Calculus: dy/dx= ?
(d^2y)/(dx^2)= ?
d^2y/dx^2= 0 at point (x,y)=?
​

ln(5y) = 4xy

Differentiate both sides with respect to x, taking y = y(x) :

d/dx [ln(5y)] = d/dx [4xy]

1/(5y) d/dx [5y] = 4y + 4x dy/dx

5/(5y) dy/dx = 4y + 4x dy/dx

1/y dy/dx = 4y + 4x dy/dx

(1/y - 4x) dy/dx = 4y

dy/dx = 4y / (1/y - 4x)

Since y = 0 is outside the domain (ln(0) is undefined), we can multiply the right side by y/y :

dy/dx = 4y ² / (1 - 4xy)

To compute the second derivative, differentiate both sides again:

d/dx [dy/dx] = d/dx [4y ² / (1 - 4xy)]

d²y/dx ² = ((1 - 4xy) d/dx [4y ²] - 4y ² d/dx [1 - 4xy]) / (1 - 4xy)²

d²y/dx ² = ((1 - 4xy) 8y dy/dx - 4y ² (0 - 4y - 4x dy/dx)) / (1 - 4xy)²

d²y/dx ² = ((8y - 32xy ²) dy/dx + 16y ³ + 16xy ² dy/dx) / (1 - 4xy)²

d²y/dx ² = ((8y - 16xy ²) dy/dx + 16y ³) / (1 - 4xy)²

Substitute the first derivative founder earlier:

d²y/dx ² = ((8y - 16xy ²) (4y ² / (1 - 4xy)) + 16y ³) / (1 - 4xy)²

d²y/dx ² = ((8y - 16xy ²) 4y ² + 16y ³ (1 - 4xy)) / (1 - 4xy)³

d²y/dx ² = (16y ³ (3 - 8xy)) / (1 - 4xy)³

The second derivative vanishes wherever the numerator vanishes (so long as 4xy ≠ 1) :

16y ³ (3 - 8xy) = 0

16y ³ = 0   or   3 - 8xy = 0

y = 0   or   8xy = 3

As pointed out earlier, y = 0 is outside this function's domain, so we're left with points on the curve

y = 3/(8x)

Substituting this into the original function gives

ln(5 (3/(8x))) = 4x (3/(8x))

ln(15 / (8x)) = 12/8

ln(15) - ln(8x) = 3/2

ln(8x) = ln(15) - 3/2

8x = exp(ln(15) - 3/2)

8x = 15 exp(-3/2)

x = 15/8 exp(-3/2)   →   y = 1/5 exp(3/2)

or approximately (0.4184, 0.8963).

(where exp(x) = e ˣ)

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Find the product: (0.45)•(0.008)

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A)80
B)85
C)75
D)60

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7 million of these Canadian people spoke French as their first language.
(a) Express 7 million as a percentage of 32 million.
Give your answer correct to 1 decimal place.

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A physical education teacher wanted to know the effects of various activity levels on body composition. She surveyed her physical education classes and categorized 30 students into five activity levels based on the amount of daily exercise in which the students participated: inactive, semiactive, normal, active, and very active. She measured percent body fat values using skinfold calipers on all subjects with the following results.

Inactive Semiactive Normal Active Very Active
30.2 29.4 22.9 17.6 10.9
29.6 17.6 25.4 13.4 13.7
35.2 26.4 19.6 20.3 12.8
19.1 25.3 18.7 19.6 14.7
26.3 22.5 21.8 15.1 9.3
22.4 28.6 24.9 10.7 12.7

Required:
a. What is the independent variable in this study? What is the dependent variable?
b. What are the means and standard deviations for each group?
c. Does a significant difference exist among any of the groups?

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what is the difference in length between the longest meteor and the shortest meteor ​

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A triangle has side lengths of 23 cm, 6 cm, and 28 cm. Classify it as right,acute, or obtuse


Obtuse
Right
Acute

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Jorge was studying average life expectancy in Europe and how it varies across different countries. He wanted to discover what variables were good predictors of life expectancy and identified 3 that he believed could possibly be good predictors:
employment rate,
access to healthcare, and
number of car accidents per year.
He conducted a multiple linear regression analysis using these variables.
When looking over the results of this analysis, he saw
that the scatterplot did not show a linear pattern between each of the explanatory variables and the response variable.
that there was not a funnel shape to the residual vs. predicted plot. that the histogram of the normal quantile plot was fairly normal.
Based on all of these findings,
he can say that the linearity condition _________
the equal variance condition ________
the normality condition _____________
the 10% condition ____________
and the success failure condition __________
Options for each:
A) is not met
B) does not need to be checked for multiple linear regression
C) is met

A fish is swimming at a constant rate toward the ocean floor. The equation

y = -7x -3 can be used to represent this situation, where y is the depth of the fish in meters below sea level and x is the number of seconds the fish has been swimming.

Which statement best describes the depth of this fish, given this equation? (DOK 3)

A
From a starting position of 3 meters below sea level, the fish is going down( descending) 7 meters per second

B
From a starting position of 7 meters below sea level, the fish is ascending 3 meters per second

C
From a starting position of 7 meters below sea level, the fish is descending 3 meters per second

D
From a starting position of 3 meters below sea level, the fish is ascending 7 meters per second

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To assess the effectiveness of a kindergarten-readiness program, 15 children from a random sample were each given a diagnostic assessment before beginning the program and a follow-up assessment after completing the program. For each child, the difference in the score points between the two assessments was calculated and used to create the 95 percent confidence interval (20.1,23.9)(20.1,23.9).
Assuming all conditions for inference are met, which of the following is a correct interpretation of the interval?
A) For all children in the program, 95 percent of the children will have a mean difference in scores of between 20.1 points and 23.9 points.
B) There is a 0.95 probability that the mean difference in scores for all children in the sample is between 20.1 points and 23.9 points.
C) There is a 0.95 probability that the mean difference in scores for the children in the program is between 20.1 points and 23.9 points.
D) We are 95 percent confident that the mean difference in scores for all children in the program is between 20.1 points and 23.9 points.
E) We are 95 percent confident that the mean difference in scores for the children in the sample is between 20.1 points and 23.9 points.

4x²y+2xy² simplefid in algebra​

Which fraction in simplest from is equivalent to 4/6?
A. 1/3
B. 1/4
C. 2/3
D. 1/3

This net folds into a cube. Which two letters appear on opposite faces?