I need help with a calculus 2 exercise, with a good explanation to what I'm trying to figure out.
1 answer:
<h3>Explanation:</h3>
When writing rational expressions, you need to be aware that ...
1/4x = (1/4)x ≠ 1/(4x)
Parentheses around the denominator are required, unless you're typesetting the expression and can use a fraction bar for grouping.
The derivative of the curve expression is ...
y' = x - 1/(4x) . . . . . parentheses added to what you wrote
and the expression (1 -(y')^2) can be written ...
1 -(y')^2 = x^2 +1/2 +1/(16x^2) . . . . . parentheses added to what you wrote
The first and last terms of this trinomial are both perfect squares, so you might suspect the whole trinomial is a perfect square. You recall that ...
(a +b)^2 = a^2 + 2ab + b^2
This is a good "pattern" to remember. Using it is a matter of <em>pattern recognition</em>, as is the case with a lot of math.
Here, you have ...
a = x
b = 1/(4x)
In order for your trinomial to be a perfect square, the product 2ab must equal the middle term of your trinomial. (Spoiler: it does.)
2ab = 2(x)(1/(4x)) = (2x)/(4x) = 1/2 . . . . . matches the middle term of 1 -(y')^2
Hence your trinomial can be written as the square ...
1 -(y')^2 = (x +1/(4x))^2
_____
This is convenient because you want to integrate the square root of this. Your integral then becomes ...
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Answer:
Step-by-step explanation:
Let's answer these questions in an all-encompassing kind of explanation. If you plot the vertex and the directrix, you see that the vertex is below the directrix. Because of the fact that a parabola opens AWAY from the directrix, and wraps itself around the focus, we know it's an upside down parabola of the form
The p value from the equation is a distance, specifically the distance between either the vertex and the directrix, or the vertex and the focus. The vertex is exactly in the middle of the directrix and the focus. So that tells us that the focus is 3 units below the vertex (because the directrix is 3 units above the vertex). We also know from this that p = 3.
Filling in the equation with a vertex of (0, 0) which is our h and k respectively:
which simplifies to
and multiplying both sides by -1:
. This is not standard form, but it matches what your equation is in the choices. So to sum up:
The focus is located at (0, -3) and the first choice is true.
The parabola opens upside down and the second choice is not true.
The p value is found by counting the units between the vertex and the directrix, so the third choice is not true.
We solved the equation by filling in the values for h, k, and p and got that the equation in the fourth choice is true.
So the fifth choice is not true.