Name the place of the highlighted digit. Then write the value of the digit
1 answer:
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Answer:
-338
Step-by-step explanation:
So we have the sequence:
5, -2, -9, -16...
First, note that this is an arithmetic sequence.
This is because each individual term is the previous term <em>added</em> by a common difference.
We can see that this common difference is -7, because each subsequent term is 7 <em>less</em> than the previous one. For example, 5 minus 7 is -2, -2 minus 7 is -9, and so on.
So, to find the 50th term, we can write an explicit formula for our sequence.
The standard form for the explicit formula for an arithmetic sequence is:
Where a is the initial term, d is the common difference, and n is the nth term.
We can see that our initial term a is 5. And we also already determined that the common difference d is -7. So, substitute:
Now, to find the 50th term, all we have to do is to substitute 50 for n. So:
Subtract within the parentheses:
Multiply:
Subtract:
So, the 50th term is -338.
And we're done!
You're trying to minimize the distance between the point 
and an arbitrary point on the curve, 
.
The distance between two such points is given by the function
so this is the function whose derivative you should check.
But before you do that, it's helpful to know that
is minimized at the same point as the modified distance function 
.
Differentiating, you have
Set this to zero and solve for the critical points.
You can use the rational root theorem to find some potential candidates for roots to the cubic. The constant term has factors
, while the leading coefficient has factors 
. The only candidates for rational roots are 
. The only one of these that works is 
, so 
is a root to the cubic above.
Polynomial division reveals that we can factor the cubic as
which has only one real root at. Checking the value of the derivative of 
to the left and right of this point confirms that a minimum occurs here.
Therefore the closest point on the curve to is 
.
The distance between two such points is given by the function
so this is the function whose derivative you should check.
But before you do that, it's helpful to know that
Differentiating, you have
Set this to zero and solve for the critical points.
You can use the rational root theorem to find some potential candidates for roots to the cubic. The constant term has factors
Polynomial division reveals that we can factor the cubic as
which has only one real root at
Therefore the closest point on the curve to