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Ind the slope m of the tangent to the curve y = 9 + 4x2 − 2x3 at the point where x =a. m = 8a−6a2 (b) find equations of the tangent lines at the points (1, 11) and (2, 9). y(x) = 2x+11 (at the point (1, 11)) y(x) = (at the point (2, 9))

Answers

Answer 1

Answer:

This is about slope of tangents via derivatives.

A) m = 8a- 6a²

B) At (1, 11), equation of tangent is y = 2x + 9

At (2, 9), equation of tangent is y = 2x + 4

The slope of a tangent to a curve is simply gotten by finding the first derivative of the function representing that curve.

We are given the function; y = 9 + 4x² - 2x³

The slope will be;

m = y' = 8x - 6x²

A) At x = a; To find the slope we will just put a for x in the slope function;

Slope; m = 8a- 6a²

B) Slope is 8x - 6x²

At the point (1, 11);

Slope = 8(1) - 6(1²) = 2

Thus, equation of tangent line is; y - 11 = 2(x - 1)

⇒ y - 11 = 2x - 2

⇒ y = 2x - 2 + 11

⇒ y = 2x + 9

At the point (2, 9);

Slope = 8(2) - 6(2²) = -8

Thus, equation of tangent line is; y - 8 = 2(x - 2)

⇒ y - 8 = 2x - 4

⇒ y = 2x - 4 + 8

⇒ y = 2x + 4

Read more at; brainly.com/question/18476507

Answer 2

Answer: Find the derivative:-
y' = 8x - 6x^2
This is the slope in terms of x.
When x = a , the slope is
8a - 6a^2

(b) equation of tangent line at x1,y1
is y- y1 = (8x - 6x^2)( x - x1) , so at (1,11) it is
y - 11 = (8(1) - 6((1)^2) ) (x - 1)
y = 2(x - 1) + 11
y = 2x + 9 (answer)

At (2,9)
y - 9 = (16-24)(x - 2)
y - 9 = -8x + 16
y = -8x + 25 answer

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Amanda buye a book for $7.85. If the tax added on to the price lis $0.75 how much will the book cost in total? What if Amanda donates SLSO to the Love of Reading Foundation? How will thie affect the total coet of the book? (Hint: there is no tax added on to the donation). Estimate what the total cost will be and then check your answer.

Answers

Answer:

$8.60 I don't know what SLSO means

Step-by-step explanation:

One question in the survey asked how much time per year the children spent in volunteer activities. The sample mean was 14.76 hours and the sample standard deviation was 16.54 hours.Required:

a. Based on the reported sample mean and sample standard deviation, explain why it is not reasonable to think that the distribution of volunteer times for the population of South Korean middle school students is approximately normal.
b. The sample size was not given in the paper, but the sample size was described as large. Suppose that the sample size was 500. Explain why it is reasonable to use a one-sample t confidence interval to estimate the population mean even though the population distribution is not approximately normal.
c. Calculate and interpret a confidence interval for the mean number of hours spent in volunteer activities per year for South Korean middle school children.

Answers

Answer:

a. If the distribution was normal, many values would be negative, what is incompatible with the response variable (hours dedicated to volunteer activities).

b. If the sample is big, accordingly to the Central Limit Theorem, the sampling distribution shape tends to be normally-like, so we can apply a one-sample t-test.

c. The 95% confidence interval for the mean is (13.307, 16.213).

Step-by-step explanation:

a. If the distribution was normal, the values with one or more standard deviation below the mean would be negative, what is incoherent for this case. This, in a normal distribution, represents approximately 16% of the values.

If we calculate the probabilty for a normal distribution with the sample parameters, the probability of having "negative hours" is 18.6% (see picture attached).

b. If the sample is big, accordingly to the Central Limit Theorem, the sampling distribution shape tends to be normally-like, so we can apply a one-sample t-test.

The sampling distribution standard deviation is also reduced by a factor of 1/√n.

c. We have to calculate a 95% confidence interval for the mean.

The population standard deviation is not known, so we have to estimate it from the sample standard deviation and use a t-students distribution to calculate the critical value.

The sample mean is M=14.76.

The sample size is N=500.

When σ is not known, s divided by the square root of N is used as an estimate of σM:

$s_M=(s)/(√(N))=(16.54)/(√(500))=(16.54)/(22.3607)=0.7397$

The t-value for a 95% confidence interval is t=1.965.

The margin of error (MOE) can be calculated as:

$MOE=t\cdot s_M=1.965 \cdot 0.7397=1.453$

Then, the lower and upper bounds of the confidence interval are:

$LL=M-t \cdot s_M = 14.76-1.453=13.307\n\nUL=M+t \cdot s_M = 14.76+1.453=16.213$

The 95% confidence interval for the mean is (13.307, 16.213).

Let S be the sphere of radius 1 centered at (2, 4, 6). Find the distance from S to the plane x + y + z = 0.

Answers

Answer:

5.928

Step-by-step explanation:

Given that:

The relation of the plane x+y+z= 0

Suppose (x,y,z) is any point on the plane.

Then the difference between (2,4,6) to (x,y,z) is:

$d^2 = (x-2)^2 + (y -4)^2 + ( z -6)^2 \n \n d^2 = (x^2 -4x+4) + ( y^2-8y +16) +(z^2 -12z + 36)$

$d^2 = x^2 + y^2 +z^2 -4x -8y -12z +4 +16 +36$

$d^2 = x^2 +y^2 + z^2 -4x -8y -12z +56$

∴

$f(x,y,z) =d^2 = x^2 + y^2 + z^2 - 4x -8y - 12 z +56 - - - (1)$

To estimate the maximum and minimum values of the function f(x,y,z) subject to the constraint g(x,y,z) = x+y+z =0

By applying Lagrane multipliers;

If we differentiate equation (1) with respect to x; we have:

f(x,y,z) = 2x -4

If we differentiate equation (1) with respect to y; we have:

f(x,y,z) = 2y - 8

If we differentiate equation (1) with respect to z; we have:

f(x,y,z) = 2z - 12

Differentiating g(x,y,z) with respect to x, we have:

$g_x(x,y,z) = 1$

Differentiating g(x,y,z) with respect to y, we have:

$g_y(x,y,z) = 1$

Differentiating g(x,y,z) with respect to z, we have:

$g_z(x,y,z) = 1$

Calculating the equations $\bigtriangledown f = \lambda \bigtriangleup g \ \ \ \& \ \ \ g(x,y,z) =0$

$f_x = \lambda g_x\n$

$2x - 4 = \lambda (1)$

$2x= 4 + \lambda$

$x= 2 + (\lambda )/(2) --- (2)$

$f_y = \lambda g_y$

$2x -8 = \lambda(1)$

$2x = 8+ \lambda$

$x = 4+(\lambda)/(2) --- (3)$

$f_z = \lambda g_z$

$2x -12 = \lambda (1)$

$x = 6 + (\lambda )/(2) --- (4)$

x+y+z = 0 - - - (5)

replacing x, y, z values in the given constraint

x + y + z = 0

$2+(\lambda)/(2)+4+(\lambda)/(2)+6+(\lambda)/(2)=0$

$12 + (3 \lambda )/(2)=0$

$(3 \lambda )/(2)=-12$

$3 \lambda=-12 * 2$

$3 \lambda=-24$

$\lambda=(-24)/(3)$

$\lambda=-8$

Therefore, from equation (2)

$x=2 +( \lambda )/(2)$

$x=2 +( -8 )/(2)$

x = 2 - 4

x = - 2

From equation (3)

$x=4 +( \lambda )/(2)$

$x=4 +( -8 )/(2)$

x = 4 - 4

x = 0

From equation (3)

$x=6 +( \lambda )/(2)$

$x=6 +( -8 )/(2)$

x = 6 -4

x = 2

i.e (x,y,z) = (-2, 0, 2)

∴

$d^2 = (x-2)^2 +(y-4)^2 + (z -6)^2$

$d^2 = (-2-2)^2 +(0-4)^2 + (2 -6)^2$

$d^2 = 16 +16 + 16$

$d^2 =48$

$d =√(48)$

$d= \pm 6.928$

since we are taking only the positive integer because distance cannot be negative, then:

The distance from the center of the sphere to the plane is 6.928.

However, the distance from the surface S to the plane is:

6.928 - radius of the sphere.

where;

the radius of the sphere is given as 1

Then:

the distance from the surface S to the plane is:

6.928 - 1

= 5.928

Solve the following equation for B. b Over 3 equals M

Answers

Answer:

b=3m

Step-by-step explanation:

$(b)/(3) =m\n\n3((b)/(3))=3(m)\n\n(b)/(1) =3m\n\nb=3m$

S = the team's scoreg = the number of goals a team has scored
Which of the variables is independent and which is dependent?

Answers

Answer:

s would be the dependent variable, and g the independent.

Step-by-step explanation:

Whatever the score is, it would depend on how many goals the team has scored.

What is the lowest common multiple of 20 and 60

Answers

Answer:

Step-by-step explanation:

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