Karen made a scale drawing of her
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We're given the implicit equation
x² + 2x + y² - 4y = 12
where "implicit" means that one of the variables is considered a function of the other. In this case, the variable y = y(x) depends on the value of the variable x - in other words, the possible values of y are implied by the value of x. We then say that this equation defines y as an implicit function of x.
We're also given the implicit derivative,
dy/dx = (-2x - 2)/(2y - 4)
which tells us the rate of change y as a function of both x and y. The value of dy/dx corresponds to the slope of the tangent line to the curve defined by the implicit equation above at some point (x, y).
The tangent line is horizontal when its slope is zero. This happens when
(-2x - 2)/(2y - 4) = 0
If y ≠ 2, then we can eliminate the denominator and we're left with
-2x - 2 = 0
Solve for x :
-2x = 2
x = -1
This tells us that the points on the curve with x-coordinate -1 have a tangent line to the curve that is horizontal.
The tangent line is vertical when its slope is undefined/infinite, or equivalently when the denominator of dy/dx is zero:
2y - 4 = 0
Solve for y :
2y = 4
y = 2
So any point on the curve with y-coordinate 2 will have a tangent line there that is vertical.
By completing the square in the implicit equation, we can easily identify where these points are located.
x² + 2x + y² - 4y = 12
x² + 2x + 1 + y² - 4y + 4 = 17
(x + 1)² + (y - 2)² = 17
This equation defines a circle centered at (-1, 2) with radius √17. When x = -1, we have
(y - 2)² = 17 ⇒ y = 2 ± √17
and when y = 2, we have
(x + 1)² = 17 ⇒ x = -1 ± √17
See the attached plot to see the circle and the tangents at these points.
Answer:
The standard deviation of the given data is 11.34.
Step-by-step explanation:
We have to find the standard deviation for the given data;
Grade on No. of students
test (X) with that score (f)
50 1 50 1073.218 1073.2176
57 2 114 663.578 1327.1552
60 4 240 518.018 2072.0704
65 3 195 315.418 946.2528
72 3 216 115.778 347.3328
75 12 900 60.218 722.6112
77 10 770 33.178 331.776
81 6 486 3.098 18.5856
83 6 498 0.058 0.3456
88 9 792 27.458 247.1184
90 12 1080 52.418 629.0112
92 12 1104 85.378 1024.5312
95 2 190 149.818 299.6352
99 4 396 263.738 1054.9504
100 <u> 5 </u> <u> 500 </u> 297.218 <u> 1486.088 </u>
Total <u> 91 </u> <u> 7531 </u> <u> 11580.68 </u>
Firstly, the mean of the above data is given by;
Mean, =
= = 82.76
Now, the standard deviation of the data is given by;
Standard deviation, S.D. =
=
= 11.34
Hence, the standard deviation of the given data is 11.34.