavePhysics1
Modern Physics Tutorials: http://bit.ly/ProfDavePhysics2
General Chemistry Tutorials: http://bit.ly/ProfDaveGenChem
Organic Chemistry Tutorials: http://bit.ly/ProfDaveOrgChem
Biochemistry Tutorials: http://bit.ly/ProfDaveBiochem
Biology Tuto...
published: 21 Feb 2018
Calculus 1: Limits & Derivatives (1 of 27) The Tangent Line and The Secant Line - Reviewed
Visit http://ilectureonline.com for more math and science lectures!
In this video I will review the tangent and secant line with respect to a function.
Next video in the series can be seen at:
http://youtu.be/723jsxCO7Oc
published: 20 Aug 2015
Slope and Equation of Normal & Tangent Line of Curve at Given Point - Calculus Function & Graphs
This calculus video shows you how to find the slope and the equation of the tangent line and normal line to the curve/function at a given point. This video contains plenty of example problems.
Derivatives - Fast Review: https://www.youtube.com/watch?v=5yfh5cf4-0w&t=1s
Equation of the Tangent Line:
https://www.youtube.com/watch?v=5NyeGzbBJQM
Derivatives - Horizontal Tangent Line:
https://www.youtube.com/watch?v=aNfoxbMUOHk
The Equation of The Normal Line:
https://www.youtube.com/watch?v=lEF2mmR3CWU
The Equation of The Secant Line:
https://www.youtube.com/watch?v=680o1GL-LRg
_________________________________
Average and Instantaneous Velocity:
https://www.youtube.com/watch?v=AwxT1xjMP9g
Instantaneous Rate o...
published: 05 Sep 2016
TANGENT LINES AND CIRCLES EXPLAINED!
Let's explore, identify, and describe the relationship between a circle and a tangent line and why the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
★ Want more FREE weekly math videos? Subscribe Now: http://bit.ly/1OVJnyh
★★ Why Math? Download YOUR Free eBook and sign-up for our FREE weekly newsletter: http://www.mashupmath.com/resources/
Understand and apply theorems about circles.
Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
What is a tangent line to a circle?
Why is a tagent line and ...
published: 05 Oct 2016
How To Find The Equation of The Tangent Line With Derivatives
This calculus video tutorial explains how to find the equation of the tangent line with derivatives. It explains how to write the equation of the tangent line in point slope form and slope intercept form.
Derivatives - Fast Review: https://www.youtube.com/watch?v=5yfh5cf4-0w&t=1s
Equation of the Tangent Line:
https://www.youtube.com/watch?v=5NyeGzbBJQM
Derivatives - Horizontal Tangent Line:
https://www.youtube.com/watch?v=aNfoxbMUOHk
The Equation of The Normal Line:
https://www.youtube.com/watch?v=lEF2mmR3CWU
The Equation of The Secant Line:
https://www.youtube.com/watch?v=680o1GL-LRg
_________________________________
Average and Instantaneous Velocity:
https://www.youtube.com/watch?v=AwxT1xjMP9g
Instan...
published: 24 Feb 2018
How to Find the Equation of a Tangent Line with Derivatives (NancyPi)
MIT grad shows how to find the tangent line equation using a derivative (Calculus). To skip ahead: 1) For a BASIC example, skip to time 0:44. 2) For an example that uses the QUOTIENT RULE, skip to time 5:58. 3) For an example that uses TRIG FUNCTIONS and the CHAIN RULE for finding the equation of a tangent line, skip to time 11:35. Nancy formerly of MathBFF explains the steps.
For HOW TO FIND DERIVATIVES, jump to: https://youtu.be/QqF3i1pnyzU
Follow me on Instagram! https://instagram.com/nancypi
Follow me on Twitter: https://twitter.com/nancypi
What is a tangent line? It is a line that is tangent to a curve at one point. You can use calculus for finding the tangent line equation, by taking the derivative of the given function.
Finding the EQUATION OF A TANGENT LINE using a DERIVATIVE:...
published: 15 May 2018
Apollonius proved that a tangent line never crosses a curve at the point of tangency.
Apollonius proved that a tangent line never crosses a curve at the point of tangency. Proposition. (I. 17, 32)
The great geometer Apollonius fortunately proved this is true for any conic and hence ANY other curve by implication. The idiots of mainstream mathematics academia think they know better and have arrogantly "dismissed" Apollonius whose genius far surpasses any of the morons who came after Euclid. Just imagine! Mainstream idiots followed the ideas of that stupid bum Gottfried Leibniz who claimed a tangent line passes through two infinitely close points and therefore can cross at the point of tangency.
Find the article used in the presentation here:
https://www.academia.edu/121458742/The_concept_of_tangent_line_was_defined_in_the_Conics_of_Apollonius_Proposition_11_I_17_32_
Lear...
This calculus video tutorial shows you how to find the equation of a tangent line with derivatives. Techniques include the power rule, product rule, and implic...
This calculus video tutorial shows you how to find the equation of a tangent line with derivatives. Techniques include the power rule, product rule, and implicit differentiation. This video shows you how to write the equation of the tangent line in point slope form and slope intercept form. This video contains plenty of examples and practice problems.
Application of Derivatives - Free Formula Sheet:
https://bit.ly/4eV6r1b
_____________________________
Derivatives - Fast Review:
https://www.youtube.com/watch?v=5yfh5cf4-0w&t=1s
Equation of the Tangent Line:
https://www.youtube.com/watch?v=5NyeGzbBJQM
Derivatives - Horizontal Tangent Line:
https://www.youtube.com/watch?v=aNfoxbMUOHk
The Equation of The Normal Line:
https://www.youtube.com/watch?v=lEF2mmR3CWU
The Equation of The Secant Line:
https://www.youtube.com/watch?v=680o1GL-LRg
_________________________________
Average and Instantaneous Velocity:
https://www.youtube.com/watch?v=AwxT1xjMP9g
Instantaneous Rate of Change:
https://www.youtube.com/watch?v=dvQdQLTnDpk
Derivatives of Rational Functions:
https://www.youtube.com/watch?v=qknoFdPwEco
Derivatives of Radical Functions:
https://www.youtube.com/watch?v=B1YkgNDbx5o
Derivatives of Fractions:
https://www.youtube.com/watch?v=qrBkQ8Ci2fM
________________________________
Derivatives - Higher Order:
https://www.youtube.com/watch?v=s7rd9YPJrNc
Simplifying Derivatives:
https://www.youtube.com/watch?v=3lUOtjkqfQo
Derivatives - The Product Rule:
https://www.youtube.com/watch?v=17X5g9QArTc
Derivatives - The Quotient Rule:
https://www.youtube.com/watch?v=8jVDEcQ0wXk
Derivatives - The Chain Rule:
https://www.youtube.com/watch?v=HaHsqDjWMLU
_______________________________________
Final Exams and Video Playlists:
https://www.video-tutor.net/
Full-Length Videos and Worksheets:
https://www.patreon.com/MathScienceTutor/collections
Derivatives - Formula Sheet:
https://bit.ly/4dThzf1
This calculus video tutorial shows you how to find the equation of a tangent line with derivatives. Techniques include the power rule, product rule, and implicit differentiation. This video shows you how to write the equation of the tangent line in point slope form and slope intercept form. This video contains plenty of examples and practice problems.
Application of Derivatives - Free Formula Sheet:
https://bit.ly/4eV6r1b
_____________________________
Derivatives - Fast Review:
https://www.youtube.com/watch?v=5yfh5cf4-0w&t=1s
Equation of the Tangent Line:
https://www.youtube.com/watch?v=5NyeGzbBJQM
Derivatives - Horizontal Tangent Line:
https://www.youtube.com/watch?v=aNfoxbMUOHk
The Equation of The Normal Line:
https://www.youtube.com/watch?v=lEF2mmR3CWU
The Equation of The Secant Line:
https://www.youtube.com/watch?v=680o1GL-LRg
_________________________________
Average and Instantaneous Velocity:
https://www.youtube.com/watch?v=AwxT1xjMP9g
Instantaneous Rate of Change:
https://www.youtube.com/watch?v=dvQdQLTnDpk
Derivatives of Rational Functions:
https://www.youtube.com/watch?v=qknoFdPwEco
Derivatives of Radical Functions:
https://www.youtube.com/watch?v=B1YkgNDbx5o
Derivatives of Fractions:
https://www.youtube.com/watch?v=qrBkQ8Ci2fM
________________________________
Derivatives - Higher Order:
https://www.youtube.com/watch?v=s7rd9YPJrNc
Simplifying Derivatives:
https://www.youtube.com/watch?v=3lUOtjkqfQo
Derivatives - The Product Rule:
https://www.youtube.com/watch?v=17X5g9QArTc
Derivatives - The Quotient Rule:
https://www.youtube.com/watch?v=8jVDEcQ0wXk
Derivatives - The Chain Rule:
https://www.youtube.com/watch?v=HaHsqDjWMLU
_______________________________________
Final Exams and Video Playlists:
https://www.video-tutor.net/
Full-Length Videos and Worksheets:
https://www.patreon.com/MathScienceTutor/collections
Derivatives - Formula Sheet:
https://bit.ly/4dThzf1
In calculus, you’ll often hear “The derivative is the slope of the tangent line.” But what is a tangent line? The definition is trickier than you might think....
In calculus, you’ll often hear “The derivative is the slope of the tangent line.” But what is a tangent line? The definition is trickier than you might think.
Tangent lines are important because they are the best way to approximate a curve using a line. We can then use the slope of the line as a way to measure the “slope” of the curve.
Be sure to subscribe so you don't miss videos from Socratica!
http://bit.ly/1ixuu9W
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Ways to support our channel:
► Join our Patreon : https://www.patreon.com/socratica
► Make a one-time PayPal donation: https://www.paypal.me/socratica
► We also accept Bitcoin @ 1EttYyGwJmpy9bLY2UcmEqMJuBfaZ1HdG9
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Written and Produced by Michael Harrison
Michael Harrison received his BS in math from Caltech, and his MS from the University of Washington where he studied algebraic number theory. After teaching math for a few years, Michael worked in finance both as a developer and a quantitative analyst (quant). He then worked at Google as a software engineer for over 5 years before leaving to found Socratica.
You can follow Michael on Twitter: http://www.twitter.com/mlh496
♦♦♦♦♦♦♦♦♦♦
In calculus, you’ll often hear “The derivative is the slope of the tangent line.” But what is a tangent line? The definition is trickier than you might think.
Tangent lines are important because they are the best way to approximate a curve using a line. We can then use the slope of the line as a way to measure the “slope” of the curve.
Be sure to subscribe so you don't miss videos from Socratica!
http://bit.ly/1ixuu9W
♦♦♦♦♦♦♦♦♦♦
Ways to support our channel:
► Join our Patreon : https://www.patreon.com/socratica
► Make a one-time PayPal donation: https://www.paypal.me/socratica
► We also accept Bitcoin @ 1EttYyGwJmpy9bLY2UcmEqMJuBfaZ1HdG9
Thank you!
♦♦♦♦♦♦♦♦♦♦
Connect with us!
Facebook: https://www.facebook.com/SocraticaStudios/
Instagram: https://www.instagram.com/SocraticaStudios/
Twitter: https://twitter.com/Socratica
♦♦♦♦♦♦♦♦♦♦
Written and Produced by Michael Harrison
Michael Harrison received his BS in math from Caltech, and his MS from the University of Washington where he studied algebraic number theory. After teaching math for a few years, Michael worked in finance both as a developer and a quantitative analyst (quant). He then worked at Google as a software engineer for over 5 years before leaving to found Socratica.
You can follow Michael on Twitter: http://www.twitter.com/mlh496
♦♦♦♦♦♦♦♦♦♦
The first operation in calculus that we have to understand is differentiation. So what is it, exactly? Well there are a couple of ways of looking at it. The fir...
The first operation in calculus that we have to understand is differentiation. So what is it, exactly? Well there are a couple of ways of looking at it. The first one involves finding the equation of a tangent line for some point on a curve. We can't do this with algebra, because for that we would need two points on the line, and we only have one. Enter calculus!
Watch the whole Calculus playlist: http://bit.ly/ProfDaveCalculus
Watch the whole Mathematics playlist: http://bit.ly/ProfDaveMath
Classical Physics Tutorials: http://bit.ly/ProfDavePhysics1
Modern Physics Tutorials: http://bit.ly/ProfDavePhysics2
General Chemistry Tutorials: http://bit.ly/ProfDaveGenChem
Organic Chemistry Tutorials: http://bit.ly/ProfDaveOrgChem
Biochemistry Tutorials: http://bit.ly/ProfDaveBiochem
Biology Tutorials: http://bit.ly/ProfDaveBio
EMAIL► [email protected]
PATREON► http://patreon.com/ProfessorDaveExplains
Check out "Is This Wi-Fi Organic?", my book on disarming pseudoscience!
Amazon: https://amzn.to/2HtNpVH
Bookshop: https://bit.ly/39cKADM
Barnes and Noble: https://bit.ly/3pUjmrn
Book Depository: http://bit.ly/3aOVDlT
The first operation in calculus that we have to understand is differentiation. So what is it, exactly? Well there are a couple of ways of looking at it. The first one involves finding the equation of a tangent line for some point on a curve. We can't do this with algebra, because for that we would need two points on the line, and we only have one. Enter calculus!
Watch the whole Calculus playlist: http://bit.ly/ProfDaveCalculus
Watch the whole Mathematics playlist: http://bit.ly/ProfDaveMath
Classical Physics Tutorials: http://bit.ly/ProfDavePhysics1
Modern Physics Tutorials: http://bit.ly/ProfDavePhysics2
General Chemistry Tutorials: http://bit.ly/ProfDaveGenChem
Organic Chemistry Tutorials: http://bit.ly/ProfDaveOrgChem
Biochemistry Tutorials: http://bit.ly/ProfDaveBiochem
Biology Tutorials: http://bit.ly/ProfDaveBio
EMAIL► [email protected]
PATREON► http://patreon.com/ProfessorDaveExplains
Check out "Is This Wi-Fi Organic?", my book on disarming pseudoscience!
Amazon: https://amzn.to/2HtNpVH
Bookshop: https://bit.ly/39cKADM
Barnes and Noble: https://bit.ly/3pUjmrn
Book Depository: http://bit.ly/3aOVDlT
Visit http://ilectureonline.com for more math and science lectures!
In this video I will review the tangent and secant line with respect to a function.
Next v...
Visit http://ilectureonline.com for more math and science lectures!
In this video I will review the tangent and secant line with respect to a function.
Next video in the series can be seen at:
http://youtu.be/723jsxCO7Oc
Visit http://ilectureonline.com for more math and science lectures!
In this video I will review the tangent and secant line with respect to a function.
Next video in the series can be seen at:
http://youtu.be/723jsxCO7Oc
This calculus video shows you how to find the slope and the equation of the tangent line and normal line to the curve/function at a given point. This video con...
This calculus video shows you how to find the slope and the equation of the tangent line and normal line to the curve/function at a given point. This video contains plenty of example problems.
Derivatives - Fast Review: https://www.youtube.com/watch?v=5yfh5cf4-0w&t=1s
Equation of the Tangent Line:
https://www.youtube.com/watch?v=5NyeGzbBJQM
Derivatives - Horizontal Tangent Line:
https://www.youtube.com/watch?v=aNfoxbMUOHk
The Equation of The Normal Line:
https://www.youtube.com/watch?v=lEF2mmR3CWU
The Equation of The Secant Line:
https://www.youtube.com/watch?v=680o1GL-LRg
_________________________________
Average and Instantaneous Velocity:
https://www.youtube.com/watch?v=AwxT1xjMP9g
Instantaneous Rate of Change:
https://www.youtube.com/watch?v=dvQdQLTnDpk
Derivatives of Rational Functions:
https://www.youtube.com/watch?v=qknoFdPwEco
Derivatives of Radical Functions:
https://www.youtube.com/watch?v=B1YkgNDbx5o
Derivatives of Fractions:
https://www.youtube.com/watch?v=qrBkQ8Ci2fM
________________________________
Derivatives - Higher Order:
https://www.youtube.com/watch?v=s7rd9YPJrNc
Simplifying Derivatives:
https://www.youtube.com/watch?v=3lUOtjkqfQo
Derivatives - The Product Rule:
https://www.youtube.com/watch?v=17X5g9QArTc
Derivatives - The Quotient Rule:
https://www.youtube.com/watch?v=8jVDEcQ0wXk
Derivatives - The Chain Rule:
https://www.youtube.com/watch?v=HaHsqDjWMLU
_______________________________________
Final Exams and Video Playlists:
https://www.video-tutor.net/
Full-Length Videos and Worksheets:
https://www.patreon.com/MathScienceTutor/collections
This calculus video shows you how to find the slope and the equation of the tangent line and normal line to the curve/function at a given point. This video contains plenty of example problems.
Derivatives - Fast Review: https://www.youtube.com/watch?v=5yfh5cf4-0w&t=1s
Equation of the Tangent Line:
https://www.youtube.com/watch?v=5NyeGzbBJQM
Derivatives - Horizontal Tangent Line:
https://www.youtube.com/watch?v=aNfoxbMUOHk
The Equation of The Normal Line:
https://www.youtube.com/watch?v=lEF2mmR3CWU
The Equation of The Secant Line:
https://www.youtube.com/watch?v=680o1GL-LRg
_________________________________
Average and Instantaneous Velocity:
https://www.youtube.com/watch?v=AwxT1xjMP9g
Instantaneous Rate of Change:
https://www.youtube.com/watch?v=dvQdQLTnDpk
Derivatives of Rational Functions:
https://www.youtube.com/watch?v=qknoFdPwEco
Derivatives of Radical Functions:
https://www.youtube.com/watch?v=B1YkgNDbx5o
Derivatives of Fractions:
https://www.youtube.com/watch?v=qrBkQ8Ci2fM
________________________________
Derivatives - Higher Order:
https://www.youtube.com/watch?v=s7rd9YPJrNc
Simplifying Derivatives:
https://www.youtube.com/watch?v=3lUOtjkqfQo
Derivatives - The Product Rule:
https://www.youtube.com/watch?v=17X5g9QArTc
Derivatives - The Quotient Rule:
https://www.youtube.com/watch?v=8jVDEcQ0wXk
Derivatives - The Chain Rule:
https://www.youtube.com/watch?v=HaHsqDjWMLU
_______________________________________
Final Exams and Video Playlists:
https://www.video-tutor.net/
Full-Length Videos and Worksheets:
https://www.patreon.com/MathScienceTutor/collections
Let's explore, identify, and describe the relationship between a circle and a tangent line and why the radius of a circle is perpendicular to the tangent where ...
Let's explore, identify, and describe the relationship between a circle and a tangent line and why the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
★ Want more FREE weekly math videos? Subscribe Now: http://bit.ly/1OVJnyh
★★ Why Math? Download YOUR Free eBook and sign-up for our FREE weekly newsletter: http://www.mashupmath.com/resources/
Understand and apply theorems about circles.
Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
What is a tangent line to a circle?
Why is a tagent line and a radius perpendicular?
How many times does a tangent line cross a circle?
Image Credit: Emojis provided free by http://emojione.com
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Let's explore, identify, and describe the relationship between a circle and a tangent line and why the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
★ Want more FREE weekly math videos? Subscribe Now: http://bit.ly/1OVJnyh
★★ Why Math? Download YOUR Free eBook and sign-up for our FREE weekly newsletter: http://www.mashupmath.com/resources/
Understand and apply theorems about circles.
Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
What is a tangent line to a circle?
Why is a tagent line and a radius perpendicular?
How many times does a tangent line cross a circle?
Image Credit: Emojis provided free by http://emojione.com
✔✔✔SUBSCRIBE FOR FREE: https://www.youtube.com/channel/UCtBt...
✔✔JOIN OUR MAILING LIST: http://bit.ly/1TqV94V
✔FOLLOW US ON TWITTER: http://www.twitter.com/mashupmath
✔FOLLOW US ON INSTAGRAM: http://www.instagram.com/mashupmath
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This calculus video tutorial explains how to find the equation of the tangent line with derivatives. It explains how to write the equation of the tangent line ...
This calculus video tutorial explains how to find the equation of the tangent line with derivatives. It explains how to write the equation of the tangent line in point slope form and slope intercept form.
Derivatives - Fast Review: https://www.youtube.com/watch?v=5yfh5cf4-0w&t=1s
Equation of the Tangent Line:
https://www.youtube.com/watch?v=5NyeGzbBJQM
Derivatives - Horizontal Tangent Line:
https://www.youtube.com/watch?v=aNfoxbMUOHk
The Equation of The Normal Line:
https://www.youtube.com/watch?v=lEF2mmR3CWU
The Equation of The Secant Line:
https://www.youtube.com/watch?v=680o1GL-LRg
_________________________________
Average and Instantaneous Velocity:
https://www.youtube.com/watch?v=AwxT1xjMP9g
Instantaneous Rate of Change:
https://www.youtube.com/watch?v=dvQdQLTnDpk
Derivatives of Rational Functions:
https://www.youtube.com/watch?v=qknoFdPwEco
Derivatives of Radical Functions:
https://www.youtube.com/watch?v=B1YkgNDbx5o
Derivatives of Fractions:
https://www.youtube.com/watch?v=qrBkQ8Ci2fM
________________________________
Derivatives - Higher Order:
https://www.youtube.com/watch?v=s7rd9YPJrNc
Simplifying Derivatives:
https://www.youtube.com/watch?v=3lUOtjkqfQo
Derivatives - The Product Rule:
https://www.youtube.com/watch?v=17X5g9QArTc
Derivatives - The Quotient Rule:
https://www.youtube.com/watch?v=8jVDEcQ0wXk
Derivatives - The Chain Rule:
https://www.youtube.com/watch?v=HaHsqDjWMLU
_______________________________________
Final Exams and Video Playlists:
https://www.video-tutor.net/
Full-Length Videos and Worksheets:
https://www.patreon.com/MathScienceTutor/collections
This calculus video tutorial explains how to find the equation of the tangent line with derivatives. It explains how to write the equation of the tangent line in point slope form and slope intercept form.
Derivatives - Fast Review: https://www.youtube.com/watch?v=5yfh5cf4-0w&t=1s
Equation of the Tangent Line:
https://www.youtube.com/watch?v=5NyeGzbBJQM
Derivatives - Horizontal Tangent Line:
https://www.youtube.com/watch?v=aNfoxbMUOHk
The Equation of The Normal Line:
https://www.youtube.com/watch?v=lEF2mmR3CWU
The Equation of The Secant Line:
https://www.youtube.com/watch?v=680o1GL-LRg
_________________________________
Average and Instantaneous Velocity:
https://www.youtube.com/watch?v=AwxT1xjMP9g
Instantaneous Rate of Change:
https://www.youtube.com/watch?v=dvQdQLTnDpk
Derivatives of Rational Functions:
https://www.youtube.com/watch?v=qknoFdPwEco
Derivatives of Radical Functions:
https://www.youtube.com/watch?v=B1YkgNDbx5o
Derivatives of Fractions:
https://www.youtube.com/watch?v=qrBkQ8Ci2fM
________________________________
Derivatives - Higher Order:
https://www.youtube.com/watch?v=s7rd9YPJrNc
Simplifying Derivatives:
https://www.youtube.com/watch?v=3lUOtjkqfQo
Derivatives - The Product Rule:
https://www.youtube.com/watch?v=17X5g9QArTc
Derivatives - The Quotient Rule:
https://www.youtube.com/watch?v=8jVDEcQ0wXk
Derivatives - The Chain Rule:
https://www.youtube.com/watch?v=HaHsqDjWMLU
_______________________________________
Final Exams and Video Playlists:
https://www.video-tutor.net/
Full-Length Videos and Worksheets:
https://www.patreon.com/MathScienceTutor/collections
MIT grad shows how to find the tangent line equation using a derivative (Calculus). To skip ahead: 1) For a BASIC example, skip to time 0:44. 2) For an example ...
MIT grad shows how to find the tangent line equation using a derivative (Calculus). To skip ahead: 1) For a BASIC example, skip to time 0:44. 2) For an example that uses the QUOTIENT RULE, skip to time 5:58. 3) For an example that uses TRIG FUNCTIONS and the CHAIN RULE for finding the equation of a tangent line, skip to time 11:35. Nancy formerly of MathBFF explains the steps.
For HOW TO FIND DERIVATIVES, jump to: https://youtu.be/QqF3i1pnyzU
Follow me on Instagram! https://instagram.com/nancypi
Follow me on Twitter: https://twitter.com/nancypi
What is a tangent line? It is a line that is tangent to a curve at one point. You can use calculus for finding the tangent line equation, by taking the derivative of the given function.
Finding the EQUATION OF A TANGENT LINE using a DERIVATIVE:
There are four main steps to find the equation of a tangent line with derivatives:
1) TAKE DERIVATIVE: The first step is to take the derivative of the given equation, with respect to x. For instance, to find the equation of the line tangent to f(x) = x^2 + 3 at x = 4, first take the derivative of f(x), which is 2x, by the power rule.
2) PLUG IN X-VALUE INTO DERIVATIVE TO GET SLOPE: The second step is to plug the given x-value into the derivative of f(x). The value you get is the slope of the tangent line, m.
3) FIND Y-VALUE OF POINT WITH ORIGINAL EQUATION: The third step is to find the y-value of the point. You get this by plugging the given x-value into the original equation to find the corresponding y-value. Since you want the tangent line at x = 4, plug x = 4 into the original f(x), and you get a y-value of 7. Now that you have both the x and y coordinates, this means that the full point is (4,7).
4) PLUG THE SLOPE AND X,Y POINT VALUES INTO POINT-SLOPE EQUATION: The last step is to put the x and y point values, as well as the slope, m, into the point slope equation of a line. The point slope form is x - x1 = m (y - y1). Plug the x and y point values into this equation for x1 and y1. Plug the slope you found in for m. This is the equation of the tangent line. The equation will still have x and y variables in it.
You can also rearrange the equation you get so that it is in "y equals" or slope intercept form, if you want.
You may need to find the derivative with other derivative rules, such as the quotient rule or chain rule. Even if you need to use other rules when taking derivatives, or even if you need to differentiate something like a trigonometry function (ex. secant) or a fraction form with a numerator and denominator (rational function), the four main steps to find the equation of the tangent line will be the same.
For more math help, check out: http://nancypi.com
MIT grad shows how to find the tangent line equation using a derivative (Calculus). To skip ahead: 1) For a BASIC example, skip to time 0:44. 2) For an example that uses the QUOTIENT RULE, skip to time 5:58. 3) For an example that uses TRIG FUNCTIONS and the CHAIN RULE for finding the equation of a tangent line, skip to time 11:35. Nancy formerly of MathBFF explains the steps.
For HOW TO FIND DERIVATIVES, jump to: https://youtu.be/QqF3i1pnyzU
Follow me on Instagram! https://instagram.com/nancypi
Follow me on Twitter: https://twitter.com/nancypi
What is a tangent line? It is a line that is tangent to a curve at one point. You can use calculus for finding the tangent line equation, by taking the derivative of the given function.
Finding the EQUATION OF A TANGENT LINE using a DERIVATIVE:
There are four main steps to find the equation of a tangent line with derivatives:
1) TAKE DERIVATIVE: The first step is to take the derivative of the given equation, with respect to x. For instance, to find the equation of the line tangent to f(x) = x^2 + 3 at x = 4, first take the derivative of f(x), which is 2x, by the power rule.
2) PLUG IN X-VALUE INTO DERIVATIVE TO GET SLOPE: The second step is to plug the given x-value into the derivative of f(x). The value you get is the slope of the tangent line, m.
3) FIND Y-VALUE OF POINT WITH ORIGINAL EQUATION: The third step is to find the y-value of the point. You get this by plugging the given x-value into the original equation to find the corresponding y-value. Since you want the tangent line at x = 4, plug x = 4 into the original f(x), and you get a y-value of 7. Now that you have both the x and y coordinates, this means that the full point is (4,7).
4) PLUG THE SLOPE AND X,Y POINT VALUES INTO POINT-SLOPE EQUATION: The last step is to put the x and y point values, as well as the slope, m, into the point slope equation of a line. The point slope form is x - x1 = m (y - y1). Plug the x and y point values into this equation for x1 and y1. Plug the slope you found in for m. This is the equation of the tangent line. The equation will still have x and y variables in it.
You can also rearrange the equation you get so that it is in "y equals" or slope intercept form, if you want.
You may need to find the derivative with other derivative rules, such as the quotient rule or chain rule. Even if you need to use other rules when taking derivatives, or even if you need to differentiate something like a trigonometry function (ex. secant) or a fraction form with a numerator and denominator (rational function), the four main steps to find the equation of the tangent line will be the same.
For more math help, check out: http://nancypi.com
Apollonius proved that a tangent line never crosses a curve at the point of tangency. Proposition. (I. 17, 32)
The great geometer Apollonius fortunately proved...
Apollonius proved that a tangent line never crosses a curve at the point of tangency. Proposition. (I. 17, 32)
The great geometer Apollonius fortunately proved this is true for any conic and hence ANY other curve by implication. The idiots of mainstream mathematics academia think they know better and have arrogantly "dismissed" Apollonius whose genius far surpasses any of the morons who came after Euclid. Just imagine! Mainstream idiots followed the ideas of that stupid bum Gottfried Leibniz who claimed a tangent line passes through two infinitely close points and therefore can cross at the point of tangency.
Find the article used in the presentation here:
https://www.academia.edu/121458742/The_concept_of_tangent_line_was_defined_in_the_Conics_of_Apollonius_Proposition_11_I_17_32_
Learn more about ratios here:
https://www.academia.edu/68520938/Book_5_of_the_Elements
A great math channel for kids!
https://www.youtube.com/@The_Math_Path
Download the free eBook on the Single Variable New Calculus:
https://www.academia.edu/41616655/An_Introduction_to_the_Single_Variable_New_Calculus
Learn about the Holy Grail of Calculus:
https://www.academia.edu/105576431/The_Holy_Grail_of_Calculus
Learn about the Gabriel Polynomial (GP) which is a very advanced concept not possible using mainstream derivative or integral definitions. The GP is always a closed form polynomial and has no Error term:
https://www.academia.edu/45229087/The_Gabriel_Polynomial
Download the best book ever written on the number concept:
https://www.academia.edu/105399167/The_Ultimate_Book_of_Numbers
Get the true Origins and History of Calculus:
https://www.academia.edu/106488069/The_Non_fictional_Origins_and_History_of_Calculus
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Apollonius proved that a tangent line never crosses a curve at the point of tangency. Proposition. (I. 17, 32)
The great geometer Apollonius fortunately proved this is true for any conic and hence ANY other curve by implication. The idiots of mainstream mathematics academia think they know better and have arrogantly "dismissed" Apollonius whose genius far surpasses any of the morons who came after Euclid. Just imagine! Mainstream idiots followed the ideas of that stupid bum Gottfried Leibniz who claimed a tangent line passes through two infinitely close points and therefore can cross at the point of tangency.
Find the article used in the presentation here:
https://www.academia.edu/121458742/The_concept_of_tangent_line_was_defined_in_the_Conics_of_Apollonius_Proposition_11_I_17_32_
Learn more about ratios here:
https://www.academia.edu/68520938/Book_5_of_the_Elements
A great math channel for kids!
https://www.youtube.com/@The_Math_Path
Download the free eBook on the Single Variable New Calculus:
https://www.academia.edu/41616655/An_Introduction_to_the_Single_Variable_New_Calculus
Learn about the Holy Grail of Calculus:
https://www.academia.edu/105576431/The_Holy_Grail_of_Calculus
Learn about the Gabriel Polynomial (GP) which is a very advanced concept not possible using mainstream derivative or integral definitions. The GP is always a closed form polynomial and has no Error term:
https://www.academia.edu/45229087/The_Gabriel_Polynomial
Download the best book ever written on the number concept:
https://www.academia.edu/105399167/The_Ultimate_Book_of_Numbers
Get the true Origins and History of Calculus:
https://www.academia.edu/106488069/The_Non_fictional_Origins_and_History_of_Calculus
Discover closed -form trigonometric functions never realised before:
https://www.academia.edu/109334669/Ancient_Greek_trigonometric_formulas_better_than_anything_ever_known
Follow me on Academia.edu:
https://independent.academia.edu/JohnGabriel30
Donate here:
https://gofund.me/af8a5312
or here:
https://wise.com/pay/me/johng3933
All my YouTube videos are backed up here:
https://odysee.com/@NewCalculus:1
The scum of YouTube are constantly threatening and looking for excuses to shut down my channel.
Want to get instant updates for the newest math around? Join our discord server! https://discord.gg/CJ9Ks3WerR
Merchandise Store:
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This calculus video tutorial shows you how to find the equation of a tangent line with derivatives. Techniques include the power rule, product rule, and implicit differentiation. This video shows you how to write the equation of the tangent line in point slope form and slope intercept form. This video contains plenty of examples and practice problems.
Application of Derivatives - Free Formula Sheet:
https://bit.ly/4eV6r1b
_____________________________
Derivatives - Fast Review:
https://www.youtube.com/watch?v=5yfh5cf4-0w&t=1s
Equation of the Tangent Line:
https://www.youtube.com/watch?v=5NyeGzbBJQM
Derivatives - Horizontal Tangent Line:
https://www.youtube.com/watch?v=aNfoxbMUOHk
The Equation of The Normal Line:
https://www.youtube.com/watch?v=lEF2mmR3CWU
The Equation of The Secant Line:
https://www.youtube.com/watch?v=680o1GL-LRg
_________________________________
Average and Instantaneous Velocity:
https://www.youtube.com/watch?v=AwxT1xjMP9g
Instantaneous Rate of Change:
https://www.youtube.com/watch?v=dvQdQLTnDpk
Derivatives of Rational Functions:
https://www.youtube.com/watch?v=qknoFdPwEco
Derivatives of Radical Functions:
https://www.youtube.com/watch?v=B1YkgNDbx5o
Derivatives of Fractions:
https://www.youtube.com/watch?v=qrBkQ8Ci2fM
________________________________
Derivatives - Higher Order:
https://www.youtube.com/watch?v=s7rd9YPJrNc
Simplifying Derivatives:
https://www.youtube.com/watch?v=3lUOtjkqfQo
Derivatives - The Product Rule:
https://www.youtube.com/watch?v=17X5g9QArTc
Derivatives - The Quotient Rule:
https://www.youtube.com/watch?v=8jVDEcQ0wXk
Derivatives - The Chain Rule:
https://www.youtube.com/watch?v=HaHsqDjWMLU
_______________________________________
Final Exams and Video Playlists:
https://www.video-tutor.net/
Full-Length Videos and Worksheets:
https://www.patreon.com/MathScienceTutor/collections
Derivatives - Formula Sheet:
https://bit.ly/4dThzf1
In calculus, you’ll often hear “The derivative is the slope of the tangent line.” But what is a tangent line? The definition is trickier than you might think.
Tangent lines are important because they are the best way to approximate a curve using a line. We can then use the slope of the line as a way to measure the “slope” of the curve.
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Written and Produced by Michael Harrison
Michael Harrison received his BS in math from Caltech, and his MS from the University of Washington where he studied algebraic number theory. After teaching math for a few years, Michael worked in finance both as a developer and a quantitative analyst (quant). He then worked at Google as a software engineer for over 5 years before leaving to found Socratica.
You can follow Michael on Twitter: http://www.twitter.com/mlh496
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The first operation in calculus that we have to understand is differentiation. So what is it, exactly? Well there are a couple of ways of looking at it. The first one involves finding the equation of a tangent line for some point on a curve. We can't do this with algebra, because for that we would need two points on the line, and we only have one. Enter calculus!
Watch the whole Calculus playlist: http://bit.ly/ProfDaveCalculus
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Visit http://ilectureonline.com for more math and science lectures!
In this video I will review the tangent and secant line with respect to a function.
Next video in the series can be seen at:
http://youtu.be/723jsxCO7Oc
This calculus video shows you how to find the slope and the equation of the tangent line and normal line to the curve/function at a given point. This video contains plenty of example problems.
Derivatives - Fast Review: https://www.youtube.com/watch?v=5yfh5cf4-0w&t=1s
Equation of the Tangent Line:
https://www.youtube.com/watch?v=5NyeGzbBJQM
Derivatives - Horizontal Tangent Line:
https://www.youtube.com/watch?v=aNfoxbMUOHk
The Equation of The Normal Line:
https://www.youtube.com/watch?v=lEF2mmR3CWU
The Equation of The Secant Line:
https://www.youtube.com/watch?v=680o1GL-LRg
_________________________________
Average and Instantaneous Velocity:
https://www.youtube.com/watch?v=AwxT1xjMP9g
Instantaneous Rate of Change:
https://www.youtube.com/watch?v=dvQdQLTnDpk
Derivatives of Rational Functions:
https://www.youtube.com/watch?v=qknoFdPwEco
Derivatives of Radical Functions:
https://www.youtube.com/watch?v=B1YkgNDbx5o
Derivatives of Fractions:
https://www.youtube.com/watch?v=qrBkQ8Ci2fM
________________________________
Derivatives - Higher Order:
https://www.youtube.com/watch?v=s7rd9YPJrNc
Simplifying Derivatives:
https://www.youtube.com/watch?v=3lUOtjkqfQo
Derivatives - The Product Rule:
https://www.youtube.com/watch?v=17X5g9QArTc
Derivatives - The Quotient Rule:
https://www.youtube.com/watch?v=8jVDEcQ0wXk
Derivatives - The Chain Rule:
https://www.youtube.com/watch?v=HaHsqDjWMLU
_______________________________________
Final Exams and Video Playlists:
https://www.video-tutor.net/
Full-Length Videos and Worksheets:
https://www.patreon.com/MathScienceTutor/collections
Let's explore, identify, and describe the relationship between a circle and a tangent line and why the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
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Understand and apply theorems about circles.
Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
What is a tangent line to a circle?
Why is a tagent line and a radius perpendicular?
How many times does a tangent line cross a circle?
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This calculus video tutorial explains how to find the equation of the tangent line with derivatives. It explains how to write the equation of the tangent line in point slope form and slope intercept form.
Derivatives - Fast Review: https://www.youtube.com/watch?v=5yfh5cf4-0w&t=1s
Equation of the Tangent Line:
https://www.youtube.com/watch?v=5NyeGzbBJQM
Derivatives - Horizontal Tangent Line:
https://www.youtube.com/watch?v=aNfoxbMUOHk
The Equation of The Normal Line:
https://www.youtube.com/watch?v=lEF2mmR3CWU
The Equation of The Secant Line:
https://www.youtube.com/watch?v=680o1GL-LRg
_________________________________
Average and Instantaneous Velocity:
https://www.youtube.com/watch?v=AwxT1xjMP9g
Instantaneous Rate of Change:
https://www.youtube.com/watch?v=dvQdQLTnDpk
Derivatives of Rational Functions:
https://www.youtube.com/watch?v=qknoFdPwEco
Derivatives of Radical Functions:
https://www.youtube.com/watch?v=B1YkgNDbx5o
Derivatives of Fractions:
https://www.youtube.com/watch?v=qrBkQ8Ci2fM
________________________________
Derivatives - Higher Order:
https://www.youtube.com/watch?v=s7rd9YPJrNc
Simplifying Derivatives:
https://www.youtube.com/watch?v=3lUOtjkqfQo
Derivatives - The Product Rule:
https://www.youtube.com/watch?v=17X5g9QArTc
Derivatives - The Quotient Rule:
https://www.youtube.com/watch?v=8jVDEcQ0wXk
Derivatives - The Chain Rule:
https://www.youtube.com/watch?v=HaHsqDjWMLU
_______________________________________
Final Exams and Video Playlists:
https://www.video-tutor.net/
Full-Length Videos and Worksheets:
https://www.patreon.com/MathScienceTutor/collections
MIT grad shows how to find the tangent line equation using a derivative (Calculus). To skip ahead: 1) For a BASIC example, skip to time 0:44. 2) For an example that uses the QUOTIENT RULE, skip to time 5:58. 3) For an example that uses TRIG FUNCTIONS and the CHAIN RULE for finding the equation of a tangent line, skip to time 11:35. Nancy formerly of MathBFF explains the steps.
For HOW TO FIND DERIVATIVES, jump to: https://youtu.be/QqF3i1pnyzU
Follow me on Instagram! https://instagram.com/nancypi
Follow me on Twitter: https://twitter.com/nancypi
What is a tangent line? It is a line that is tangent to a curve at one point. You can use calculus for finding the tangent line equation, by taking the derivative of the given function.
Finding the EQUATION OF A TANGENT LINE using a DERIVATIVE:
There are four main steps to find the equation of a tangent line with derivatives:
1) TAKE DERIVATIVE: The first step is to take the derivative of the given equation, with respect to x. For instance, to find the equation of the line tangent to f(x) = x^2 + 3 at x = 4, first take the derivative of f(x), which is 2x, by the power rule.
2) PLUG IN X-VALUE INTO DERIVATIVE TO GET SLOPE: The second step is to plug the given x-value into the derivative of f(x). The value you get is the slope of the tangent line, m.
3) FIND Y-VALUE OF POINT WITH ORIGINAL EQUATION: The third step is to find the y-value of the point. You get this by plugging the given x-value into the original equation to find the corresponding y-value. Since you want the tangent line at x = 4, plug x = 4 into the original f(x), and you get a y-value of 7. Now that you have both the x and y coordinates, this means that the full point is (4,7).
4) PLUG THE SLOPE AND X,Y POINT VALUES INTO POINT-SLOPE EQUATION: The last step is to put the x and y point values, as well as the slope, m, into the point slope equation of a line. The point slope form is x - x1 = m (y - y1). Plug the x and y point values into this equation for x1 and y1. Plug the slope you found in for m. This is the equation of the tangent line. The equation will still have x and y variables in it.
You can also rearrange the equation you get so that it is in "y equals" or slope intercept form, if you want.
You may need to find the derivative with other derivative rules, such as the quotient rule or chain rule. Even if you need to use other rules when taking derivatives, or even if you need to differentiate something like a trigonometry function (ex. secant) or a fraction form with a numerator and denominator (rational function), the four main steps to find the equation of the tangent line will be the same.
For more math help, check out: http://nancypi.com
Apollonius proved that a tangent line never crosses a curve at the point of tangency. Proposition. (I. 17, 32)
The great geometer Apollonius fortunately proved this is true for any conic and hence ANY other curve by implication. The idiots of mainstream mathematics academia think they know better and have arrogantly "dismissed" Apollonius whose genius far surpasses any of the morons who came after Euclid. Just imagine! Mainstream idiots followed the ideas of that stupid bum Gottfried Leibniz who claimed a tangent line passes through two infinitely close points and therefore can cross at the point of tangency.
Find the article used in the presentation here:
https://www.academia.edu/121458742/The_concept_of_tangent_line_was_defined_in_the_Conics_of_Apollonius_Proposition_11_I_17_32_
Learn more about ratios here:
https://www.academia.edu/68520938/Book_5_of_the_Elements
A great math channel for kids!
https://www.youtube.com/@The_Math_Path
Download the free eBook on the Single Variable New Calculus:
https://www.academia.edu/41616655/An_Introduction_to_the_Single_Variable_New_Calculus
Learn about the Holy Grail of Calculus:
https://www.academia.edu/105576431/The_Holy_Grail_of_Calculus
Learn about the Gabriel Polynomial (GP) which is a very advanced concept not possible using mainstream derivative or integral definitions. The GP is always a closed form polynomial and has no Error term:
https://www.academia.edu/45229087/The_Gabriel_Polynomial
Download the best book ever written on the number concept:
https://www.academia.edu/105399167/The_Ultimate_Book_of_Numbers
Get the true Origins and History of Calculus:
https://www.academia.edu/106488069/The_Non_fictional_Origins_and_History_of_Calculus
Discover closed -form trigonometric functions never realised before:
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In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve y = f (x) at a point x = c on the curve if the line passes through the point (c, f (c)) on the curve and has slope f'(c) where f' is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space.
As it passes through the point where the tangent line and the curve meet, called the point of tangency, the tangent line is "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point.
Similarly, the tangent plane to a surface at a given point is the plane that "just touches" the surface at that point. The concept of a tangent is one of the most fundamental notions in differential geometry and has been extensively generalized; see Tangent space.
1 Oregon. Longo’s quote that now appears like the beginning of the end of his stint in Madison came as a tangent to an answer about play-calling when the offensive line is controlling the game. People are also reading… ... .
One that comes to my mind is, “Watch out for tangents,” from artist Robert Hunt. It’s about keeping your design lines clean and uncluttered. In life, and art, tangents can distract, so it’s helped me keep things focused.
Apparently they’ve double-checked the math and all the cosines line up properly with the hectares and tangents and whatnot and, still, the algorithmics and hypotenuses all add up slick as a whistle, right down to the nth isosceles or some such ...Chaos.
And Trump repeated his line that Harris is a “low IQ individual”, followed by an incoherent tangent seemingly imagining her struggling to sleep ...Elizabeth Slaby, 81, was the first in line at Harris’s Allentown rally, arriving at about 6am ... .
Trump's accusation may be why Harris spokesperson James Singer decided Thursday to share the following line from CNN's report on Fox News. "Fox’s edits omitted numerous Trump tangents and exaggerations.".
his films are lush and transgressive, but in interviews, he's just as likely to talk about that as to go on a tangent about the beauty of his lead actor's ears. Nevertheless, the filmmaker knows when to draw the line.
this W is for ‘weird’) ... If we start calling them all out on it, the currency’s going to devalue fast ... ‘And your point is?’ But of course, windy Donald will go off on a tangled tangent. Maybe he’ll reprise his baffling ‘late, great Hannibal Lecter’ line.
Great-grandmothers. We all have them ...FaithAvery ... The digging ... Because every pile of documents contains multiple stories, the key to a coherent narrative is locating a through line that addresses the biggest conundrums while identifying the tangents ... 1.
Let X be the point on line BC different from C such that the line through X parallel to AC is tangent to ω . Similarly, let Y be the point on line BC different from B such that the line through Y parallel to AB is tangent to ω ... ....
READ MORE ... She raises the question ... The Atlantic writer concludes, "Biden’s departure allows the Democrats to turn their opponents’ best attack line back on them. Maybe old men whose sentences go off on weird tangents shouldn’t run for president?" ... .
Biden’s departure allows the Democrats to turn their opponents’ best attack line back on them ... Now the defining clip so far of their potential VP is a hack line about cat ladies that would have sounded sexist in 1974? Ouch.
... the United States.LINE (9A. Word before "dancing" or "drawing") Sometimes I go on silly tangents, like when I read this clue and thought, "What about doing a LINE drawing of LINE dancing?"MENSA (13A.
... above the water line ... In an interesting tangent, the authors found that rather than rising forever, the S-type proportion rises and falls in line with cycles of supercontinent formation and collapse.