How do you find the points where the graph of the function #y = x - (x / 18)^2# has horizontal tangents and what is the equation?

Answer 1

#y=81# at #(162,81)#

First, we can simplify the function.

#y=x-x^2/324#

We can find the derivative of this through the power rule:

#y'=1-x/162#
Horizontal tangents will occur when #y'=0#.
#1-x/162=0#
#x/162=1#
#x=162#
We can find the equation of the tangent line by plugging in #162# to the original equation.
#y=162-(162/18)^2#
#y=162-81#
#y=81#

Graphed are the tangent line and original function:

graph{(x-(x/18)^2-y)(y-81+0x)=0 [-119.8, 488.8, -146.2, 158]}

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Answer 2

To find the points where the graph of the function y = x - (x / 18)^2 has horizontal tangents, we need to find the values of x where the derivative of the function is equal to zero.

First, we find the derivative of the function y = x - (x / 18)^2 with respect to x.

The derivative is given by: dy/dx = 1 - (2x / 18^2)

Next, we set the derivative equal to zero and solve for x:

1 - (2x / 18^2) = 0

Simplifying the equation, we have:

1 - (2x / 324) = 0

Multiplying both sides by 324, we get:

324 - 2x = 0

Solving for x, we find:

x = 162

Therefore, the point where the graph of the function has a horizontal tangent is (162, y), where y is the corresponding value of y when x = 162.

The equation of the horizontal tangent line is y = x - (x / 18)^2, evaluated at x = 162.

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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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