How do you find the tangent line to #x=cos(y/4)# at #y=π#?
We have:
Power rule:
Chain rule:
That is the answer!
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To find the tangent line to (x = \cos\left(\frac{y}{4}\right)) at (y = \pi), we first need to find the derivative of (x) with respect to (y), which will give us the slope of the tangent line. Then, we can evaluate this derivative at (y = \pi) to find the slope of the tangent line at that point. Finally, we can use the point-slope form of a line to write the equation of the tangent line.
First, we differentiate (x = \cos\left(\frac{y}{4}\right)) with respect to (y) using the chain rule:
[\frac{dx}{dy} = -\frac{1}{4}\sin\left(\frac{y}{4}\right)]
Now, we evaluate this derivative at (y = \pi):
[\frac{dx}{dy} \bigg|_{y=\pi} = -\frac{1}{4}\sin\left(\frac{\pi}{4}\right) = -\frac{1}{4}]
So, the slope of the tangent line at (y = \pi) is (-\frac{1}{4}).
Now, to find the equation of the tangent line, we use the point-slope form of a line:
[y - y_1 = m(x - x_1)]
Where (m) is the slope and ((x_1, y_1)) is the point on the curve.
At (y = \pi), (x = \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}).
Substituting (m = -\frac{1}{4}), (x_1 = \frac{\sqrt{2}}{2}), and (y_1 = \pi) into the point-slope form, we get:
[y - \pi = -\frac{1}{4}\left(x - \frac{\sqrt{2}}{2}\right)]
This is the equation of the tangent line to (x = \cos\left(\frac{y}{4}\right)) at (y = \pi).
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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