Question
upstudy study bank question image url

12. [-/1 Points] DETAILS MY NOTES SCALCET9 3.5.032. Use implicit differentiation to find an equation of the tangent line to the curve at the given point. \[ x^{2}+6 x y+12 y^{2}=28, \quad(2,1) \quad \text { (ellipse) } \] Sten

Ask by Lowe Ortiz. in the United States
Mar 18,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The equation of the tangent line at the point \((2, 1)\) is: \[ y = -\frac{5}{18}x + \frac{14}{9} \]

Solution

Sign in to Unlock Answers for Free!

A Learning Platform Trusted by Millions of Real Students and Teachers.

star-icon Unlock

Answered by UpStudy AI and reviewed by a Professional Tutor

Bonus Knowledge

To tackle this problem, let's first differentiate both sides of the equation \( x^{2}+6xy+12y^{2}=28 \) implicitly with respect to \( x \). Doing so gives us \( 2x + 6(y + x \frac{dy}{dx}) + 24y\frac{dy}{dx} = 0 \). Now, we can isolate \( \frac{dy}{dx} \) by solving for it and substituting the coordinates of the point (2, 1). Once evaluated, you’ll find that the slope \( \frac{dy}{dx} \) at the point (2, 1) is equal to a specific number, say \( m \). Using the point-slope form of the linear equation, you can easily derive the equation of the tangent line. For a deeper understanding of implicit differentiation, consider exploring classic instances from calculus - like studying the curves defined by equations you wouldn’t typically solve for \( y \) directly. Implicit differentiation is like a magical key that opens up the secrets of these curves without ever having to rearrange them into explicit forms! Moreover, exploring real-world applications of implicit differentiation can be incredibly fascinating. This technique has critical roles in fields such as physics and economics, where relationships between variables can't always be defined explicitly. For instance, calculating the slope of curves describing supply and demand in economics often requires the careful application of implicit differentiation to understand shifts and changes—helping make sense of real-life market behaviors!

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy