How do you find the points where the graph of #y^2+2y=4x^3−16x−1# is the tangent line vertical?

Answer 1

The coordinates of the points that have vertical tangents are at #(-2, -1)#, #(0, -1)#, and #(2, -1)#.

If

#y^2+2y=4x^3-16x-1#

Then by implicit differentiation we have

#2y(dy)/(dx)+2(dy)/(dx)=12x^2-16#.

Now we can solve for (dy)/(dx).

#(dy)/(dx)=(12x^2-16)/(2y+2)#

The tangent line will be vertical when #(dy)/(dx)# becomes infinite. This will happen when

#2y+2=0#.

So the tangent line will be vertical when #y=-1#.

Now we need to calculate the #x#-coordinate(s) when #y=-1#. Our equation becomes

#(-1)^2+2(-1)=-1=4x^3-16x-1#.

#4x^3-16x=0#

#4x(x^2-4)=4x(x-2)(x+2)=0#

This equation has three solutions. They are #x=-2#, #x=0#, and #x=2#. We note that all of these values makes #12x^2-16# (the numerator of our expression for the derivative) finite so all of these #x#-values have vertical tangents. So the coordinates of the points that have vertical tangents are at #(-2, -1)#, #(0, -1)#, and #(2, -1)#.

Of course, looking at an actual plot of our relation (this is NOT a function) confirms that we have the correct answer.

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Answer 2

To find the points where the graph of the equation y^2 + 2y = 4x^3 - 16x - 1 is tangent to the vertical line, we need to determine the values of x that make the derivative of the equation equal to infinity.

First, we differentiate the equation with respect to x:

d/dx (y^2 + 2y) = d/dx (4x^3 - 16x - 1)

2y * dy/dx + 2 * dy/dx = 12x^2 - 16

Next, we solve for dy/dx:

dy/dx = (12x^2 - 16) / (2y + 2)

To find the points where the graph is tangent to the vertical line, we set dy/dx equal to infinity:

(12x^2 - 16) / (2y + 2) = ∞

Since infinity is not a real number, we can conclude that there are no points where the graph is tangent to the vertical line.

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Answer from HIX Tutor

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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