Finding the tangent line to $y= 7 \sin( \pi x+y)$ at the point $(-1,0)$

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Hint: Use Implicit Differentiation. We get $$\frac{dy}{dx}=7\left(\pi+\frac{dy}{dx}\right)\cos(\pi x+y).$$

Added: Now we could solve for $\frac{dy}{dx}$ and substitute. I prefer to substitute and then solve. So let $m$ be the value of $\frac{dy}{dx}$ at $(-1,0)$. Substituting, we obtain $$m=7(\pi +m)(-1).$$ Solve for $m$. We get $m=-\frac{7\pi}{8}$.

This $m$ is the slope of our tangent line. Now I expect you can find the equation of the line that passes through $(-1,0)$ and has slope $-\frac{7\pi}{8}$.

For the equation of the normal, note that the slope of the normal is $\frac{8}{7\pi}$.

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Updated on July 29, 2022

Comments

  • neloy
    neloy less than a minute

    Find the lines that are tangent and normal to the curve at the given point $$y= 7 \sin( \pi x+y), \qquad (-1,0)$$ The line tangent to the curve $y= 7 \sin (\pi x+y)$ at $(-1,0)$ is $y=\ ?$

    How would I solve this?

    • Admin
      Admin over 8 years
      Do you know the definition of the tangent line, and its relationship with the derivative at the point?
    • TMM
      TMM over 8 years
      Note that the constant $\pi$ is not written as "pie" but as "pi."
  • neloy
    neloy over 8 years
    that seems pretty different from where i ended up halfway...
  • neloy
    neloy over 8 years
    i got dy/dx= 7(pie) cos( (pie)x + y) + 3 cos( (pie)x+y) dy/dx
  • André Nicolas
    André Nicolas over 8 years
    Well, continue. Set $y=0$, $x=1$, and use the fact that $\cos(-\pi)=-1$. So at $(-1,0)$ the derivative (slope) $m$ satisfies $m=7(\pi +m)(-1)$. Solve for $m$.
  • neloy
    neloy over 8 years
    after that i don't know what to do. in the examples it says, collect the terms with dy/dx on one side of the equation , and then solve for dy/dx. This is where i get super confused. Then it says evaluate the derivative at the point (-1,0) to find the slope of the tangent. Then I'm supposed to substitute the values of the slope and the given point to find the y-intercept. This is so hard :'(
  • André Nicolas
    André Nicolas over 8 years
    I will add some more to the post.
  • neloy
    neloy over 8 years
    ok, i think i know what to do now. gonna find y-intercept. ok, put it in point slope formula it is
  • neloy
    neloy over 8 years
    y=8/7pi + 8/7pi. ty so much for the help.
  • neloy
    neloy over 8 years
    darn i was wrong…….noooooooooooooo