A satellite following the equation #y = 1/2x^2 - 4#, where #x# and #y# are in millions of kilometres, is surveying a far away planet, located at the origin (0,0). How close to the planet does the satellite get?
Essentially, the problem is this:
By the distance formula, we have:
We're now going to find the critical points by
We can see very quickly that there exists no real solution to this equation.
Now for the other set of critical points.
However, we must find the distance, therefore:
Hopefully this helps!
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Alternative approach, if you are happy with the Lagrange Multiplier.
In simple units, actual distance units quoted above...
We wish to optimise:
Basic idea;
In terms of showing this to be a min, note from geometry that:
graph{1/2x^2 - 4 [-10, 10, -5, 5]}
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The satellite gets as close as 8 million kilometers to the planet.
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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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