1. Describe how the line of best fit and the correlation coefficient can be used to determine the correlation between the two variables on your graph.


2. Describe the type of correlation between the two variables on your graph. How do you know?


3.Does the correlation between the variables imply causation? Explain.


4.How do you calculate the residuals for a scatterplot?
50 POINTS.

The value after one year will be $535.

To explain in the simplest form, interest is calculated as a percent of the principal. For example, assume that you have borrowed $100 from your friend and you have promised to repay it with 5% interest, then the amount of interest you would pay along with the actual amount would just be 5% of 100 which is $100(5/100) = $5.

An annual percentage of the amount of a loan is known as interest. For example, when you deposit your money in a high-yield savings account, the bank will pay interest. Now, according to the question

Given the amount = $500

interest rate is given as 7% on 3-year certificates of deposit.

Therefore, the value after one year will be

= 500 x 7% + 500

=500 x 0.07 + 500

= 35 + 500

= $535

Hence, the value will be $535.

To learn more about the interest rate;

https://brainly.com/question/25720319

#SPJ4

Answer:

x=4

Step-by-step explanation:

Based on the given conditions, formulate: 28(6-x)+30x = 176

Apply the Distributive Property: 168 - 28x+30x = 176

Combine like terms: 168+2x=176

Rearrange variables to the left side of the equation: 2x=176-168

Calculate the sum or difference: 2x=8

Divide both sides of the equation by the coefficient of variable:x=8/2

Cross out the common factor: x=4

Each piece of wood that Ren cuts will be 3/8 of a meter long.

To solve the problem, we need to divide 3/4 by 2. This can be written as:

3/4 ÷ 2

To model this using fraction bars, we can start by drawing a bar to represent the whole piece of wood, which is 3/4 of a meter long:

___________________

|___________________|

        3/4

Next, we need to divide this bar into 2 equal parts. We can do this by drawing a line down the middle of the bar:

_______ _______

|_______|_______|

  3/4     3/4

Now we can see that we have two equal pieces of wood, each of which is 3/4 ÷ 2 = 3/8 of a meter long.

To calculate this, we can divide the numerator (3) by 2 to get 1.5, and then write this as a fraction with a denominator of 8:

1.5 ÷ 2 = 0.75

0.75 = 3/4

3/4 ÷ 2 = 3/8

To learn more about wood

https://brainly.com/question/10967023

#SPJ4

Sample size is inversely related to the tolerable deviation rate, a larger sample size is needed to provide a more accurate estimate of the population parameter.

Step-by-step explanation:
1. Sample size refers to the number of observations or units included in a study or analysis to represent a population.
2. Desired level of confidence refers to the degree of certainty that the estimate obtained from the sample accurately represents the population parameter. It is directly related to sample size, as a higher level of confidence generally requires a larger sample.
3. Expected population deviation rate refers to the anticipated rate of deviation or error in a population. It is also directly related to sample size, as a higher expected deviation rate requires a larger sample to ensure accuracy.
4. Tolerable deviation rate, on the other hand, is the maximum rate of deviation that can be accepted in the sample without affecting the overall conclusions. This is inversely related to sample size because as the tolerable deviation rate decreases, a larger sample size is needed to provide a more accurate estimate of the population parameter.

Learn more about deviation here:

brainly.com/question/475676

#SPJ11

Since the difference is negative (-8), the sequence is monotonic decreasing.

A monotonic function in mathematics is a function between ordered sets that maintains or flips the given order. Calculus was where this idea initially surfaced, and it was later applied to the more abstract context of order theory.  If the variables Yj can be arranged so that if Yj is missing, then all variables Yk with k>j are likewise missing, then the pattern of missing data is said to be monotone.

This happens, for instance, in drop-out-prone longitudinal research. The pattern is said to as non-monotone or generic if it is not monotonous.

Using the difference test to calculate the nth term of the sequence, we get:

[tex]a_n - a_{n-1} \\= 6 - 8n - (6 - 8(n-1)) \\= -8[/tex]

Since the difference is negative (-8), the sequence is monotone decreasing.

Learn more about monotonic Visit: brainly.com/question/30627393

#SPJ4

Of the joint probability function

(a) The value of the constant c is approximately 0.0238.

(b) P(X=0,Y=3) ≈ 0.0714.

(c) P(X≥ 0,Y≤ 1) ≈ 0.4524.

(d) The given statement "X and Y are not independent" is False.

(a) To find the value of the constant c, we need to use the fact that the sum of the probabilities over all possible values of X and Y must be equal to 1:

∑∑f(x,y) = 1

∑x=[tex]0^2[/tex] ∑y=[tex]0^3[/tex] c(2x+y) = 1

c(0+1+2+3+2+3+4+5+4+5+6+7) = 1

c(42) = 1

c = 1/42 ≈ 0.0238 (rounded to three decimal places)

(b) P(X=0,Y=3) = f(0,3) = c(2(0)+3) = 3c = 3(1/42) ≈ 0.0714 (rounded to three decimal places)

(c) P(X≥0,Y≤1) = f(0,0) + f(0,1) + f(1,0) + f(1,1) + f(2,0) + f(2,1)

= c(2(0)+0) + c(2(0)+1) + c(2(1)+0) + c(2(1)+1) + c(2(2)+0) + c(2(2)+1)

= c(1+3+2+4+4+5) = 19c = 19(1/42) ≈ 0.4524 (rounded to three decimal places)

(d) We can check whether X and Y are independent by verifying if P(X=x,Y=y) = P(X=x)P(Y=y) for all possible values of X and Y. Let's check this for some cases:

P(X=0,Y=0) = f(0,0) = c(2(0)+0) = 0

P(X=0) = f(0,0) + f(0,1) + f(0,2) + f(0,3) = c(0+1+2+3) = 6c

P(Y=0) = f(0,0) + f(1,0) + f(2,0) = c(0+2+4) = 6c

P(X=0)P(Y=0) = [tex]36c^2[/tex]

Since P(X=0,Y=0) ≠ P(X=0)P(Y=0), X and Y are not independent. Therefore, the answer is (C) false.

To know more about joint probability function, refer to the link below:

https://brainly.com/question/31129873#

#SPJ11

The number of fifth graders are 170 and the number of sixth graders are 150.

Given that, the ratio of the number of fifth grade voter to the number of sixth grade voter was 17:15.

Here, number of fifth grader are 17x and the number of sixth grader are 15x.

The ratio would have been 8:7 if 90 fewer fifth graders and 80 fewer 6th graders had taken part.

Now, the number of fifth graders are 17x-90 and the number of sixth graders are 15x-80.

The new ratio is 17x-90/15x-80 = 8/7

Now, 7(17x-90)=8(15x-80)

119x-630=120x-640

120x-119x=640-630

x=10

Number of fifth graders = 17x=170

Number of sixth graders = 15x = 150

Therefore, the number of fifth graders are 170 and the number of sixth graders are 150.

To learn more about the ratio visit:

brainly.com/question/13419413.

#SPJ4

You must not pass on a curve or the crest of a hill if you cannot see at least 500 feet ahead.

This statement is referring to a basic safety rule of driving. Passing on a curve or the crest of a hill can be very dangerous since visibility is limited, and the driver may not be able to see approaching vehicles until it is too late to avoid a collision.

The amount of distance that a driver must be able to see ahead before passing depends on various factors such as the speed of the vehicles, road conditions, and weather conditions.

However, a general rule of thumb is that a driver should be able to see at least 400 feet ahead before passing. This distance allows the driver enough time to react if another vehicle suddenly appears or if there is an obstacle on the road.

Learn more about safe driving at:

brainly.com/question/25292145

#SPJ11

The value of y for the solution set to the given system of equations is :

(e) y = -3

To use Cramer's rule, we need to find the determinant of the coefficient matrix and several other determinants obtained by replacing one column of the coefficient matrix with the constant terms. The coefficient matrix is:

{{-2, 3, -1}, {3, -1, 2}, {-2, 2, -1}}

The determinant of this matrix is:

|-2  3 -1|
| 3 -1  2|
|-2  2 -1| = -12

Now we replace the first column with the constants:

{{-2, 3, -1}, {-1, -1, 2}, {-1, 2, -1}}

The determinant of this matrix is:

|-2  3 -1|
|-1 -1  2|
|-1  2 -1| = 9

Next, we replace the second column with the constants:

{{-2, -2, -1}, {3, -1, 2}, {-2, -1, -1}}

The determinant of this matrix is:

|-2 -2 -1|
| 3 -1  2|
|-2 -1 -1| = 12

Finally, we replace the third column with the constants:

{{-2, 3, -2}, {3, -1, -1}, {-2, 2, -1}}

The determinant of this matrix is:

|-2  3 -2|
| 3 -1 -1|
|-2  2 -1| = -18

Now we can use Cramer's rule to find the value of y. The solution is:

y = D2 / D = 9 / (-12) = -3/4

Therefore, the answer is e) y = -3.

To learn more about Crammer's rule visit : https://brainly.com/question/20354529

#SPJ11

The vertical asymptote is x = -4 and the horizontal asymptote of the function is y = 2.

To find the vertical asymptote of the function f(x) = 2x ÷ (x+4), we need to look for any value of x that makes the denominator equal to zero. In this case, we have: x + 4 = 0

x = -4

Therefore, the vertical asymptote is x = -4.

f(x) = (2x ÷ x) ÷ (x ÷ x + 4 ÷ x)

f(x) = 2 ÷ (1 + 4/x)

As x becomes very large, the term 4/x becomes very small and can be neglected.

Therefore, as x → ∞, f(x) → 2/1 = 2.

Similarly, as x becomes very small (i.e., negative), the term 4/x becomes very large and can be neglected. Therefore, as

x → -∞, f(x) → 2/1 = 2.

To learn more about vertical follow the link:

https://brainly.com/question/27029296

#SPJ1

The complete question is:

Determine the equations of the vertical and horizontal asymptotes, if any, for f(x) = 2x / x + 4.

There will be 0.0451 seconds the peak response lag behind the input peak

The peak response of the system occurs at the same frequency as the input torque, which is given as w_f = 16 rad/s.

The amplitude of the steady-state response can be found using the given equation:

A = T_o / sqrt((Jw² - b²)² + (bw)²)

Substituting the given values, we get:

A = 154 / sqrt((3*(16)² - 58²)² + (58*16)²) ≈ 0.574

The phase angle between the input and output can be found using the equation:

tan(o) = bw / (Jw² - b²)

Substituting the given values, we get:

tan(o) = (5816) / (3(16)² - 58²) ≈ 0.908

Therefore, the phase lag between the input and output is given by:

o = arctan(0.908) ≈ 0.725 radians

To find the time lag, we divide the phase lag by the angular frequency:

t_lag = o / w_f ≈ 0.0451 seconds

Therefore, the peak response lags behind the input peak by approximately 0.0451 seconds.

Learn more about momentum: brainly.com/question/13767949

#SPJ11

Toby is correct. A trapezoid is a quadrilateral with most effective one pair of parallel facets.

Therefore, it has pairs of opposite aspects that are not equal in length. The non-parallel facets of a trapezoid are called the legs, whilst the parallel facets are referred to as the bases.

The gap among the 2 bases is known as the height or altitude of the trapezoid. some other properties of trapezoids encompass: the sum of the indoors angles is 360 tiers, the midsegment is parallel to the bases and is half the sum in their lengths, and the place is given via the method: A = (b1 + b2)h/2, wherein b1 and b2 are the lengths of the two bases and h is the height.

Learn more about trapezoid:-

https://brainly.com/question/26487711

#SPJ4

(a)  (AB)1,2 = (1)(-1) + (2)(3) + (-2)(1) = -1 + 6 - 2 = 3. (b)  (AB)2,3 and (AB)3,2 do not exist. (c) To determine if BA exists, we need to check if the number of columns in matrix B is equal to the number of rows in matrix A. B has 2 columns and A has 4 rows, so BA does not exist. (d) Since we don't have matrix C, we cannot find CA.

(a) To find (AB)1,2 without computing the whole matrix, we only need to compute the dot product of the first row of matrix A and the second column of matrix B.
A = | 1  2 |
   |-2  3 |
   | 2  4 |
   |10  4 |
B = | 3 -1 |
   | 1  5 |
   | 3  1 |
   | 2  2 |
(AB)1,2 = (1 * -1) + (2 * 5) = -1 + 10 = 9
(b) (AB)2,3 and (AB)3,2 do not exist because matrix A has 2 columns and matrix B has 3 rows. For these elements to exist, matrix A should have 3 columns and matrix B should have 3 rows.
(c) BA does not exist because matrix A has 2 columns and matrix B has 3 rows. For matrix multiplication to be possible, the number of columns in matrix A must match the number of rows in matrix B.
(d) To find matrix CA where Cϵ R, we need to know the values of matrix C. Since the matrix C is not provided, we cannot compute CA.

learn more about matrix multiplication here: brainly.com/question/13006200

#SPJ11

The Kermack and McKendrick model of an epidemic proposes that the population can be divided into three classes: healthy, sick, and dead. The total population remains constant in size, except for deaths due to the epidemic. The model is kxy, where k and l are positive constants. The equations are based on the assumptions that healthy people get sick at a rate proportional to the product of x and y, and sick people die at a constant rate l.

The given model consists of three variables: x(t), y(t), and z(t), representing the number of healthy, sick, and dead people, respectively, in a population. The model has two assumptions:

1. Healthy people get sick at a rate proportional to the product of x and y (kxy).
2. Sick people die at a constant rate l.

We are given the following system of equations:

dx/dt = -kxy
dy/dt = kxy - ly
dz/dt = ly

Now, our goal is to reduce this third-order system to a first-order system that can be analyzed by our methods.

First, we notice that the total population N is constant except for deaths due to the epidemic, so we have:

N = x(t) + y(t) + z(t)

Since the total population remains constant (ignoring deaths due to the epidemic), we have:

dN/dt = dx/dt + dy/dt + dz/dt = 0

Substituting the given equations into the equation above, we get:

(-kxy) + (kxy - ly) + ly = 0

Notice that the terms involving kxy and ly cancel each other out. As a result, the system of equations is already reduced to a first-order system:

dx/dt = -kxy
dy/dt = kxy - ly

Now you can analyze this first-order system using the appropriate methods for first-order differential equations.

Learn more about :

first-order system : brainly.com/question/13260151

#SPJ11

The requreid local baseball team sold 99 adult tickets, 66 child tickets, and 22 senior tickets for the game.

Let A, C, and S represent the number of adult, child, and senior tickets sold, respectively.

A + C + S = 187 (the total number of tickets sold)

A:C = 3:2 (the ratio of adult to child tickets)

A:S = 9:2 (the ratio of adult to senior tickets)

We can use the ratios to write:

A = 3x (where x is a common factor)

C = 2x

S = (2/9)A = (2/9)(3x) = (2/3)x

Now we can substitute these expressions into the first equation:

A + C + S = 3x + 2x + (2/3)x = (9/3)x + (6/3)x + (2/3)x = 17x/3 = 187

x = 187(3/17) ≈ 33

Therefore, we can find the number of adults, children, and senior tickets sold by multiplying x by the appropriate ratio factors:

A = 3x ≈ 99

C = 2x ≈ 66

S = (2/3)x ≈ 22

Learn more about ratios here:

https://brainly.com/question/13419413

#SPJ1

Answer:

Anna will get 350 ZED

Step-by-step explanation:

since jenny is getting 3/8ths of the money, we can find how much money Jenny is getting and subtract that amount from the original total. to find this, take the original amount divided by the denominator then multiplied by the numerator.

for example: 560 / 8 = 70 × 3 = 210

560 - 210 = 350

350 is how much anna will get.

The pdf of Y, fy(y), is e^y * e^(-e^y) for y > 0.

(a) To find the probability density function (pdf) of Y = X^2, we can use the method of transformation. First, let's find the cumulative distribution function (CDF) of Y and then differentiate it to obtain the pdf.

To find the CDF of Y, we need to evaluate P(Y ≤ y), where y is a positive value.

Since Y = X^2, we can rewrite the inequality as X^2 ≤ y. Taking the square root of both sides (note that X and y are positive), we get X ≤ √y.

Using the pdf of X, fX(x) = e^(-x), we can write the probability as:

P(Y ≤ y) = P(X^2 ≤ y) = P(X ≤ √y) = ∫[0,√y] e^(-x) dx

Integrating the expression, we get:

P(Y ≤ y) = ∫[0,√y] e^(-x) dx = [-e^(-x)] [0,√y] = -(e^(-√y) - e^0) = 1 - e^(-√y)

To find the pdf of Y, fy(y), we differentiate the CDF with respect to y:

fy(y) = d/dy [1 - e^(-√y)] = 0.5 * e^(-√y) / √y

So, the pdf of Y, fy(y), is 0.5 * e^(-√y) / √y for y > 0.

(b) To find the pdf of Y = ln(X), we can again use the method of transformation.

First, let's find the CDF of Y:

P(Y ≤ y) = P(ln(X) ≤ y)

To simplify the inequality, we exponentiate both sides:

e^(ln(X)) ≤ e^y

X ≤ e^y

Using the pdf of X, fX(x) = e^(-x), we can write the probability as:

P(Y ≤ y) = P(X ≤ e^y) = ∫[0,e^y] e^(-x) dx

Integrating the expression, we get:

P(Y ≤ y) = ∫[0,e^y] e^(-x) dx = [-e^(-x)] [0,e^y] = -(e^(-e^y) - e^0) = 1 - e^(-e^y)

To find the pdf of Y, fy(y), we differentiate the CDF with respect to y:

fy(y) = d/dy [1 - e^(-e^y)] = e^y * e^(-e^y)

So, the pdf of Y, fy(y), is e^y * e^(-e^y) for y > 0.

To know more about probability density function refer here

https://brainly.com/question/31039386#

#SPJ11

Answer:

If the perimeter of a square picture frame is 12 inches, then each side of the square frame must be 3 inches long, since 4 x 3 = 12.

To find the area of the picture frame, we need to subtract the area of the picture from the area of the frame. Since the frame is a square with 3-inch sides, its area is 3 x 3 = 9 square inches.

However, we don't know the size of the picture, so we can't calculate its area directly. Instead, we can use the fact that the picture and the frame together form a larger square with sides that are 12 inches long (since the perimeter of the whole thing is 12 inches).

The area of this larger square is 12 x 12 = 144 square inches.

Since the area of the frame is 9 square inches, the area of the picture must be 144 - 9 = 135 square inches.

Therefore, the area of the picture frame is 9 square inches, and the area of the picture is 135 square inches.

If we assume that each plant needs one seed to grow, then the number of seeds needed will be equal to the number of plants. The farmer will need 36,000 corn seeds to plant one acre of land.

To find the number of seeds the farmer needs, we first need to determine the area of one acre. One acre is equal to 43,560 square feet. The field shown in the question may have a different area, but we'll assume it's one acre for the purposes of this problem.

Now, we know that the farmer wants to plant 36,000 corn plants per acre. If we assume that each plant needs one seed to grow, then the number of seeds needed will be equal to the number of plants.

Therefore, the farmer will need 36,000 corn seeds to plant one acre of land.we know that the farmer wants to plant 36,000 corn plants per acre.

Visit here to learn more about  number : https://brainly.com/question/17429689
#SPJ11

In the "algebraic-expression" 3/(x+3) = {2/(2(x+3))} - {1/(x-2)}., the value of "x" is 1/3.

The "Algebraic-Expression" is defined as a mathematical phrase which contain numbers, variables, and are joined by operations such as addition, subtraction, multiplication, and division.

The Algebraic expression is ⇒ 3/(x+3) = 2/(2(x+3)) - 1/(x-2),

We first simplify the expression on the "right-hand" side by finding a common denominator;

⇒ 2/(2(x+3)) - 1/(x-2),

⇒ (2(x-2) - 2(x+3))/(2(x+3)(x-2)),

⇒ (-10)/(2(x+3)(x-2))

⇒ -5/(x+3)(x-2),

We substituting this back into the original equation,

We get,

⇒ 3/(x+3) = -5/(x+3)(x-2),

To solve for x, we can cross-multiply;

⇒ 3(x-2) = -5,

⇒ 3x - 6 = -5,

⇒ 3x = 1,

⇒ x = 1/3.

Therefore, the value of x that satisfies the equation is x = 1/3.

Learn more about Expression here

https://brainly.com/question/17513857

#SPJ1

The given question is incomplete, the complete question is

Find the value of "x" in the algebraic expression 3/(x+3) = {2/(2(x+3))} - {1/(x-2)}.

To find the radius of convergence, we can use the ratio test. The ratio test states that if the limit of the absolute value of the ratio of consecutive terms of a series is L, then the series converges if L < 1 and diverges if L > 1.

The given series is:

∑ [(6 · 13 · 20 · ⋯ · (7n − 1)) / n!] * xn     (n starts from 1)

Using the ratio test, we take the absolute value of the ratio of the (n+1)th term to the nth term:

|(6 · 13 · 20 · ⋯ · (7(n+1) − 1)) / (n+1)! * x^(n+1)| / |(6 · 13 · 20 · ⋯ · (7n − 1)) / n! * xn|

Simplifying, we get:

|[(7n + 6) / (n+1)] * x| / |(7n − 1)|

Now, we take the limit as n approaches infinity:

lim(n→∞) |[(7n + 6) / (n+1)] * x| / |(7n − 1)|

Using the limit properties, we can simplify this expression further:

lim(n→∞) |(7 + 6/n) * x| / 7

Since the series involves x^n, we want the limit to be in terms of x. Therefore, we take the absolute value of x out of the limit:

|x| * lim(n→∞) |(7 + 6/n)| / 7

The term lim(n→∞) |(7 + 6/n)| / 7 is equal to 1, so we have:

|x| * 1

Therefore, the limit expression simplifies to:

|r|

Now, we know that for the series to converge, the absolute value of r must be less than 1. Thus, we have:

|r| < 1

This means that the radius of convergence is 1. Now, to find the interval of convergence, we need to check the endpoints of the interval.

When |x| = 1, the series becomes:

∑ [(6 · 13 · 20 · ⋯ · (7n − 1)) / n!] * x^n

Since the ratio test is inconclusive at the endpoints, we need to check for convergence or divergence separately.

For x = 1, the series becomes:

∑ [(6 · 13 · 20 · ⋯ · (7n − 1)) / n!]

This series is known as the "alternating harmonic series" and is convergent.

For x = -1, the series becomes:

∑ [(-1)^n * (6 · 13 · 20 · ⋯ · (7n − 1)) / n!]

This series also converges.

Therefore, the interval of convergence is -1 ≤ x ≤ 1.

In summary:

Radius of convergence (r) = 1

Interval of convergence (i) = -1 ≤ x ≤ 1

To know more about convergence refer here

https://brainly.com/question/29258536#

#SPJ11

The problem involves finding the area of a shaded region formed by two semicircles drawn on adjacent sides of a square. To solve this problem, we need to find the area of the square and subtract the area of the two semicircles from it.

To find the area of the square, we can simply square the length of its side which is given as 1 unit. So, the area of the square is 1 x 1 = 1 square unit.

Now, to find the area of the shaded region, we need to subtract the area of the two semicircles from the area of the square. The diameter of each semicircle is equal to the length of one of the sides of the square.

Thus, the radius of each semicircle is 1/2 units. Therefore, the area of one semicircle is (π/2) x (1/2)² = π/8 square units. Since there are two semicircles, the total area of the shaded region is (2 x π/8) = π/4 square units. Finally, we can subtract this area from the area of the square to obtain the area of the shaded region which is 1 - π/4 = (4-π)/4 square units.

To learn more about semicircles, click here:

brainly.com/question/29140521

#SPJ1`1

Using the intermediate value theorem and Rolle's theorem, we showed that the function [tex]f(x) = ln(x^{2} ) - x + 2[/tex] has exactly one zero on the interval [4,6], which is x = 2.

To show that the function [tex]f(x) = ln(x^{2} ) - x + 2[/tex] has exactly one zero on the interval [4,6], we need to use the intermediate value theorem and Rolle's theorem.

First, we can find that the function is continuous and differentiable for x > 0. Taking the derivative of f(x), we get [tex]f'(x) = (2/x) - 1[/tex]. Setting f'(x) = 0, we get x = 2.

Now, let's evaluate f(4) and f(6). We have [tex]f(4) = ln(16) - 4 + 2 = ln(16) - 2[/tex]and [tex]f(6) = ln(36) - 6 + 2 = ln(36) - 4[/tex]. Using a calculator, we find that f(4) < 0 and f(6) > 0.

By the intermediate value theorem, since f(x) is continuous on [4,6] and takes on values of opposite signs at the endpoints, there exists at least one zero of f(x) on the interval.

Finally, to show that there is only one zero, we use Rolle's theorem. Since f(x) is differentiable on (4,6) and has a zero on this interval, there must exist at least one point c in (4,6) such that f'(c) = 0.

From earlier, we know that f'(x) = (2/x) - 1, so we have [tex]f'(c) = (2/c) - 1 = 0[/tex], which implies c = 2. Therefore, the only zero of f(x) on [4,6] is x = 2.

In summary, using the intermediate value theorem and Rolle's theorem, we showed that the function [tex]f(x) = ln(x^{2} ) - x + 2[/tex] has exactly one zero on the interval [4,6], which is x = 2.

To know more about Rolle's theorem refer here:

https://brainly.com/question/31331894#

#SPJ11

The vector space to ||f||=4, ||g||=[tex]\sqrt{13}[/tex].

The angle θf,g between f(x) and g(x) is given by θf,g ≈ 1.893 radians.

Using the given inner product in the vector space C0[0,1], we can find the norms and angle between f(x) and g(x) as follows:

The norm of f(x) is given by:

||f|| = [tex]\sqrt{( < f, f > )[/tex] = [tex]\sqrt{(\int10 f(x)^2 dx)[/tex]

Substituting f(x) = [tex]5x^2 - 9[/tex], we get:

||f|| = [tex]\sqrt{(\int10 (5x^2 - 9)^2 dx)[/tex]

= [tex]\sqrt{(\int10 25x^4 - 90x^2 + 81 dx)[/tex]

= [tex]\sqrt{( [25/5]x^5 - [90/3]x^3 + [81]x |_0^1)[/tex]

= [tex]\sqrt{(25/5 - 90/3 + 81)[/tex]

= [tex]\sqrt{(16)[/tex]

= 4

Similarly, the norm of g(x) is given by:

||g|| = [tex]\sqrt{( < g, g >[/tex]) = [tex]\sqrt{(\int10 g(x)^2 dx)[/tex]

Substituting g(x) = -9x + 2, we get:

||g|| = [tex]\sqrt{(\int10 (-9x + 2)^2 dx)[/tex]

= [tex]\sqrt{(\int10 81x^2 - 36x + 4 dx)[/tex]

= [tex]\sqrt{( [81/3]x^3 - [36/2]x^2 + [4]x |_0^1)[/tex]

=[tex]\sqrt{(81/3 - 36/2 + 4)[/tex]

= [tex]\sqrt(13)[/tex]

The inner product of f(x) and g(x) is given by:

<f, g> = ∫10 f(x) g(x) dx

Substituting f(x) = [tex]5x^2 - 9[/tex] and g(x) = -9x + 2, we get:

<f, g> = [tex]\int10 (5x^2 - 9)(-9x + 2) dx[/tex]

= [tex]\int10 -45x^3 + 10x^2 + 81x - 18 dx[/tex]

[tex]= [-45/4]x^4 + [10/3]x^3 + [81/2]x^2 - 18x |_0^1[/tex]

= -45/4 + 10/3 + 81/2 - 18

= -9/4

The angle between f(x) and g(x) is given by:

cos(θf,g) = <f, g> / (||f|| ||g||)

[tex]= (-9/4) / (4 \times \sqrt(13))[/tex]

[tex]= -9 / (16 \sqrt(13))[/tex]

Using a calculator, we can find that:

cos(θf,g) ≈ -0.3112

The angle θf,g between f(x) and g(x) is given by:

[tex]\theta f,g \approx cos^{-1}(-0.3112)[/tex]

θf,g ≈ 1.893 radians

For similar questions on vector space

https://brainly.com/question/13258990

#SPJ11

The curve r = 5 + 4 cos(theta) the area enclosed by the curve is 32.5π square units.

The curve you've provided is given by the polar equation r = 5 + 4 cos(theta). This curve represents a limaçon, a specific type of polar curve.

To find the area enclosed by the curve, you can use the polar area formula: Area = (1/2) ∫[r^2 d(theta)], where the integral is evaluated over the range of theta for one full rotation.

In this case, r = 5 + 4 cos(theta), and theta ranges from 0 to 2π: Area = (1/2) ∫[(5 + 4 cos(theta))^2 d(theta)] from 0 to 2π. Evaluating this integral, we get: Area = (1/2) * (65π) = 32.5π square units.

Visit here to learn more about Integral:

brainly.com/question/30094386

#SPJ11

Check the picture below, so the parabola looks more or less like so, with a positive "p" distance of 2, with the vertex half-way between the directrix and the focus point.

[tex]\textit{horizontal parabola vertex form with focus point distance} \\\\ 4p(x- h)=(y- k)^2 \qquad \begin{cases} \stackrel{vertex}{(h,k)}\qquad \stackrel{focus~point}{(h+p,k)}\qquad \stackrel{directrix}{x=h-p}\\\\ p=\textit{distance from vertex to }\\ \qquad \textit{ focus or directrix}\\\\ \stackrel{p~is~negative}{op ens~\supset}\qquad \stackrel{p~is~positive}{op ens~\subset} \end{cases} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\begin{cases} h=8\\ k=7\\ p=2 \end{cases}\implies 4(2)(~~x-8~~) = (~~y-7~~)^2 \implies 8(x-8)=(y-7)^2 \\\\\\ x-8=\cfrac{1}{8}(y-7)^2\implies {\Large \begin{array}{llll} x=\cfrac{1}{8}(y-7)^2+8 \end{array}}[/tex]