# How do you find the point on the line #y=4x + 7# that is closest to the point (0,-3)?

The distance between and arbitrary point

Minimizing the squared distance will occur at the same value of

That this value of

When

Here's a picture for this situation. The line segment intersects the given line at a right angle.

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To find the point on the line ( y = 4x + 7 ) that is closest to the point ( (0, -3) ), you can follow these steps:

- Calculate the perpendicular distance from the given point ( (0, -3) ) to the line ( y = 4x + 7 ).
- Use the formula for the perpendicular distance from a point ( (x_1, y_1) ) to a line ( Ax + By + C = 0 ): [ \text{Distance} = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} ]
- Substitute the values from the equation of the line and the coordinates of the point ( (0, -3) ) into the formula.
- Minimize the distance function to find the point on the line that is closest to ( (0, -3) ).

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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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