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lidiya [134]
3 years ago
13

find the coordinates of the point P on the parabola y=1-x^2 with domain 0≤x≤1 that minimize the area of the triangle enclosed by

the tangent line at P, the x-axis, and y-axis
Mathematics
1 answer:
coldgirl [10]3 years ago
6 0

Let point P be with coordinates (x_0,y_0). Find the equation of the  tangent line.

1. If y=1-x^2, then y'=-2x.

2. The equation of the tangent line at point P is

y-y_0=-2x_0(x-x_0).

Find x-intercept and y-intercept of this line:

  • when x=0, then y=y_0+2x_0^2;
  • when y=0, then x=\dfrac{y_0}{2x_0}+x_0=\dfrac{y_0+2x_0^2}{2x_0}.

The area of the triangle enclosed by the tangent line at P, the x-axis, and y-axis is

A=\dfrac{1}{2}\cdot (2x_0^2+y_0)\cdot \left(\dfrac{y_0+2x_0^2}{2x_0}\right)=\dfrac{(y_0+2x_0^2)^2}{4x_0}.

Since point P is on the parabola, then y_0=1-x_0^2 and

A=\dfrac{(1-x_0^2+2x_0^2)^2}{4x_0}=\dfrac{(1+x_0^2)^2}{4x_0}.

Find the derivative A':

A'=\dfrac{2(1+x_0^2)\cdot 2x_0\cdot 4x_0-4(1+x_0^2)^2}{16x_0^2}=\dfrac{12x_0^4+8x_0^2-4}{16x_0^2}.

Equate this derivative to 0, then

12x_0^4+8x_0^2-4=0,\\ \\3x_0^4+2x_0^2-1=0,\\ \\D=2^2-4\cdot 3\cdot (-1)=16,\ \sqrt{D}=4,\\ \\x_0^2_{1,2}=\dfrac{-2\pm4}{6}=-1,\dfrac{1}{3},\\ \\x_0^2=\dfrac{1}{3}\Rightarrow x_0_{1,2}=\pm\dfrac{1}{\sqrt{3}}.

And

y_0=1-\left(\pm\dfrac{1}{\sqrt{3}}\right)^2=\dfrac{2}{3}.

Answer: two points: P_1\left(-\dfrac{1}{\sqrt{3}},\dfrac{2}{3}\right), P_2\left(\dfrac{1}{\sqrt{3}},\dfrac{2}{3}\right).

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Answer:

A

Step-by-step explanation:

Point A:

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Dylan and his friend Lamar volunteered to bring sandwiches for lunch at a homeless shelter. They bought 8 loaves of bread. Each
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Answer:

64 slices

Step-by-step explanation:

The first step to solving this is to find out how many slices of bread there were in total. We know that each loaf of bread had 16 slices, and we know that there were 8 loaves. To get the amount of slices, we multiply the amount of slices in each loaf by the number of loaves there are.

16⋅8=128 slices of bread in total

Now that we know the total number of slices, we need to find how many sandwiches they made. If they used two slices per sandwich, we would divided the total number of bread slices by two.

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True or false <br> A trapezium is a parallelogram
jek_recluse [69]
True
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One of the tallest living men has a height of 236 cm. One of the tallest living women is 224 cm tall. Heights of men have a mean
Roman55 [17]

Answer:

A. The woman is relatively taller because the z score for her height is greater than the z score for the man​'s height.

Step-by-step explanation:

Whoever has the highest z-score, is taller relative to the population of the same gender.

The z-score of a value X in a set with mean \mu and standard deviation /sigma is given by:

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Solution:

Heights of men have a mean of 170 cm and a standard deviation of 6 cm. One of the tallest living men has a height of 236 cm. So the z-score of his height is:

Z = \frac{X - \mu}{\sigma} = \frac{236-170}{6} = 11

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irinina [24]

Answer:

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Now simplify that and you’ll Get

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