# What is the shortest distance from the point #(-3,3)# to the curve #y=(x-3)^3#?

Please see below.

By reducing the radicand, we can reduce the distance:

Distinguish:

Utilize technology or an approximation technique to obtain

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Determine the curve's first derivative:

The normal line's slope is:

At the point of nomalcy, the y coordinate, y_1, is:

The equation of the normal line can be found using the point-slope form of the equation of a line:

This fifth order equation was solved by me using WolframAlpha:

The associated y-coordinate is:

Apply the formula for distance:

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To find the shortest distance from the point (-3,3) to the curve y = (x - 3)^3, we can use the distance formula. The distance between a point (x1, y1) and a point (x2, y2) is given by the formula:

Distance = √[(x2 - x1)^2 + (y2 - y1)^2]

Substituting (-3, 3) for (x1, y1) and (x, (x - 3)^3) for (x2, y2), we get:

Distance = √[(x - (-3))^2 + ((x - 3)^3 - 3)^2]

Now, to minimize the distance, we differentiate the distance formula with respect to x and set it equal to zero to find the critical points. After finding the critical points, we can determine which one yields the minimum distance.

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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.

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