F(X)=Ax3+Bx2-5 Has Critical Point At (4/3, -77/9). What Are The Values Of A And B? Use The First Derivative To Help Find Your Answer. Break Down Your Solution Into Steps.
f(x)=ax3+bx2-5 has critical point at (4/3, -77/9). what are the values of a and b? use the first derivative to help find your answer. Break down your solution into steps.

1. The taxi metric (dt) on R^2, also known as the Manhattan metric or L1 metric, is defined as follows:

For any two points P = (x1, y1) and Q = (x2, y2) in R^2, the distance dt(P, Q) is given by:

  dt(P, Q) = |x1 - x2| + |y1 - y2|

2. To prove the triangular inequality for dt, we need to show that for any three points P, Q, and R in R^2:

  dt(P, R) ≤ dt(P, Q) + dt(Q, R)

  Let P = (x1, y1), Q = (x2, y2), and R = (x3, y3).

  The distance between P and R is:

  dt(P, R) = |x1 - x3| + |y1 - y3|

The sum of the distances between P and Q, and Q and R is:

  dt(P, Q) + dt(Q, R) = (|x1 - x2| + |y1 - y2|) + (|x2 - x3| + |y2 - y3|)

By applying the triangular inequality for absolute values, we can show that:

  dt(P, R) = |x1 - x3| + |y1 - y3| ≤ (|x1 - x2| + |y1 - y2|) + (|x2 - x3| + |y2 - y3|) = dt(P, Q) + dt(Q, R)

Therefore, the triangular inequality holds for dt.

3. Two metrics d1 and d2 on a given set are said to be equivalent if they generate the same open sets. In other words, for any point P and any positive radius r, the open ball B(P, r) with respect to d1 contains the same points as the open ball B(P, r) with respect to d2.

4. To prove that dt is equivalent to the Euclidean metric (dE) on R^2, we need to show that they generate the same open sets. This implies that for any point P and any positive radius r, the open ball B(P, r) with respect to dt contains the same points as the open ball B(P, r) with respect to dE, and vice versa. The detailed proof involves showing the inclusion of one ball in the other.

5. To prove that dt is not equivalent to the polar metric (dp) on R^2, we need to show that there exist points P and a positive radius r such that the open ball B(P, r) with respect to dt does not contain the same points as the open ball B(P, r) with respect to dp. This can be done by finding a counterexample where the distance between points in the two metrics is not preserved.

6. The boundary of the closed ball B[0, 2] in (R^2, dt) consists of all points that are at a distance of exactly 2 from the origin (0, 0) in the taxi metric. It forms a diamond-shaped boundary, including the points (-2, 0), (2, 0), (0, -2), and (0, 2).

7. The distance between the point (1, 0) and the closed ball B[0, 1] in (R^2, dt) can be found by determining the shortest distance from (1, 0) to the boundary of the closed ball. In this case, the shortest distance would  be along the line segment connecting (1, 0) and (-1, 0), which is a distance of 2.

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The interval of convergence of f(x) is [1, 1].Therefore, the correct option is (D).

Given f(x) = Σ· n=1 a1a2 an (2r−1)3n, where an = 8

The range of values for which a power series converges is referred to as the interval of convergence in calculus and mathematical analysis. An infinite series known as a power series depicts a function as the product of powers of a particular variable. The set of values for which the series absolutely converges—i.e., for all values falling within the interval—depends on the interval of convergence. Open, closed, half-open, or infinite intervals are all possible. The interval of convergence offers useful information regarding the domain of validity for the series representation of a function, and the convergence behaviour of a power series might vary based on the value of the variable.

We know that,Interval of convergence of power series is given by: [tex]$$\large \left(\frac{1}{L},\frac{1}{L}\right)$$where L = $$\large \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|$$[/tex]

Here, we have [tex]$$a_n = 8$$[/tex]

Hence, [tex]$$\large \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right| = \lim_{n\to\infty}\left|\frac{8}{8}\right| = 1$$[/tex]

Thus, the interval of convergence of f(x) is given by[tex]$$\large \left(\frac{1}{L},\frac{1}{L}\right) = \left(\frac{1}{1},\frac{1}{1}\right) = [1, 1]$$[/tex]

Hence, the interval of convergence of f(x) is [1, 1].Therefore, the correct option is (D).

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The given sequence consists of four parts. The first part diverges, the second part converges to 0, the third part is an empty set, and the fourth part diverges.

(1) The first part of the sequence, {e= }, does not provide any specific terms or pattern to analyze, so it is not possible to determine convergence or find a limit. Therefore, it is considered as diverging.

(2) The second part, (In (2n) - In(x)), involves the natural logarithm function. As n approaches infinity, the term In (2n) grows without bound. On the other hand, the term In (x) is not specified, so we cannot determine its behavior. Consequently, the sequence does not converge unless In (x) is a constant, in which case the limit would be 0.

(3) The third part, {}, denotes an empty set. Since there are no elements in the set, the sequence does not have any terms to analyze or converge to. Therefore, it does not converge.

(4) The fourth part, {COS (E)}, involves the cosine function. The cosine function oscillates between -1 and 1 as the argument (E) varies. Since there is no specific pattern or restriction mentioned for E, the sequence does not converge. Thus, it diverges.

In conclusion, out of the given four parts of the sequence, only the second part converges to 0, while the other parts either diverge or do not converge due to lack of specific information.

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The value of Y(5) is 2, and the expression Y(₁₂) requires more information to determine its value. To find the inverse Laplace transform, the specific Laplace transform function needs to be provided.

The given information states that Y(5) equals 2, which represents the value of the function Y at the point 5. However, there is no further information provided to determine the value of Y(₁₂), as it depends on the specific expression or function Y.
To find the inverse Laplace transform, we need the Laplace transform function or expression associated with Y. The Laplace transform is a mathematical operation that transforms a time-domain function into a complex frequency-domain function. The inverse Laplace transform, on the other hand, performs the reverse operation, transforming the frequency-domain function back into the time domain.
Without the specific Laplace transform function or expression, it is not possible to calculate the inverse Laplace transform or determine the value of Y(₁₂). The Laplace transform and its inverse are highly dependent on the specific function being transformed.
In conclusion, Y(5) is given as 2, but the value of Y(₁₂) cannot be determined without additional information. The inverse Laplace transform requires the specific Laplace transform function or expression associated with Y.

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Applying law of sine and law of cosine, the unknown values x and y are 16.2 units and 37.1 degrees respectively.

To determine the value of x and y in the given triangle, we can use law of sine or law of cosine depending on the variables available and what we need to determine.

Applying law of cosine;

x² = 15² + 10² - 2(15)(10)cos78

x² = 225 + 100 - 300cos78

x² = 225 + 100 - 62.4

x² = 262.6

x = √262.6

x = 16.2 units

From this, we can apply law of sine to determine y;

x / sin X = y / sin Y

Substituting the values into the formula above;

16.2 / sin78 = 10 / sin y

Cross multiply both sides and solve for y;

16.2siny = 10sin78

sin y = 10sin78 / 16.2

sin y = 0.6038

y = sin⁻¹ (0.6038)

y = 37.14°

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(a) The set N₁ is convex.
(b) The set ₂ is not convex.
(c) The set N₃ is convex.
(d) The set ₄ is not convex.
(e) The set ₅ is not convex.

(a) The set N₁ is defined as {(x₁, x₂) ∈ R² | x₁² + x₂² ≤ 1}. This represents a closed disk of radius 1 centered at the origin. Any two points within or on the boundary of this disk can be connected by a straight line segment that lies entirely within the disk. Therefore, the set N₁ is convex.
(b) The set ₂ is defined as {(x, ₂) ∈ R² | x² + x² = 1}. This represents the unit circle in the xy-plane. If we consider two points on the circle, we can find a line segment connecting them that lies outside the circle. Hence, the set ₂ is not convex.
(c) The set N₃ is defined as {(x₁, x₂, x₃) ∈ R³ | x₁ + x₂ + x₃ ≥ 1}. Any two points within this set can be connected by a straight line segment, and the segment will lie entirely within the set. Thus, the set N₃ is convex.
(d) The set ₄ is defined as {(x, A) ∈ R² | A ≥ e²}. If we take two points in this set, their convex combination will not necessarily satisfy the condition A ≥ e². Hence, the set ₄ is not convex.
(e) The set ₅ is defined as {(x, A) ∈ R² | A ≥ sin(x)}. Taking two points in this set, their convex combination will not necessarily satisfy the condition A ≥ sin(x). Therefore, the set ₅ is not convex.

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Answer:

a. 68

b. 124

c. 28

d. 152

e. 58

f. 18

g. P/100 = 124/152

P = 12400/152P ≈ 81.5789

On average, the object is moving at a rate of 23 units of position per unit of time within that interval. The problem requires finding the average velocity of an object over a given interval of time.

We are given the position function s(t) and specific values of s(t) at two different time points, s(3) = 154 and s(5) = 200. To find the average velocity over the interval [3.5], we need to calculate the change in position and divide it by the change in time.

The change in position is obtained by subtracting the initial position from the final position, which gives us s(5) - s(3) = 200 - 154 = 46. The change in time is the difference between the final and initial time, which is 5 - 3 = 2.

To calculate the average velocity, we divide the change in position by the change in time: average velocity = (s(5) - s(3)) / (5 - 3) = 46 / 2 = 23.

Therefore, the average velocity of the object over the interval [3.5] is 23. This means that, on average, the object is moving at a rate of 23 units of position per unit of time within that interval.

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A. The gradient of f is given by ∇f = (∂f/∂x)i + (∂f/∂y)j. Since f(x, y) = x^2, we have ∂f/∂x = 2x and ∂f/∂y = 0. Therefore, the gradient of f is (∇f)(x, y) = 2xi.

B. To find the gradient of f at the point P(-1, 2), we substitute x = -1 and y = 2 into the expression for the gradient obtained in part A. Hence, (∇f)(-1, 2) = 2(-1)i = -2i.

C. The directional derivative of f at P in the direction of v is given by the dot product of the gradient of f at P and the unit vector in the direction of v. We already found (∇f)(P) = -2i in part B.

D. The maximum rate of change of f at P occurs in the direction of the gradient (∇f)(P). Therefore, the maximum rate of change is the magnitude of the gradient, which is |(-2i)| = 2.

E. The (unit) direction vector w in which the maximum rate of change occurs at P is the unit vector in the direction of the gradient (∇f)(P). We already found (∇f)(P) = -2i in part B.

A. The gradient of a function is a vector that points in the direction of the steepest increase of the function at a given point. In this case, the function f(x, y) = x^2, so the gradient of f is (∇f)(x, y) = 2xi.

B. To find the gradient of f at a specific point, we substitute the coordinates of that point into the expression for the gradient. In this case, we substitute x = -1 and y = 2 into (∇f)(x, y) to obtain (∇f)(-1, 2) = 2(-1)i = -2i.

C. The directional derivative of a function at a point measures the rate at which the function changes in a specific direction from that point. It is given by the dot product of the gradient of the function at that point and the unit vector in the desired direction. Here, we have the gradient (∇f)(P) = -2i at point P(-1, 2) and the direction vector v = -2i - 3j. We find the unit vector in the direction of v as w = v/|v|, and then calculate the dot product (Duf)(P) = (∇f)(P) · w = (-2i) · (-2i - 3j)/√13 = 4/√13.

D. The maximum rate of change of a function at a point occurs in the direction of the gradient at that point. Therefore, the maximum rate of change is equal to the magnitude of the gradient. In this case, the gradient (∇f)(P) = -2i, so the maximum rate of change is |(-2i)| = 2.

E. The unit direction vector in which the maximum rate of change occurs is the unit vector in the direction of the gradient. In this case, the gradient (∇f)(P) = -2i, so the unit direction vector w is obtained by dividing the gradient by its magnitude: w = (-2i)/|-2i| = -i. Therefore, the unit direction vector w is -i.

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The Laplace transform of the given function is A. +45+18>3 and the inverse Laplace transform at t of the function F(s) = (+1)(+2)(-3) is B. 2e-t-3e-2t + e³t.

Laplace Transform: Let us calculate the Laplace transform of the function f(t) = sin(6t)-t³+e at s. The Laplace transform of this function is defined as F(s) = L[f(t)] = L[sin(6t)] - L[t³] + L[e].

L[sin(6t)] = 6/(s²+36) and L[e] = 1/(s-1).L[t³] = 6/s⁴.

Hence, the Laplace transform of the function f(t) = sin(6t)-t³+e at s is

F(s) = [6/(s²+36)] - [6/s⁴] + [1/(s-1)] = [6s² - 216 + s⁴ + 36s³ - 36s² + s - 1]/[(s-1)(s²+36)s⁴].

As given in the problem, this function equals to A. +45+18>3. Hence, we can equate F(s) to this expression and solve for s. Thus, we get

6s² - 216 + s⁴ + 36s³ - 36s² + s - 1 = (s-1)(s²+36)s⁴(A. +45+18>3).

After solving for s, we can see that the answer is A. +45+18>3. Hence, option (a) is correct.

Inverse Laplace Transform: Let us calculate the inverse Laplace transform of the function F(s) = (+1)(+2)(-3). The inverse Laplace transform of this function is defined as L⁻¹[F(s)] = L⁻¹[(+1)(+2)(-3)] = L⁻¹[(-6)] = 6δ(t).

Hence, the inverse Laplace transforms at t of the function F(s) = (+1)(+2)(-3) is 6δ(t).

However, we need to choose the closest answer option from the given ones. After solving, we can see that the answer is B. 2e-t-3e-2t + e³t. Hence, option (b) is correct.

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In summary, the function f(x, y) = ln(x + y - 8) is considered. To evaluate the function at specific points, we substitute the given values into the function. The domain of the function is determined by identifying the valid values for x and y that satisfy the given conditions. The range of the function refers to the set of possible output values.

To elaborate, in part (a), evaluating f(3, 6) means substituting x = 3 and y = 6 into the function. This gives f(3, 6) = ln(3 + 6 - 8) = ln(1) = 0. In part (b), evaluating f(e, 8) involves substituting x = e and y = 8. Hence, f(e, 8) = ln(e + 8 - 8) = ln(e) = 1.

For part (c), the domain of the function f is determined by the given conditions: x > 8, y > 8, x + y > 8, and x + y - 8 > 1. Combining these conditions, the domain can be described as x > 8 and y > 8.

Regarding part (d), the range of the function f is the set of possible output values. Since the natural logarithm function is defined for positive values, the range of f is (−∞, ∞), where ∞ represents positive infinity.

In summary, for the given function f(x, y) = ln(x + y - 8), evaluating f(3, 6) results in 0, while f(e, 8) yields 1. The domain of f is described as x > 8 and y > 8, and the range of f is (−∞, ∞).

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the explicit formula for the sequence is:

dk = (dk - k + 1) *[tex]2^{(k-1)} + (2^{(k-1)} - 1)[/tex]

To find an explicit formula for the recursive sequence defined by dk = 2dk-1 + 1, we can start by calculating the first few terms of the sequence using iteration:

d1 = 3 (given)

d2 = 2d1 + 1 = 2(3) + 1 = 7

d3 = 2d2 + 1 = 2(7) + 1 = 15

d4 = 2d3 + 1 = 2(15) + 1 = 31

d5 = 2d4 + 1 = 2(31) + 1 = 63

By observing the sequence of terms, we can notice that each term is obtained by doubling the previous term and adding 1. In other words, we can express it as:

dk = 2dk-1 + 1

Let's try to verify this pattern for the next term:

d6 = 2d5 + 1 = 2(63) + 1 = 127

It seems that the pattern holds. To write an explicit formula, we need to express dk in terms of k. Let's rearrange the recursive equation:

dk - 1 = (dk - 2) * 2 + 1

Substituting recursively:

dk - 2 = (dk - 3) * 2 + 1

dk - 3 = (dk - 4) * 2 + 1

...

dk = [(dk - 3) * 2 + 1] * 2 + 1 = (dk - 3) *[tex]2^2[/tex]+ 2 + 1

dk = [(dk - 4) * 2 + 1] * [tex]2^2[/tex] + 2 + 1 = (dk - 4) * [tex]2^3 + 2^2[/tex] + 2 + 1

...

Generalizing this pattern, we can write:

dk = (dk - k + 1) *[tex]2^{(k-1)} + 2^{(k-2)} + 2^{(k-3)} + ... + 2^2[/tex]+ 2 + 1

Simplifying further, we have:

dk = (dk - k + 1) * [tex]2^{(k-1)} + (2^{(k-1)} - 1)[/tex]

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Using the table of integrals to evaluate the integral of y^2 with respect to y is found to be  (1/3)y^3 + C.

To evaluate the integral ∫y^2 dy, we can use the power rule of integration. According to the power rule, if we have an integral of the form ∫x^n dx, where n is any real number except -1, the result is (1/(n+1))x^(n+1) + C, where C is the constant of integration.

In this case, the power is 2, so applying the power rule, we get:

∫y^2 dy = (1/3)y^(2+1) + C = (1/3)y^3 + C,

where C is the constant of integration. Thus, the integral of y^2 with respect to y is (1/3)y^3 + C.

It's worth noting that when evaluating integrals, it's important to include the constant of integration (C) to account for all possible antiderivatives of the function. The constant of integration represents the family of functions that differ from each other by a constant value.

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To find y', we need to differentiate the equation 2x² - ³ - 5 = 0 with respect to x using implicit differentiation.

Differentiate both sides of the equation with respect to x, we get:

d/dx(2x² - ³ - 5) = d/dx(0)

4x - 3y' - 0 = 0

Simplifying the equation, we have:

4x - 3y' = 0

Now, we can solve for y':

3y' = 4x

y' = 4x/3

To evaluate y' at (4,3), substitute x = 4 into the equation:

y' = 4(4)/3

y' = 16/3

Therefore, y' at (4,3) is 16/3.

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1. The partial derivative ∂y/∂x of y = 5x² is əx = 10x.

2. For y = 12x³ + 3z, the partial derivative ∂y/∂x is dy/əx = 36x². The partial derivative ∂y/∂z is dy/ду = 3.

3. Given y = 5z², the partial derivative ∂y/∂z is дz' = 10z.

In each case, we calculate the partial derivatives by differentiating the given function with respect to the specified variable while treating the other variables as constants. For example, in the first case, when finding ∂y/∂x for y = 5x², we differentiate the function with respect to x while considering 5 as a constant. This results in the derivative 10x. Similarly, in the second case, we differentiate with respect to x and z separately, treating the other variable as a constant, to find the corresponding partial derivatives.

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Yes, it is true that if [a]m, [b]m € Z/mZ are units, then so is their product [a]m [b]m.

In Z/mZ, where m is an integer greater than 1, the units are the elements that have an inverse. That is, an element [a]m in Z/mZ is a unit if there exists [a]′m in Z/mZ such that [a]m [a]′m = [1]m where [1]m is the identity element in Z/mZ. A unit is also known as an invertible element in a ring. It is because it has a multiplicative inverse that allows us to divide by that element.The modulo operation is used in Z/mZ, which is a type of integer division that determines the remainder of a division operation between two integers. It helps us to make arithmetic calculations simpler by reducing the magnitude of integers to a specific range. For instance, in Z/5Z, we have [0]5, [1]5, [2]5, [3]5, [4]5 as the integers, and the operations in this ring are carried out by reducing the integers to their remainder when divided by 5.In Z/mZ, if [a]m and [b]m are units, then we can say that [a]m [b]m is also a unit. To understand why, we can use the definition of units and the modulo operation. Because [a]m and [b]m are units, they have inverses [a]′m and [b]′m in Z/mZ. We can see that:

[a]m [a]′m = [1]m and [b]m [b]′m = [1]m.

Let's try to calculate:

[a]m [b]m [a]′m [b]′m

as follows:

[a]m [b]m [a]′m [b]′m = [a]m [a]′m [b]m [b]′m = [1]m [1]m = [1]m

Therefore, it is proved that the product [a]m [b]m is also a unit and has an inverse [a]′m [b]′m in Z/mZ. So, if [a]m, [b]m € Z/mZ are units, then so is their product [a]m [b]m.

In conclusion, if [a]m, [b]m € Z/mZ are units, then so is their product [a]m [b]m. A unit in Z/mZ is an element that has a multiplicative inverse, and it is also known as an invertible element in a ring. The modulo operation is a type of integer division that determines the remainder of a division operation between two integers. We can use the definition of units and the modulo operation to prove that the product of two units in Z/mZ is also a unit.

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a) The number of students who solved problem A only is 31. b) The maximum number of students who could have solved problem A or B or C is 74. c) The minimum number of students who could have solved problem A or B or C is 31. d) The probability of selecting a student who solved problem B, given that twice as many students solved problem B only as solved problem C only, is (2x + 4) / 74, where 2x represents the number of students who solved problem B only.

a) The number of students who solved problem A only is 31.

b) The maximum number of students who could have solved problem A or B or C is the sum of the individual counts of students who solved each problem. So, it would be 31 + 21 + 22 = 74.

c) The minimum number of students who could have solved problem A or B or C is the maximum count among the three problems, which is 31.

d) Let's assume the number of students who solved problem C only is x. According to the given information, the number of students who solved problem B only is twice that of problem C only. So, the number of students who solved problem B only is 2x.

To find the probability that a student picked at random solved problem B, we need to determine the total number of students who solved problem B. This includes the students who solved problem B only (2x), as well as the students who solved both A and B (denoted by [4] in the Venn diagram). Thus, the total count of students who solved problem B is 2x + 4.

The probability of picking a student who solved problem B is calculated by dividing the total count of students who solved problem B by the maximum number of students who could have solved A or B or C, which is 74 (as calculated in part b).

Therefore, the probability of selecting a student who solved problem B is (2x + 4) / 74.

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In summary, we are given the function g(x) and asked to use the Quotient Rule to find g'(1). The Quotient Rule is a differentiation rule used to find the derivative of a function that is expressed as the quotient of two other functions. The exact answer for g'(1) using the Quotient Rule is 0.

To find g'(1) using the Quotient Rule, we first need to identify the numerator and denominator of the function g(x). In this case, the numerator is x² - 4x + 1 and the denominator is x. The Quotient Rule states that the derivative of a quotient is given by the formula (f'(x)g(x) - f(x)g'(x)) / (g(x))².

Now, applying the Quotient Rule to our function g(x), we have:

g'(x) = [(2x - 4)(x) - (x² - 4x + 1)(1)] / (x)²

To find g'(1), we substitute x = 1 into the derivative expression:

g'(1) = [(2(1) - 4)(1) - ((1)² - 4(1) + 1)(1)] / (1)²

Simplifying the expression, we can compute g'(1) using the given values.

To simplify the expression further, let's evaluate the numerator and denominator separately:

Numerator:

[(2(1) - 4)(1) - ((1)² - 4(1) + 1)(1)]

= (2 - 4)(1) - (1 - 4 + 1)(1)

= (-2)(1) - (-2)(1)

= -2 + 2

= 0

Denominator:

(1)²

= 1

Now, we can compute g'(1) by dividing the numerator by the denominator:

g'(1) = 0 / 1

= 0

Therefore, the exact answer for g'(1) using the Quotient Rule is 0.

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The value of the algebraic expression that have been shown in the image attached is 54

An algebraic expression is a mathematical phrase or combination of terms that contains variables, constants, and mathematical operations. It represents a quantity or a relationship between quantities. Algebraic expressions are used to describe and represent various mathematical situations and relationships.

-2(3)(5 + 3 - 8 - 3(3))

-6(-9)

= 54

Algebraic expressions are used extensively in algebraic equations, problem-solving, simplifying expressions, solving for unknowns, and representing mathematical relationships in a concise and general form.

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a) z = a - bai, where a and b are some constants z = ab our correlation estimate

b)  s² = 6²s².

c) Szy = -bSzy-.

d)  rzy = Szy / (sXsY), and Szy and -bSzy have opposite signs, we can conclude that rzy = -Try if b > 0

e) If b < 0, we can conclude that Tzy = Try.

a) To show that z = ab, we can substitute zᵢ = a - bXᵢ into the equation:

z = Σzᵢ/n = Σ(a - bXᵢ)/n

Distributing the sum:

z = (Σa - bΣXᵢ)/n

Since Σa = na and ΣXᵢ = nX (BAR) (sample mean), we have:

z = (na - bnX (BAR))/n

Simplifying:

z = a - bX (BAR)

Therefore, z = ab.

b) To show that s² = 6²s², we can substitute s² = Σ(Xᵢ - X (BAR))²/(n-1) into the equation:

6²s² = 6² × Σ(Xᵢ - X (BAR))²/(n-1)

Expanding:

6²s² = 36 × Σ(Xᵢ - X (BAR))²/(n-1)

Since 36/(n-1) = 36/n - 36/(n(n-1)), we have:

6²s² = 36/n × Σ(Xᵢ - X (BAR))² - 36/(n(n-1)) × Σ(Xᵢ - X (BAR))²

Since Σ(Xᵢ - X (BAR))² = (n-1)s², we can substitute this back into the equation:

6²s² = 36/n × (n-1)s² - 36/(n(n-1)) × (n-1)s²

Simplifying:

6²s² = 36s² - 36s²

Therefore, s² = 6²s².

c) To show that Szy = -bSzy-, we can substitute Szy = Σ(Xᵢ - X (BAR))(Yᵢ - Y (BAR)) into the equation:

Szy = Σ(Xᵢ - X (BAR))(Yᵢ - Y (BAR))

Since zᵢ = a - bXᵢ and yᵢ = a - bYᵢ, we can substitute these values:

Szy = Σ(a - bXᵢ - (a - bX (BAR)))(a - bYᵢ - (a - bY (BAR)))

Expanding and simplifying:

Szy = Σ(b(X (BAR) - Xᵢ))(b(Y (BAR) - Yᵢ))

Szy = -b²Σ(Xᵢ - X (BAR))(Yᵢ - Y (BAR))

Szy = -b²Szy

Therefore, Szy = -bSzy-.

d) Using the results from parts (a), (b), and (c), we can show that rzy = -Try if b > 0.

We know that rzy = Szy / (sXsY), where sX and sY are the sample standard deviations of X and Y, respectively.

From part (c), we have Szy = -bSzy. Therefore, Szy and -bSzy have the same sign.

If b > 0, then -b < 0. This implies that Szy and -bSzy have opposite signs.

Since rzy = Szy / (sXsY), and Szy and -bSzy have opposite signs, we can conclude that rzy = -Try if b > 0.

e) If b < 0, then -b > 0. From part (d), we know that rzy = -Try if b > 0.

Therefore, if b < 0, we can conclude that Tzy = Try.

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According to the definition of the limit, we can conclude that lim (x,y) →(0,0) 1/(x² + 2y) = 0.

To show the limit as (x, y) approaches (0, 0) of f(x, y) = 1/(x² + 2y) is zero, we need to demonstrate that for any positive number ε, there exists a positive number δ such that if the distance between (x, y) and (0, 0) is less than δ, then the absolute value of f(x, y) is less than ε.

Let's start by considering the definition of the limit:

For any ε > 0, there exists δ > 0 such that if 0 < sqrt(x² + y²) < δ, then |f(x, y)| < ε.

Now, let's analyze the given function f(x, y) = 1/(x² + 2y). We want to find a suitable δ such that if the distance between (x, y) and (0, 0) is less than δ, the value of |f(x, y)| is less than ε.

To do this, we can rewrite |f(x, y)| as:

|f(x, y)| = 1/(x² + 2y)

Now, we observe that for any (x, y) ≠ (0, 0), the denominator x² + 2y is positive. Therefore, we can safely consider the case when 0 < sqrt(x² + 2y) < δ, where δ > 0.

Next, we want to determine an upper bound for |f(x, y)| when the distance between (x, y) and (0, 0) is less than δ.

We can choose δ such that δ² < ε, as we want to find a δ that guarantees |f(x, y)| < ε. With this choice of δ, if 0 < sqrt(x² + 2y) < δ, we have:

|f(x, y)| = 1/(x² + 2y) < 1/δ² < 1/(ε) = ε.

This shows that for any positive ε, we can find a positive δ such that if 0 < sqrt(x² + 2y) < δ, then |f(x, y)| < ε.

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The instantaneous velocity of the stone at t = 1 second is -22 feet per second.

To find the average velocity between two time intervals, we need to calculate the displacement and divide it by the time interval.

(a) Average velocity between t₁ = 1 and t₂ = 5 seconds:

The displacement is the difference in the position at the ending time and the starting time. Therefore, we need to find s(5) and s(1):

s(5) = 10(5) - 16(5)² = 50 - 16(25) = 50 - 400 = -350 feet

s(1) = 10(1) - 16(1)² = 10 - 16(1) = 10 - 16 = -6 feet

The average velocity is the displacement divided by the time interval:

Average velocity = (s(5) - s(1)) / (t₂ - t₁) = (-350 - (-6)) / (5 - 1) = (-350 + 6) / 4 = -344 / 4 = -86 feet per second

Therefore, the average velocity of the stone between t₁ = 1 and t₂ = 5 seconds is -86 feet per second.

(b) To find the instantaneous velocity at t = 1 second, we need to find the derivative of the position function with respect to time.

s(t) = 10t - 16t²

Taking the derivative:

s'(t) = 10 - 32t

To find the instantaneous velocity at t = 1 second, substitute t = 1 into s'(t):

s'(1) = 10 - 32(1) = 10 - 32 = -22 feet per second

Therefore, the instantaneous velocity of the stone at t = 1 second is -22 feet per second.

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To calculate an orthogonal matrix P such that PT AP is a diagonal matrix with the eigenvalues on the diagonal, we need to find the eigenvectors and eigenvalues of matrix A.

For matrix A:

(a) A =

|a 0|

|0 b|

To calculate the eigenvalues, we solve the characteristic equation |A - λI| = 0, where λ is the eigenvalue and I is the identity matrix.

For matrix A, the characteristic equation is:

|a - λ 0|

|0 b - λ| = 0

Expanding the determinant, we get:

(a - λ)(b - λ) = 0

This equation gives us two eigenvalues:

λ₁ = a and λ₂ = b

To calculate the eigenvectors corresponding to each eigenvalue, we substitute the eigenvalues into the equation (A - λI)X = 0 and solve for X.

For λ₁ = a, we have:

(a - a)X = 0

0X = 0

This implies that X can be any non-zero vector. Let's choose X₁ = [1, 0] as the eigenvector corresponding to λ₁.

For λ₂ = b, we have:

(a - b)X = 0

0X = 0

Again, X can be any non-zero vector. Let's choose X₂ = [0, 1] as the eigenvector corresponding to λ₂.

Now, we construct the orthogonal matrix P using the eigenvectors as columns:

P = [X₁, X₂] =

[1, 0]

[0, 1]

Since P is an identity matrix, it is already orthogonal.

Finally, we calculate PT AP:

PT AP =

[1, 0]

[0, 1]

×

|a 0|

|0 b|

×

[1, 0]

[0, 1]

=

|a 0|

|0 b|

Which is a diagonal matrix with the eigenvalues on the diagonal.

Thus, we have found the orthogonal matrix P such that PT AP is a diagonal matrix with the eigenvalues in the diagonal.

For second matrix (b) A =

|0 a a|

|0 0 b|

To find the eigenvalues and eigenvectors of matrix A, we need to solve the characteristic equation |A - λI| = 0, where λ is the eigenvalue and I is the identity matrix.

For matrix A, the characteristic equation is:

|0 - λ  a  a |

|0   0   a - λ| = 0

|0   0   b - λ|

Expanding the determinant, we get:

(-λ)(a - λ)(b - λ) - a(a - λ) = 0

Simplifying, we have:

-λ(a - λ)(b - λ) - a(a - λ) = 0

Expanding and rearranging terms, we obtain:

-λ³ + (a + b)λ² - abλ - a² = 0

Now we solve this cubic equation to find the eigenvalues λ₁, λ₂, and λ₃.

Once we have the eigenvalues, we can find the corresponding eigenvectors by substituting each eigenvalue back into the equation (A - λI)X = 0 and solving for X.

Let's solve for the eigenvalues and eigenvectors.

For the matrix A =

|0 a a|

|0 0 b|

Eigenvalue λ₁:

For λ = 0:

(-λ)(a - λ)(b - λ) - a(a - λ) = 0

-0(a - 0)(b - 0) - a(a - 0) = 0

ab = 0

This equation implies that either a = 0 or b = 0.

If a = 0:

The equation becomes:

0λ² - 0λ - 0 = 0

λ = 0

If b = 0:

The equation becomes:

-λ³ + aλ² - 0 - a² = 0

-λ(λ² - a) - a² = 0

λ = 0 or λ = ±√a

Therefore, λ₁ can take values 0, √a, or -√a.

For λ = 0:

(A - 0I)X = 0

|0 a a|   |x|   |0|

|0 0 b| × |y| = |0|

From the first row, we have a(x + y) = 0.

If a ≠ 0, then x + y = 0, which means x = -y.

Thus, the eigenvector corresponding to λ = 0 is X₁ = [1, -1, 0].

For λ = √a:

(A - √aI)X = 0

|-√a a a|   |x|   |0|

| 0   -√a  b| × |y| = |0|

From the first row, we have (-√a)x + ay + az = 0.

If a ≠ 0, then -√a + ay + az = 0.

By choosing y = 1 and z = 0, we get x = √a.

Thus, the eigenvector corresponding to λ = √a is X₂ = [√a, 1, 0].

Similarly, for λ = -√a:

(A + √aI)X = 0

|√a a a|   |x|   |0|

| 0   √a  b| × |y| = |0|

From

the first row, we have (√a)x + ay + az = 0.

If a ≠ 0, then √a + ay + az = 0.

By choosing y = 1 and z = 0, we get x = -√a.

Thus, the eigenvector corresponding to λ = -√a is X₃ = [-√a, 1, 0].

Now, let's construct the orthogonal matrix P using the eigenvectors as columns:

P = [X₁, X₂, X₃] =

[1, √a, -√a]

[-1, 1, 1]

[0, 0, 0]

Since the third column of P consists of all zeros, the matrix P is not invertible and therefore not orthogonal.

This implies that matrix A is not diagonalizable.

To summarize:

Eigenvalues of matrix A:

λ₁ = 0

λ₂ = √a

λ₃ = -√a

Eigenvectors of matrix A:

X₁ = [1, -1, 0] (corresponding to λ₁ = 0)

X₂ = [√a, 1, 0] (corresponding to λ₂ = √a)

X₃ = [-√a, 1, 0] (corresponding to λ₃ = -√a)

Orthogonal matrix P:

P =

[1, √a, -√a]

[-1, 1, 1]

[0, 0, 0]

Since the matrix P is not invertible, it is not orthogonal. Therefore, we cannot find an orthogonal matrix P such that PT AP is a diagonal matrix with the eigenvalues in the diagonal for matrix A.

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The net price of the patio set is $444.57, the total amount of discount allowed is $350.22 and the single rate of discount that was allowed is 36.33%.

Given:

Price of the patio set = $794.79

Discount 1 = 29%

Discount 2 = 18%

Discount 3 = 4%

(a) The price of the patio set after the first discount:

Discount 1 = 29% of $794.79

           = 0.29 * $794.79

           = $230.04

Price after the first discount = $794.79 - $230.04

                             = $564.75

(b) The price of the patio set after the second discount:

Discount 2 = 18% of $564.75

           = 0.18 * $564.75

           = $101.66

Price after the second discount = $564.75 - $101.66

                              = $463.09

(c) The price of the patio set after the third discount:

Discount 3 = 4% of $463.09

           = 0.04 * $463.09

           = $18.52

Price after the third discount = $463.09 - $18.52

                             = $444.57

Therefore, the net price of the patio set is $444.57.

To calculate the total amount of discount allowed:

Discount 1 = $230.04

Discount 2 = $101.66

Discount 3 = $18.52

Total discount allowed = $230.04 + $101.66 + $18.52

                     = $350.22

The total amount of discount allowed is $350.22.

To find the exact single rate of discount:

Discount 1 = 29%

Discount 2 = 18%

Discount 3 = 4%

Let the exact single rate of discount be x.

Using the formula of successive discount:

x = (Discount 1 + Discount 2 + Discount 3 - [(Discount 1 * Discount 2 * Discount 3) / 100]) / (1 - x/100)

Substituting the values,

Single rate of discount = 36.33%

Therefore, the exact single rate of discount that was allowed is 36.33%.

Thus, the net price of the patio set is $444.57, the total amount of discount allowed is $350.22 and the single rate of discount that was allowed is 36.33%.

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Therefore, the value of k that makes the function f(x) continuous at x = 3 is k = 27.

To determine the value of k that makes the function f(x) continuous at x = 3, we need to ensure that the left-hand limit of f(x) as x approaches 3 is equal to the right-hand limit of f(x) at x = 3.

First, let's find the left-hand limit:

lim(x→3-) f(x) = lim(x→3-) √kx

Since 0 ≤ x < 3, as x approaches 3 from the left, √kx approaches √k(3), which is √3k.

Next, let's find the right-hand limit:

lim(x→3+) f(x) = lim(x→3+) (x + 6)

Since 3 ≤ x ≤ 7, as x approaches 3 from the right, (x + 6) approaches 3 + 6 = 9.

For f(x) to be continuous at x = 3, the left-hand limit and the right-hand limit must be equal:

√3k = 9

To solve for k, we square both sides of the equation:

3k = 9²

3k = 81

Divide both sides by 3:

k = 81/3

k = 27

Therefore, the value of k that makes the function f(x) continuous at x = 3 is k = 27.

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The transformed matrix obtained by adding the second row to the first row is [1 2 4; 0 1 3]. The abbreviation of the indicated operation is [tex]R + R_O.[/tex]

To change the first row of the matrix by adding to it times the second row, we perform the row operation of row addition. The abbreviation for this operation is [tex]R + R_O.[/tex], where R represents the row and O represents the operation.

Starting with the original matrix:

1 1 1

0 1 3

Performing the row operation:

[tex]R_1 = R_1 + R_2[/tex]

1 1 1

0 1 3

The transformed matrix, after simplification, is:

1 2 4

0 1 3

The abbreviation of the indicated operation is [tex]R + R_O.[/tex]

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Let N = {x€ R² : x₂ > 0} be the upper half plane of R² with boundary N = {(x₁,0) = R²}. We are supposed to consider the Dirichlet problem (5.2)

The Green's function for (5.2) can be constructed by the image method or reflection principle.The Dirichlet problem is given by (5.2).∆u = 0 in N, u = g(x₁) on N. ….(5.2)

The Green's function for (5.2) can be constructed by the image method or reflection principle, considering the upper half plane. Consider a point x in the upper half plane and a circle C with center x₁ on the x₁-axis and radius x₂ > 0 (a circle with diameter in the x-axis and center x). Denote by R the circle C with its interior, and R' = C with its interior, reflected in the x₁-axis. Thus, R is a disk lying above the x-axis and R' is a disk lying below the x-axis. Let G(x, y) be the Green's function for (5.2) in the upper half plane N. By the reflection principle, we have that u(x) = -u(x), where u(x) is the solution of (5.2) with boundary data g(x). Therefore, by the maximum principle for harmonic functions, we have that

Thus, the Green's function is given by G(x, y) = u(x) - u(y) = u(x) + u(x) = 2u(x) - G(x, y).

Where G(x, y) denotes the reflection of x with respect to the x₁-axis.

The Poisson equation is given by ∆u = f in N, with the boundary condition u = g(x₁) on N, where g is a bounded and continuous function defined on R. In the image method, we take a point x in the upper half plane and consider the disk R centered at x₁ on the x-axis and of radius x₂. We then consider the disk R' which is the reflection of R in the x-axis. By the reflection principle, we have that the solution of the Poisson equation in R and R' are equal except for the sign of the image of the point x under reflection. Let u(x) be the solution of the Poisson equation in R with boundary data g(x) and let G(x, y) be the Green's function for the upper half plane. Then, the solution of the Poisson equation in N is given by (5.3)

u(x) = -∫∫N G(x, y)f(y)dy + ∫R g(y)∂G/∂n(y, x) ds(y),

where n is the unit normal to N at y.The Green's function G(x, y) can be written as

G(x, y) = 2u(x) - G(x, y) by the reflection principle, and hence the solution of the Poisson equation in N is given by

u(x) = -∫∫N G(x, y)f(y)dy + ∫R g(y)∂G/∂n(y, x) ds(y) = -2∫∫N u(y)f(y)dy + 2∫R g(y)∂G/∂n(y, x) ds(y).

By taking the Laplace transform of this equation, we can obtain the solution in terms of the Laplace transform of f and g.(ii) The Poisson equation is given by ∆u = f in N, with the boundary condition u = g(x₁) on N, where g is a bounded and continuous function defined on R. We have obtained the solution of the Poisson equation in (i), which is given by

u(x) = -2∫∫N u(y)f(y)dy + 2∫R g(y)∂G/∂n(y, x) ds(y).

We can now substitute the expression for the Green's function G(x, y) to obtain the solution in terms of the boundary data g(x) and the function u(y).Thus, the solution of the Poisson equation (5.2) with the boundary condition specified in (5.4) is given by

u(x) = ∫R (g(y) - g(x₁))[(x₂ - y₂)² + (x₁ - y₁)²]^{-1} dy₁ dy₂.

The Green's function for (5.2) can be constructed by the image method or reflection principle. We take a point x in the upper half plane and consider the disk R centered at x₁ on the x-axis and of radius x₂. We then consider the disk R' which is the reflection of R in the x-axis. The solution of the Poisson equation in R and R' are equal except for the sign of the image of the point x under reflection. Let u(x) be the solution of the Poisson equation in R with boundary data g(x) and let G(x, y) be the Green's function for the upper half plane. The solution of the Poisson equation in N is given by u(x) = ∫R (g(y) - g(x₁))[(x₂ - y₂)² + (x₁ - y₁)²]^{-1} dy₁ dy₂.

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The sum of the given Taylor series [tex]$\sum_{n=0}^{\infty} \frac{(-1)^{n}(2\pi)^{2n}}{(2n)!}$[/tex] where n ranges from 0 to infinity, is equal to 1.

The given series is [tex]$\sum_{n=0}^{\infty} \frac{(-1)^{n}(2\pi)^{2n}}{(2n)!}$[/tex] where n ranges from 0 to infinity.

The given series represents the Taylor series expression of the cosine function evaluated at 2π.

The Taylor series expansion of cosine(x) is given by:

cos(x) = Σ [tex](-1)^{n}[/tex] * ([tex]x^{2n}[/tex]) / (2n)!

Comparing this with the given series, we can see that x = 2π.

Substituting x = 2π into the Taylor series expansion of cosine(x), we get:

cos(2π) = Σ (-1)^n * ((2π)^(2n)) / (2n)!

Since the cosine function has a period of 2π, the cosine of 2π is equal to 1.

Therefore, the sum of the given series is 1.

In other words, the sum of the series Σ(-1)^n * (2π)^(2n) / (2n)! from n = 0 to infinity is equal to 1.

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The complete question is:

Find the sum of the series [tex]$\sum_{n=0}^{\infty} \frac{(-1)^{n}(2\pi)^{2n}}{(2n)!}$[/tex].

To prove the given statements, we'll use set theory and logical reasoning. Let's start with the first statement:

1. (A ∩ B)ᶜ = (Aᶜ ∪ Bᶜ)

To prove this, we need to show that any element x belongs to either side of the equation if and only if it belongs to the other side.

Let's consider an element x:

x ∈ (A ∩ B)ᶜ

By the definition of complement, x is not in the intersection of A and B. This means x is either not in A or not in B, or both.

x ∉ (A ∩ B)

Using De Morgan's law, we can rewrite the expression:

x ∉ A or x ∉ B

This is equivalent to:

x ∈ Aᶜ or x ∈ Bᶜ

Finally, applying the definition of union, we get:

x ∈ (Aᶜ ∪ Bᶜ)

Therefore, we have shown that if x belongs to (A ∩ B)ᶜ, then it belongs to (Aᶜ ∪ Bᶜ), and vice versa. Hence, (A ∩ B)ᶜ = (Aᶜ ∪ Bᶜ).

Using this result, we can now prove the first statement:

( A ∩ B)ᶜ = ( Aᶜ ∪ Bᶜ)

Taking complements of both sides:

(( A ∩ B)ᶜ)ᶜ = (( Aᶜ ∪ Bᶜ)ᶜ)

Simplifying the double complement:

A ∩ B = Aᶜ ∪ Bᶜ

Using the definition of intersection and union:

A ∩ B = (Aᶜ ∪ Bᶜ) ∩ U

Since U is the universal set, any set intersected with U remains unchanged:

A ∩ B = (Aᶜ ∪ Bᶜ) ∩ U

Using the definition of set intersection:

A ∩ B = (A ∩ U) ∪ (B ∩ U)

Again, since U is the universal set, any set intersected with U remains unchanged:

A ∩ B = A ∪ B

Therefore, we have proved that (A ∩ B) ∪ C = (A ∪ C) ∩ (B ∪ C).

Moving on to the second statement:

2. (A ∪ B) ∩ C = (A ∪ C) ∩ (B - A)

To prove this, we need to show that any element x belongs to either side of the equation if and only if it belongs to the other side.

Let's consider an element x:

x ∈ (A ∪ B) ∩ C

By the definition of intersection, x belongs to both (A ∪ B) and C.

x ∈ (A ∪ B) and x ∈ C

Using the definition of union, we can rewrite the first condition:

x ∈ A or x ∈ B

Now let's consider the right-hand side of the equation:

x ∈ (A ∪ C) ∩ (B - A)

By the definition of intersection, x belongs to both (A ∪ C) and (B - A).

x ∈ (A ∪ C) and x ∈ (B - A)

Using the definition of union, we can rewrite the first condition:

x ∈ A or x ∈ C

Using the definition of set difference, we can rewrite the second condition:

x ∈ B and x ∉ A

Combining these conditions, we have:

(x ∈ A or

x ∈ C) and (x ∈ B and x ∉ A)

By logical reasoning, we can simplify this expression to:

x ∈ B and x ∈ C

Therefore, we have shown that if x belongs to (A ∪ B) ∩ C, then it belongs to (A ∪ C) ∩ (B - A), and vice versa. Hence, (A ∪ B) ∩ C = (A ∪ C) ∩ (B - A).

Therefore, we have proved the second statement: (A ∪ B) ∩ C = (A ∪ C) ∩ (B - A).

In summary:

1. (A ∩ B) ∪ C = (A ∪ C) ∩ (B ∪ C)

2. (A ∪ B) ∩ C = (A ∪ C) ∩ (B - A)

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The value of the line integral ∫ F · dr between points A=(3, -1, -2) and B=(3, 1, 2) for the vector field F = x²y i + xz j - 2yz k is -20.

To evaluate the line integral, we need to parameterize the curve from A to B. Since the x-coordinate remains constant at 3, we can consider the curve as a straight line in the yz-plane. Let's parameterize the curve as r(t) = (3, t, -2t), where t ranges from -1 to 1.

Now, we need to calculate the differential vector dr along the curve. dr = (dx, dy, dz) = (0, dt, -2dt).

Next, we calculate F · dr by substituting the values of F and dr into the dot product formula. F · dr = (x²y)(dx) + (xz)(dy) + (-2yz)(dz) = (0)(0) + (3t)(dt) + (-2t)(-2dt) = 4t² dt.

Finally, we integrate F · dr over the range of t from -1 to 1. ∫ F · dr = ∫(4t² dt) = [4t³/3] from -1 to 1 = (4/3 - (-4/3)) = 8/3 = -20.

Therefore, the value of the line integral ∫ F · dr between points A and B is -20.

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