Let u = [] { [ ] [ ] }; and let S = 2 2 2 Determine the values of r for which v is in the span of S.

a) Definition of Limit: Let f(x) be defined on an open interval containing c, except possibly at c itself.

We say that the limit of f(x) as x approaches c is L and write: 

[tex]limx→cf(x)=L[/tex]

if for every number ε>0 there exists a corresponding number δ>0 such that |f(x)-L|<ε whenever 0<|x-c|<δ.

Let's prove that f(x) = 2x³ - 1 is continuous at the point x = 2; that is, [tex]lim-2 2x³ - 1[/tex]= 15.

Let [tex]limx→2(2x³-1)[/tex]= L than for ε > 0, there exists δ > 0 such that0 < |x - 2| < δ implies

|(2x³ - 1) - 15| < ε

|2x³ - 16| < ε

|2(x³ - 8)| < ε

|x - 2||x² + 2x + 4| < ε

(|x - 2|)(x² + 2x + 4) < ε

It can be proved that δ can be made equal to the minimum of 1 and ε/13.

Then for

0 < |x - 2| < δ

|x² + 2x + 4| < 13

|x - 2| < ε

Thus, [tex]limx→2(2x³-1)[/tex]= 15.

b) (i) Definition of Contractions: Let f: [a, b] → [a, b] be a function.

We say f is a contraction if there exists a constant 0 ≤ k < 1 such that for any x, y ∈ [a, b],

|f(x) - f(y)| ≤ k |x - y| and |k|< 1.

(ii) We need to prove that h(x) = cos(sin x) is continuous at every point x = x0; that is, [tex]limx→x0[/tex] | cos(sin x)| = | cos(sin(x0)).

First, we prove that cos(x) is a contraction function on the interval [0, π].

Let f(x) = cos(x) be defined on the interval [0, π].

Since cos(x) is continuous and differentiable on the interval, its derivative -sin(x) is continuous on the interval.

Using the Mean Value Theorem, for all x, y ∈ [0, π], we have cos (x) - cos(y) = -sin(c) (x - y),

where c is between x and y.

Then,

|cos(x) - cos(y)| = |sin(c)|

|x - y| ≤ 1 |x - y|.

Therefore, cos(x) is a contraction on the interval [0, π].

Now, we need to show that h(x) = cos(sin x) is also a contraction function.

Since sin x takes values between -1 and 1, we have -1 ≤ sin(x) ≤ 1.

On the interval [-1, 1], cos(x) is a contraction, with a contraction constant of k = 1.

Therefore, h(x) = cos(sin x) is also a contraction function on the interval [0, π].

Hence, by the Contraction Mapping Theorem, h(x) = cos(sin x) is continuous at every point x = x0; that is,

[tex]limx→x0 | cos(sin x)| = | cos(sin(x0)).[/tex]

(c) (i) Given a common deviation bound of 0.00025 for both x - 2 and y - 2, we need to prove that x + y is accurate to +2 by 3 decimal places.

Let x - 2 = δ and y - 2 = ε.

Then,

x + y - 4 = δ + ε.

So,

|x + y - 4| ≤ |δ| + |ε|

≤ 0.00025 + 0.00025

= 0.0005.

Therefore, x + y is accurate to +2 by 3 decimal places.(ii) The mapping diagram is shown below:

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graph G is planar, while graph H is not planar according to Kuratowski's theorem.

Graph G:

Based on the provided graph G, it can be observed that it does not contain any edge crossings. Therefore, it can be embedded in a plane without any issues, making it a planar graph.

Graph H:

To determine whether graph H is planar or not, we need to apply Kuratowski's theorem. According to Kuratowski's theorem, a graph is non-planar if and only if it contains a subgraph that is a subdivision of K₅ (the complete graph on five vertices) or K₃,₃ (the complete bipartite graph on six vertices).

Upon examining graph H, it can be observed that it contains a subgraph that is a subdivision of K₅, specifically the subgraph formed by the five vertices in the center. This violates Kuratowski's theorem, indicating that graph H is non-planar.

Therefore, graph G is planar, while graph H is not planar according to Kuratowski's theorem.

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The maximum value of z is 54, and it occurs when x₁ = 11 and x₂ = 14.

To solve the given linear programming problem using the two-stage method, we'll first set up the initial tableau and then perform iterations to find the optimal solution. Let's begin:

Step 1: Set up the initial tableau

We introduce slack variables, s₁, s₂, s₃, and an auxiliary variable, A, to convert the problem into a standard form.

The initial set of equations and inequalities are as follows:

x₁ + 2x₂ + s₁ = 18

x₁ + 3x₂ - s₂ + A = 12

2x₁ + 2x₂ + s₃ = 28

We'll convert these equations into standard form as follows:

x₁ + 2x₂ + s₁ = 18

x₁ + 3x₂ - s₂ + A = 12

2x₁ + 2x₂ + s₃ = 28

Now, we'll construct the initial tableau:

diff

Copy code

---------------------------

| C_B | x₁ | x₂ | s₁ | s₂ | s₃ |  A |

---------------------------

|  -Z | -5 | -10|  0 |  0 |  0 |  0 |

---------------------------

|  s₁ |  1 |  2 |  1 |  0 |  0 |  0 |

---------------------------

|  A  |  1 |  3 |  0 | -1 |  1 | 12 |

---------------------------

|  s₃ |  2 |  2 |  0 |  0 |  1 | 28 |

---------------------------

Step 2: Perform iterations

We'll perform iterations to find the optimal solution by applying the simplex method.

Iteration 1:

We choose s₃ as the entering variable and x₂ as the leaving variable. Pivot on the element 2 in the s₃ column.

diff

Copy code

---------------------------

| C_B | x₁ | x₂ | s₁ | s₂ | s₃ |  A |

---------------------------

|  -Z | -5 |  0 |  0 |  0 | -2 | 56 |

---------------------------

|  s₁ |  1 |  0 |  1 |  0 | -1 | 10 |

---------------------------

|  A  |  1 |  1 |  0 | -1 |  1 | 14 |

---------------------------

|  x₂ |  1 |  1 |  0 |  0 | 1/2| 14 |

---------------------------

Iteration 2:

We choose s₂ as the entering variable and s₃ as the leaving variable. Pivot on the element -1 in the s₂ column.

diff

Copy code

---------------------------

| C_B | x₁ | x₂ | s₁ | s₂ | s₃ |  A |

---------------------------

|  -Z | -5 |  0 |  0 |  1 |  0 | 54 |

---------------------------

|  s₁ |  1 |  0 |  1 |  1 |  0 | 11 |

---------------------------

|  A  |  1 |  1 |  0 |  1 |  0 | 13 |

---------------------------

|  x₂ |  1 |  1 |  0 |  0 | 1/2| 14 |

---------------------------

Iteration 3:

No negative coefficients exist in the objective row, indicating the optimal solution has been reached.

Step 3: Read the optimal solution

From the final tableau, we can read the optimal values of the decision variables and the objective function.

x₁ = 11

x₂ = 14

Z = 54

Therefore, the maximum value of z is 54, and it occurs when x₁ = 11 and x₂ = 14.

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The expression for the function whose graph is the line segment joining the points (1, -2) and (5, 12) is f(x) = (7/2)x - 3/2.

To find an expression for the function whose graph is the line segment joining the points (1, -2) and (5, 12), we can use the point-slope form of a linear equation.

The point-slope form is given by: y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope of the line.

Given the points (1, -2) and (5, 12), we can calculate the slope:

m = (y₂ - y₁) / (x₂ - x₁) = (12 - (-2)) / (5 - 1) = 14 / 4 = 7/2

Now, let's choose one of the points, say (1, -2), and plug it into the point-slope form:

y - (-2) = (7/2)(x - 1)

Simplifying the equation, we get:

y + 2 = (7/2)(x - 1)

Next, we can rewrite the equation in slope-intercept form (y = mx + b):

y = (7/2)x - 7/2 + 2

y = (7/2)x - 7/2 + 4/2

y = (7/2)x - 3/2

Therefore, the expression for the function whose graph is the line segment joining the points (1, -2) and (5, 12) is f(x) = (7/2)x - 3/2.

To find the domain of the function, we need to consider the values of x for which the function is defined. Since f(x) is a linear function, it is defined for all real numbers. Therefore, the domain of the function f(x) = (7/2)x - 3/2 is (-∞, +∞) or (-∞, ∞) in interval notation, indicating that it is defined for all values of x.

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(i) The bipartite graph for A₁ = {1,2,3}, A₂ = {2, 3, 4}, A₃ = {1} has a system of distinct representatives.

(ii) The bipartite graph for A₁ = {1, 2, 3, 4, 5}, A₂ = {1,3}, A₃ = {1,3}, A₄ = {1, 2, 3, 4, 5}, A₅ = {1,3} does not have a system of distinct representatives.

To construct the associated bipartite graph for each family of sets, we can represent the sets as vertices and draw edges between them based on their intersection.

(i) Family of sets: A₁ = {1,2,3}, A₂ = {2, 3, 4}, A₃ = {1}

The bipartite graph for this family of sets would have two sets of vertices, one representing the sets A₁, A₂, A₃ and the other representing the elements 1, 2, 3, 4.

The vertices for sets A₁, A₂, A₃ would be connected to the vertices representing the elements that are contained in those sets. The edges would be drawn based on the intersection of the sets.

To find a system of distinct representatives, we need to find a matching where each vertex on the left side (sets) is connected to a unique vertex on the right side (elements). In this case, a system of distinct representatives exists since every set has at least one unique element.

(ii) Family of sets: A₁ = {1, 2, 3, 4, 5}, A₂ = {1,3}, A₃ = {1,3}, A₄ = {1, 2, 3, 4, 5}, A₅ = {1,3}

The bipartite graph for this family of sets would have two sets of vertices, one representing the sets A₁, A₂, A₃, A₄, A₅ and the other representing the elements 1, 2, 3, 4, 5.

The vertices for sets A₁, A₂, A₃, A₄, A₅ would be connected to the vertices representing the elements that are contained in those sets.

To find a system of distinct representatives, we need to find a matching where each vertex on the left side (sets) is connected to a unique vertex on the right side (elements). In this case, a system of distinct representatives does not exist because element 1 is shared by all the sets A₁, A₂, A₃, A₄, A₅, violating the requirement of distinctness.

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, f'(12) = 126. And f'(16) = 1/8. To find the derivative of the function f(x) = 5x² + 6x + 12, we can apply the power rule and the constant rule of differentiation.
Taking the derivative with respect to x, we have:
f'(x) = d/dx (5x²) + d/dx (6x) + d/dx (12)
      = 10x + 6 + 0
      = 10x + 6
To find f'(12), we substitute x = 12 into the derivative:
f'(12) = 10(12) + 6
      = 120 + 6
      = 126

Therefore, f'(12) = 126.

For the function f(x) = √x, we can use the power rule and chain rule to find its derivative.
Taking the derivative with respect to x, we have:
f'(x) = d/dx (√x)
      = (1/2) * (x)^(-1/2)
      = 1 / (2√x)
To find f'(16), we substitute x = 16 into the derivative:
f'(16) = 1 / (2√16)
      = 1 / (2 * 4)
      = 1/8
Therefore, f'(16) = 1/8.



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The second derivative of the function g(x) = 5√x + e³x ln(x) is [tex]-5/(4x^(3/2)) + (6 + 2e³x)/x.[/tex]

To find the second derivative, we first need to find the first derivative of g(x) and then differentiate it again. Let's start by finding the first derivative:

g'(x) = d/dx (5√x + e³x ln(x))

Using the power rule and the chain rule, we can differentiate each term separately:

[tex]g'(x) = 5(1/2)(x)^(-1/2) + e³x (ln(x))' + e³x (ln(x))'[/tex]

Simplifying further, we have:

g'(x) = 5/(2√x) + e³x (1/x) + e³x (1/x)

Next, to find the second derivative, we differentiate g'(x) with respect to x:

g''(x) = d/dx (5/(2√x) + e³x (1/x) + e³x (1/x))

Using the power rule and the product rule, we can differentiate each term:

g''(x) = -5/(4x^(3/2)) + e³x (1/x)' + e³x (1/x)' + e³x (1/x) + e³x (1/x)

Simplifying further, we have:

[tex]g''(x) = -5/(4x^(3/2)) + 2e³x/x + 2e³x/x + e³x/x + e³x/x[/tex]

Combining like terms, the second derivative of g(x) is:

[tex]g''(x) = -5/(4x^(3/2)) + (6 + 2e³x)/x[/tex]

So, the second derivative of the function g(x) = 5√x + e³x ln(x) is [tex]-5/(4x^(3/2)) + (6 + 2e³x)/x.[/tex]

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The two sets A' n B and O are not equal based on the given Venn diagrams.

In the given problem, we are asked to determine whether the two sets A' n B and O are equal using Venn diagrams. The notation A' represents the complement of set A, and n denotes the intersection of two sets. O represents the null set or the empty set.

To analyze the equality of the sets, we examine the Venn diagrams representing A' n B and O. In a Venn diagram, the intersection of two sets is represented by the overlapping region between them. However, in this case, since O represents the empty set, there is no overlap between the two sets. Thus, we can conclude that A' n B and O are not equal.

The absence of any common elements between A' n B and O indicates that the sets do not have any shared elements. In other words, the intersection of A' n B is empty, which aligns with the definition of the null set O. Therefore, based on the Venn diagrams, we can confidently state that the two sets A' n B and O are not equal.

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Consider a vector a ∈ ℝⁿ, where n ≥ 1 is an integer, and let s, r ∈ ℝ with 0 < s. In this context, the question seems to be asking for additional information or a specific task to be performed.

The given information defines a vector a ∈ ℝⁿ, where ℝⁿ represents the n-dimensional Euclidean space. Additionally, the conditions specify that s and r are real numbers with s > 0. However, the question does not indicate what needs to be done with this information. It could be asking for calculations, properties, or relationships involving the vector a or the real numbers s and r.

To provide a more specific answer, please provide additional details or clarify the task or question you would like assistance with regarding the vector a, or the real numbers s and r. This will allow for a more tailored and meaningful response.

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The equation function of cosine with an amplitude of 4, a period of 2, and a point at (0,2) is y = 4cos(2πx) + 2.

The general form of a cosine function is y = A cos(Bx - C) + D, where A represents the amplitude, B is related to the period, C indicates any phase shift, and D represents a vertical shift.

In this case, the given amplitude is 4, which means the graph will oscillate between -4 and 4 units from its centerline. The period is 2, which indicates that the function completes one full cycle over a horizontal distance of 2 units.

To incorporate the given point (0,2), we know that when x = 0, the corresponding y-value should be 2. Since the cosine function is at its maximum at x = 0, the vertical shift D is 2 units above the centerline.

Using these values, the equation function becomes y = 4cos(2πx) + 2, where 4 represents the amplitude, 2π/2 simplifies to π in the argument of cosine, and 2 is the vertical shift. This equation satisfies the given conditions of the cosine function.

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(a) The contrapositive of the given statement is: "If n is an odd number, then (n+1)² is an even number." (b) To prove the contrapositive, assume n is an odd number (n = 2k + 1), then (n+1)² = 4(k+1)², which is divisible by 2, proving the contrapositive. (c) The original statement is true because squaring an odd number results in an odd number, and if (n+1)² is odd, it implies n+1 is odd, thus n must be even due to the sum of an even and odd number being odd.

(a) The contrapositive of the given statement is: "If n is an odd number, then (n+1)² is an even number."

(b) To prove the contrapositive statement, we assume that n is an odd number. Therefore, we can write n as 2k + 1, where k is an integer.

Substituting this value of n into the expression (n+1)², we get:

(n+1)² = (2k + 1 + 1)² = (2k + 2)² = 4(k+1)²

Since (k+1) is an integer, we can represent it as m, where m = k + 1. Therefore, the expression becomes:

4(k+1)² = 4m² = 2(2m²)

We can see that the expression 2(2m²) is an even number since it is divisible by 2. Thus, we have proven that if n is an odd number, then (n+1)² is an even number, which is the contrapositive statement.

(c) The original statement is true because it is based on the properties of even and odd numbers. If we square an odd number, the result is always an odd number. On the other hand, if we square an even number, the result is always an even number.

In the given statement, if (n+1)² is an odd number, it means that n+1 is an odd number (since the square of an odd number is odd). If n+1 is odd, then n must be even because adding an odd number to an even number always results in an odd number. Therefore, the original statement holds true based on the properties of even and odd numbers.

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The integral ∫(rπ/4) tan²(θ) sec²(θ) dθ can be evaluated by applying trigonometric identities and integration techniques.

To evaluate the integral, we can start by using the trigonometric identity tan²(θ) + 1 = sec²(θ). Rearranging this equation gives tan²(θ) = sec²(θ) - 1.

Substituting this identity into the integral, we have ∫(rπ/4) (sec²(θ) - 1) sec²(θ) dθ.

Simplifying further, we get ∫(rπ/4) (sec⁴(θ) - sec²(θ)) dθ.

Now, we can integrate each term separately. The integral of sec⁴(θ) is (1/3)tan(θ)sec²(θ) + (2/3)θ + C, and the integral of sec²(θ) is tan(θ) + C, where C is the constant of integration.

Thus, the final solution to the integral is ((1/3)tan(θ)sec²(θ) + (2/3)θ - tan(θ)) evaluated over the range π/4.

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The row operations are Add 1 times row 1 to row 2,

R2 = R2 + R1 for the first two matrices and Add 1 times row 1 to row 2,

R2 = R2 + R1, R3 = R3 + R1 for the last matrix.

For each of the given matrices, we are to give the row operation that has the same result as left-multiplication by the given matrix.

Below is the solution to the given problem:

1000 0300

Add 1 times row 1 to row 2, which can be represented as

R2 = R2 + R1.0 0 1 0 0 0 1 0 0 becomes

0 0 1 1 0 0 1 0 0 00001 1050

Add 1 times row 1 to row 2, which can be represented as

R2 = R2 + R1.

0 0 1 0 0001 1 0 0 0 becomes

0001 1 0 1 0 00001 0 0 1 0 0

Add 1 times row 1 to row 2, which can be represented as

R2 = R2 + R1.

0 0 10 0 0 1 1 0 0 becomes

0 0 1 0 0 10 1 0 0 00001 0 0 10

The row operations are Add 1 times row 1 to row 2,

R2 = R2 + R1 for the first two matrices and Add 1 times row 1 to row 2,

R2 = R2 + R1, R3 = R3 + R1 for the last matrix.

Note: There are several ways to do this problem but this is one of the simplest and quickest ways.

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The general solution of the differential equation is y(x) = C₁x¹/² + 1/4 cos(x).

Given a differential equation:

4x²y" + y = x³/² sin(x)

and y(x) = x¹/²

We need to verify whether the given function is the solution of the differential equation or not.

Therefore, we will substitute the value of y(x) in the differential equation.

Let's start by finding the first and second derivatives of y(x) which will be used further.

y(x) = x¹/²y'(x)

= d/dx (x¹/²)y'(x)

= (1/2)x^(-1/2)y''(x)

= d/dx[(1/2)x^(-1/2)]y''(x)

= (-1/4)x^(-3/2)

Therefore, substituting y(x) and y" (x) in the differential equation:

4x² (-1/4)x^(-3/2) + x¹/² = x³/²sin(x)

Thus, the above equation simplifies as:-

x^(-1) + x¹/² = x³/²sin(x)

Here, we can see that the given function is not a solution of the differential equation.

However, we can find the particular solution of the differential equation by the method of variation of parameters.

Where we write the given equation in the standard form:

y'' + [1/4x⁴]y = [x¹/² sin(x)]/4x⁻²

On comparing with the standard form:

y'' + p(x) y' + q(x) y = g(x) where p(x) = 0, q(x) = 1/4x⁴ and g(x) = [x¹/² sin(x)]/4x⁻²

Now, let's calculate the Wronskian for the differential equation as:

W(y₁, y₂) = | y₁    y₂ |-1/4x²               1/4x²-1/2W(y₁, y₂)

= 1/4x³

The particular solution y₂(x) will be:

y₂(x) = -y₁(x) ∫[g(x) y₁(x)] / W(x) dx

Substituting the given value in the above equation, we get:

y₂(x) = -x¹/² ∫[x¹/² sin(x)] / (x³/² 4x⁻²) dx

y₂(x) = -1/4 ∫sin(x) dx

y₂(x) = -1/4 [-cos(x)] + C₁

y₂(x) = 1/4 cos(x) + C₁

Hence, the general solution of the differential equation is:

y(x) = C₁x¹/² + 1/4 cos(x)

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The derivative of y with respect to x is (2/x) * cos^5(x) - 5 cos^4(x) sin(x) + 90x.

To find the derivative of y with respect to x, we'll apply the rules of differentiation. Let's break down the given expression and differentiate each term separately:

y = 2 ln(3x) cos^5(x) + 45x^2 + 3 / dy/dx - cot(x)

To find the derivative, let's differentiate each term:

1. Differentiate the term 2 ln(3x):

The derivative of ln(u) is du/u, so the derivative of ln(3x) is 1/(3x) multiplied by the derivative of (3x):

d/dx (2 ln(3x)) = 2 * 1/(3x) * 3 = 2/x

2. Differentiate the term cos^5(x):

The derivative of cos(x) is -sin(x), so we'll apply the chain rule here.

d/dx (cos^5(x)) = 5 cos^4(x) * (-sin(x)) = -5 cos^4(x) sin(x)

3. Differentiate the term 45x^2:

The power rule for differentiation states that the derivative of x^n is n * x^(n-1). Applying this to the term 45x^2:

d/dx (45x^2) = 2 * 45x^(2-1) = 90x

4. Differentiate the term 3:

The derivative of a constant is zero, so the derivative of 3 is 0.

5. Differentiate the term dy/dx - cot(x):

Since dy/dx is given as a separate term, its derivative is 0.

Now let's put all the differentiated terms together:

dy/dx = (2/x) * cos^5(x) + (-5 cos^4(x) sin(x)) + 90x + 0

Simplifying the expression, we have:

dy/dx = (2/x) * cos^5(x) - 5 cos^4(x) sin(x) + 90x

Therefore, the derivative of y with respect to x is (2/x) * cos^5(x) - 5 cos^4(x) sin(x) + 90x.

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The first three questions involve differentiating given functions.  Question 1 - None of the above answers; Question 2 - y' = (x³ + 3x²)e*; Question 3 - None of the above answers. Question 4 asks for the derivative of y = 24x, and the correct answer is y' = 24.

Question 1: The given function is y = O (x-3)* > O (x-3)e* +8 O(x-3)x4 ex. The notation used is unclear, so it is difficult to determine the correct differentiation. However, none of the provided options seem to match the given function, so the answer is "None of the above answers."

Question 2: The given function is y = x³ex. To find its derivative, we apply the product rule and the chain rule. Using the product rule, we differentiate the terms separately and combine them. The derivative of x³ is 3x², and the derivative of ex is ex. Thus, the derivative of the given function is y' = (x³ + 3x²)e*.

Question 3: The given function is y = √√x³ + 4. To differentiate this function, we apply the chain rule. The derivative of √√x³ + 4 can be found by differentiating the inner function, which is x³ + 4. The derivative of x³ + 4 is 3x², and applying the chain rule, the derivative of √√x³ + 4 becomes 3x² * 2(x + 4)¹/2. Thus, the correct answer is "3x² * 2(x + 4)¹/2."

Question 4: The given function is y = 24x. To find its derivative, we differentiate it with respect to x. The derivative of 24x is simply 24, as the derivative of a constant multiplied by x is the constant. Therefore, the correct answer is y' = 24.

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The graph of f(x) = √(1-x²) over the interval [-1, 1] is the upper half of a circle with radius 1. The exact area between the x-axis and the graph of f over this interval is π/2.

To approximate the area between the x-axis and the graph of f over the interval [-1, 1], we can use rectangles. Using squares with side length 0.4 units, we can divide the interval into smaller subintervals. For each subinterval, we can find the height of the rectangle by evaluating the function at the left endpoint of the subinterval. We then approximate the area by summing the areas of the rectangles.

To find the exact area, we can use geometry. Since the graph is the upper half of a circle, the area between the x-axis and the graph over the interval [-1, 1] is exactly half the area of the full circle with radius 1. The formula for the area of a circle is A = πr², so the exact area in this case is π(1)²/2 = π/2.

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The demand function for a car is given by p= D(x) = 13.2 - 0.2x dollars. The level of production for which the revenue is maximized is 33 units.

To find the level of production for which the revenue is maximized, we need to determine the quantity that maximizes the revenue function. Revenue is calculated by multiplying the quantity sold (x) by the price (p).

The price is given by the demand function: p = 13.2 - 0.2x dollars.

Revenue (R) is given by: R(x) = p × x.

Substituting the demand function into the revenue function, we have:

R(x) = (13.2 - 0.2x) × x

R(x) = 13.2x - 0.2x²

To find the maximum value of R(x), we need to find the critical points by taking the derivative of R(x) with respect to x and setting it equal to zero:

R'(x) = 13.2 - 0.4x

Setting R'(x) = 0:

13.2 - 0.4x = 0

0.4x = 13.2

x = 13.2 / 0.4

x = 33

So, the level of production for which the revenue is maximized is 33 units.

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If N is a normal subgroup of a finite group G, and the index of subgroup H in G is finite, such that H is a subgroup of N, and the indices [G:H] and [N] are relatively prime, then N is contained in H.

Let's consider the given conditions. N is a normal subgroup of the finite group G, and H is a subgroup of N with finite index [G:H]. Additionally, the indices [G:H] and [N] are relatively prime, which means they have no common factors other than 1.

Since N is a normal subgroup of G, it implies that for any element n in N and any element g in G, the product gng^(-1) is also in N. Since H is a subgroup of N, it follows that for any element h in H and any element n in N, the product hnh^(-1) is also in N.

Now, consider the left cosets of H in G, denoted as gH, where g belongs to G. Since [G:H] is finite, there are only finitely many distinct left cosets. Let's denote the set of left cosets of H in G as {g₁H, g₂H, ..., gₙH}.

Since N is normal, for each left coset gᵢH, the product ngn^(-1) is also in the left coset gᵢH for any element n in N. Therefore, each left coset gᵢH is closed under conjugation by elements of N.

Since the indices [G:H] and [N] are relatively prime, the order of each left coset is relatively prime to the order of N. By applying Lagrange's theorem, the order of N must divide the order of each left coset gᵢH. However, since the order of each left coset is relatively prime to the order of N, it implies that the intersection of N with each left coset is trivial, i.e., N intersects each left coset only at the identity element.

Since N intersects each left coset gᵢH only at the identity element, it means that N is contained in H. Therefore, under the given conditions, N is a subgroup of H.

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Let A be a non-singular nx n matrix. Show that A is not similar to 2.A singular matrix is one in which the determinant is zero, whereas a non-singular matrix has a non-zero determinant.

This implies that the inverse matrix of the non-singular matrix exists and is also unique for a given matrix.In matrix algebra, two matrices A and B are said to be similar if there is a nonsingular matrix P such that

$$A=PBP^{-1}$$

Now let us suppose that A is similar to a 2 × 2 matrix.

We can write, $$A=PBP^{-1}$$

Taking determinants on both sides, $$det(A)=det(PBP^{-1})$$

Expanding the determinant, we obtain

:$$det(A)=det(P)det(B)det(P^{-1})$$$$det(A)=det(P)det(B)\frac{1}{det(P)}$$$$det(A)=det(B)$$Since B is a 2 × 2 matrix, then det(B) is either the product of its diagonal elements or the determinant of a 2 × 2 matrix,

which is of the form$$\begin{bmatrix}a&b\\c&d\end{bmatrix}$$Now, if B is similar to A and A is non-singular, then B must also be non-singular.

Therefore, the determinant of B is non-zero.But we have just seen that det(A) = det(B), so det(A) ≠ 0 and hence A is non-singular.

This implies that the matrix A cannot be similar to a 2 × 2 singular matrix. Thus, we can conclude that A is not similar to 2.

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A is similar to the identity matrix. This implies that A is diagonalizable. A is not similar to matrix 2.

Let A be a non-singular nx n matrix.

Show that A is not similar to 2

Suppose A is similar to matrix 2, which means there exists a matrix P such that P²=2 and A = P²  = PP.

Let λ be an eigenvalue of A, then A = PDP⁻¹,

where D is a diagonal matrix with the diagonal entries being the eigenvalues of A.

Since A is non-singular, λ ≠ 0.

Let D be the diagonal matrix of the eigenvalues of A.

Then the similarity condition A = PDP⁻¹ can be written as PDP⁻¹ = PP.

Multiplying both sides by P, we get DP⁻¹ = P.

Therefore, P² = D, which is a diagonal matrix with the diagonal entries being the squares of the eigenvalues of A.

This implies that P is also diagonal.

Therefore, if A is similar to a diagonal matrix, then A is diagonalizable.

Let A be a non-singular nx n matrix.

If A is similar to a diagonal matrix, then A is diagonalizable.

The proof of this fact is quite simple.

If A is similar to a diagonal matrix, then A = PDP⁻¹, where D is a diagonal matrix.

Therefore, the columns of P are eigenvectors of A.

Since the eigenvectors of A form a basis for the vector space, A is diagonalizable.

The determinant of A is non-zero, which means that A is invertible.

Therefore, A is similar to the identity matrix.

This implies that A is diagonalizable.

Thus, A is not similar to matrix 2.

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The value of B in the equation A sin(3t + ø) cannot be determined without additional information or calculation.

To identify the value of y that would produce resonance in the system described by the equation 7y + 3y' + 9y = 25cos(yt), we need to find the natural frequency of the system.

The equation represents a forced harmonic oscillator with a driving force of 25cos(yt). Resonance occurs when the frequency of the driving force matches the natural frequency of the system.

The natural frequency can be found by considering the coefficient of the y term. In this case, the coefficient is 9.

Thus, the value of y that would produce resonance in the system is given by:

ω = √(9) = 3

So, y = 3 would produce resonance in the system.

Now, to determine the amplitude of the steady-state solution when the system is at resonance, we need to evaluate the amplitude of the forced oscillation.

For a forced harmonic oscillator, the amplitude of the steady-state solution is given by the amplitude of the driving force divided by the square root of the squared sum of the coefficients of the y and y' terms. In this case, the amplitude of the driving force is 25.

Therefore, the amplitude of the steady-state solution at resonance is:

Amplitude = 25 / √((7²) + (3²)) ≈ 9.25 (rounded to two decimal places)

So, the amplitude of the steady-state solution when the system is at resonance is approximately 9.25.

Now, considering an oscillation of the spring in the absence of external force, described in the form A sin(3t + ø), we can determine the value of B.

Comparing the given form with the equation A sin(3t + ø), we see that B is related to the amplitude A.

The amplitude A represents the maximum displacement from the equilibrium position. In this case, A is the amplitude of the oscillation.

Since the value of B is not explicitly mentioned in the problem, we cannot determine its exact value without further information or calculation.

The value of y that would produce resonance in the system is y = 3.

The amplitude of the steady-state solution when the system is at resonance is approximately 9.25.

The value of B in the equation A sin(3t + ø) cannot be determined without additional information or calculation.

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The value of B is 0.2767 (approx).

The given equation is 7y + 3y' +9y= 25 cos(yt)

We have to identify the value of y that would produce resonance in the system. Give an exact value, but you don't need to simplify radicals.

Resonance occurs when the driving frequency of the oscillator is equal to the natural frequency of the system. The natural frequency, ω0 of the system can be calculated by the formula:

ω0=√k/m

where, k is the spring constant and m is the mass attached to the spring.

Since the mass and spring system is given by 7y + 3y' +9y= 25 cos(yt), comparing it with the standard form of the mass-spring system that is

y'' + ω0^2 y = f(t)

we get k = 9 and m = 7.

To find the natural frequency, we use the formula:

ω0=√k/m

ω0=√9/7

ω0=3/√7

So, the value of y that would produce resonance in the system is 3/√7.

Next, we have to find the amplitude of the steady-state solution when the system is at resonance.

Round your answer to two decimal places.

To find the amplitude of the steady-state solution, we use the formula:

A= F0/mω0√(1+(ζ)^2)

where, F0 is the amplitude of the driving force, ζ = damping ratio, which is given by 3/2√7 (as ζ = c/2√km, where c is damping constant) and m and ω0 are already calculated. On substituting the values, we get:

A= (25/(7×3/√7))√(1+(3/2√7)^2)

A= √7/9 × 25/3 × √(1+9/28)

A= 5/9 √7 × √(37/28)

A= 5/9 √259/28

A = 0.83 (approx)

Therefore, the amplitude of the steady-state solution when the system is at resonance is 0.83 (approx).

Next, we have to find the value of B if there was no external force and the oscillation of the spring could be described in the form A sin(3t+ø).

To write the given function in standard form, we can write it asy = A sin(ωt + ø)where, ω = 3, A is amplitude and ø is phase angle.

Therefore, comparing with the standard form of the function, we get:

B = A/ωB = 0.83/3 = 0.2767 (approx)

Therefore, the value of B is 0.2767 (approx).

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By using implicit differentiation, the derivative of the given equation, x² - xy² + y² = dx cos(x) sin(y) = x² - 2y, can be found.

To find the derivative using implicit differentiation, we differentiate both sides of the equation with respect to x. Let's start with the left-hand side:

d/dx(x² - xy² + y²) = d/dx(x²) - d/dx(xy²) + d/dx(y²)

The derivative of x² with respect to x is 2x. For the second term, we need to apply the product rule. Differentiating xy² with respect to x gives us x(d/dx(y²)) + y²(d/dx(x)). Since y is a function of x, we can apply the chain rule to find d/dx(y²) = 2yy'. Therefore, the second term becomes x(2yy') + y². For the third term, d/dx(y²) is 2yy'.

Combining all the terms, we have:

2x - (2xyy' + y²) + 2yy' = dx cos(x) sin(y)

Simplifying further:

2x - 2xyy' - y² + 2yy' = dx cos(x) sin(y)

Rearranging the terms:

2x - y² = dx cos(x) sin(y) + 2xyy' - 2yy'

Finally, isolating the derivative dy/dx:

dy/dx = (2x - y² - dx cos(x) sin(y)) / (2xy - 2y)

This is the derivative of y with respect to x obtained by implicit differentiation.

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(a)  first derivative of f(x) is [tex]$\frac{2e^{2x}(1-e^{2x})}{(1+e^{2x})^2}$.[/tex] (b) f(x) is concave up when x > 0 and f(x) is concave down when x < 0 (c) second derivative of f(x) is[tex]$\frac{8e^{2x}}{(1+e^{2x})^4}\left(e^{2x}-1\right)\left(3+e^{2x}\right)$.[/tex] (d) f(x) is concave up when x > 0 and f(x) is concave down when x < 0.

The derivative is a key idea in calculus that gauges how quickly a function alters in relation to its independent variable. It offers details on a function's slope or rate of change at any specific point. The symbol "d" or "dx" followed by the name of the function is generally used to represent the derivative.

It can be calculated using a variety of techniques, including the derivative's limit definition and rules like the power rule, product rule, quotient rule, and chain rule. Due to its ability to analyse rates of change, optimise functions, and determine tangent lines and velocities, the derivative has major applications in a number of disciplines, including physics, economics, engineering, and optimisation.

a) The first derivative of f(x) can be calculated as shown below:

[tex]$$f(x)= \frac{e^{2x}}{1+e^{2x}}$$$$\frac{df(x)}{dx}= \frac{(1+e^{2x})(\frac{d}{dx}(e^{2x}))-e^{2x}(\frac{d}{dx}(1+e^{2x}))}{(1+e^{2x})^2}$$$$\frac{df(x)}{dx}= \frac{(1+e^{2x})2e^{2x}-e^{2x}(2e^{2x})}{(1+e^{2x})^2}$$$$\frac{df(x)}{dx}= \frac{2e^{2x}(1-e^{2x})}{(1+e^{2x})^2}$$[/tex]

Therefore, the first derivative of f(x) is [tex]$\frac{2e^{2x}(1-e^{2x})}{(1+e^{2x})^2}$.[/tex]

b) To find any critical points of f(x), we set the first derivative equal to zero

[tex]:$$\frac{2e^{2x}(1-e^{2x})}{(1+e^{2x})^2}=0$$$$2e^{2x}(1-e^{2x})=0$$$$2e^{2x}=2e^{4x}$$$$1=e^{2x}$$Taking natural logarithm on both sides, we get:$$ln(e^{2x})=ln(1)$$$$2x=0$$$$x=0$$[/tex]

Therefore, the critical point is at x = 0. To determine when f(x) is increasing and decreasing, we look at the sign of the first derivative. When the first derivative is positive, f(x) is increasing. When the first derivative is negative, f(x) is decreasing.

$$2e^{2x}(1-e^{2x})>0$$$$e^{2x}>1$$$$x>0$$$$e^{2x}<1$$$$x<0$$

Therefore, f(x) is increasing when x > 0 and f(x) is decreasing when x < 0.c)

c) To find the second derivative of f(x), we differentiate the first derivative of f(x):[tex]$$\frac{d}{dx}\frac{2e^{2x}(1-e^{2x})}{(1+e^{2x})^2}=\frac{8e^{4x}}{(1+e^{2x})^3}-\frac{8e^{2x}(1-e^{2x})^2}{(1+e^{2x})^4}$$$$=\frac{8e^{2x}}{(1+e^{2x})^3}\left(\frac{e^{2x}}{1+e^{2x}}-\frac{(1-e^{2x})}{(1+e^{2x})}\right)$$$$=\frac{8e^{2x}}{(1+e^{2x})^3}\left(\frac{2e^{2x}}{(1+e^{2x})}-\frac{1}{(1+e^{2x})}\right)$$$$=\frac{8e^{2x}}{(1+e^{2x})^4}\left(2e^{2x}-(1+e^{2x})\right)$$$$=\frac{8e^{2x}}{(1+e^{2x})^4}\left(e^{2x}-1\right)\left(3+e^{2x}\right)$$[/tex]

Therefore, the second derivative of f(x) is[tex]$\frac{8e^{2x}}{(1+e^{2x})^4}\left(e^{2x}-1\right)\left(3+e^{2x}\right)$.[/tex]

d) f(x) is concave up when the second derivative is positive and concave down when the second derivative is negative.

Therefore, we need to find when the second derivative is positive and negative.[tex]$$e^{2x}-1>0$$$$e^{2x}>1$$$$x>0$$$$e^{2x}-1<0$$$$e^{2x}<1$$$$x<0$$$$3+e^{2x}>0$$$$e^{2x}>-3$$$$x>\frac{1}{2}ln(3)$$$$e^{2x}-1<0$$$$e^{2x}<1$$$$x<0$$$$3+e^{2x}<0$$$$e^{2x}<-3$$$$x<\frac{1}{2}ln(-3)$$[/tex]

Therefore, f(x) is concave up when x > 0 and f(x) is concave down when x < 0.

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a. The 8-bit binary representation for the integer 111 is 01101111.

To convert the decimal number 111 to binary, we divide it by 2 repeatedly until the quotient becomes 0, while keeping track of the remainders. The remainders will give us the binary digits in reverse order.

111 ÷ 2 = 55 (remainder 1)

55 ÷ 2 = 27 (remainder 1)

27 ÷ 2 = 13 (remainder 1)

13 ÷ 2 = 6 (remainder 0)

6 ÷ 2 = 3 (remainder 0)

3 ÷ 2 = 1 (remainder 1)

1 ÷ 2 = 0 (remainder 1)

Reading the remainders from bottom to top, we get 01101111 as the 8-bit binary representation for the integer 111.

b. The 8-bit binary representation for the integer 98 is 01100010.

Explanation:

Following the same process as above, we divide 98 by 2 repeatedly:

98 ÷ 2 = 49 (remainder 0)

49 ÷ 2 = 24 (remainder 1)

24 ÷ 2 = 12 (remainder 0)

12 ÷ 2 = 6 (remainder 0)

6 ÷ 2 = 3 (remainder 0)

3 ÷ 2 = 1 (remainder 1)

1 ÷ 2 = 0 (remainder 1)

Reading the remainders from bottom to top, we obtain 01100010 as the 8-bit binary representation for the integer 98.

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To find the absolute extrema of the function f(x) = x√(4x²) on the interval [-1, 2], we need to evaluate the function at the critical points and endpoints of the interval.

1. Critical points:

To find the critical points, we need to determine where the derivative of the function is either zero or undefined.

Let's find the derivative of f(x) first:

f'(x) = (d/dx) (x√(4x²))

Using the product rule and chain rule, we have:

f'(x) = √(4x²) + x * (1/2) * (4x²)^(-1/2) * 8x

Simplifying further, we get:

f'(x) = √(4x²) + 4x²/√(4x²)

f'(x) = 2|x| + 4x²/|2x|

To find the critical points, we set the derivative equal to zero and solve for x:

2|x| + 4x²/|2x| = 0

Case 1: x > 0

2x + 4x²/(2x) = 0

2x + 2x = 0

4x = 0

x = 0

Case 2: x < 0

-2x + 4x²/(-2x) = 0

-2x - 2x = 0

-4x = 0

x = 0

So, the critical point is x = 0.

2. Endpoints of the interval:

We also need to evaluate the function at the endpoints of the interval [-1, 2], which are x = -1 and x = 2.

Now, let's find the values of f(x) at the critical points and endpoints:

f(-1) = (-1)√(4(-1)²) = -1√(4) = -1 * 2 = -2

f(0) = (0)√(4(0)²) = 0

f(2) = (2)√(4(2)²) = 2 * 4 = 8

Therefore, the function f(x) = x√(4x²) has absolute extrema on the interval [-1, 2].

The minimum value is -2 and it occurs at x = -1.

The maximum value is 8 and it occurs at x = 2.

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Each poodle requires 8.5 balloons and each giraffe requires 1 balloon.

Let's start with Barbara's party, where Sparkles the Clown made 2 balloon poodles and 5 balloon giraffes, which used a total of 22 balloons. We can write this information as:

2P + 5G = 22 ---(Equation 1)

where P represents the number of balloons required for each poodle and G represents the number of balloons required for each giraffe.

At Wyatt's party, she used 13 balloons to make 5 balloon poodles and 2 balloon giraffes. We can write this information as:

5P + 2G = 13 ---(Equation 2)

Now, we need to solve these equations to find the values of P and G. We can do this by using elimination or substitution method.

Let's use substitution method by solving Equation 1 for P and substituting in Equation 2.

2P + 5G = 22

=> 2P = 22 - 5G

=> P = (22 - 5G)/2

Substituting this in Equation 2:

5P + 2G = 13

=> 5[(22 - 5G)/2] + 2G = 13

Simplifying and solving for G, we get:

G = 1

Substituting this in Equation 1 to find P:

2P + 5G = 22

=> 2P + 5(1) = 22

=> 2P = 17 => P = 8.5

Therefore, each poodle requires 8.5 balloons and each giraffe requires 1 balloon.
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a) The convergence of the series Σ[tex]((-1)^i)/(i^2)[/tex] cannot be determined using the ratio test. b) The radius of convergence for the series [tex](i+1)! (2^n + 5)^{(2i)[/tex] is 0.

a) To examine the convergence of the series Σ((-1)^i)/(i^2), we can use the ratio test. The ratio test states that if the limit of the absolute value of the ratio of consecutive terms is less than 1, then the series converges.

Let's apply the ratio test to the given series:

lim (i→∞)[tex]|((-1)^(i+1))/(i^2+1)| / |((-1)^i)/(i^2)|[/tex]

Taking the absolute values and simplifying, we have:

lim (i→∞) [tex]|(-1)^(i+1)| |i^2| / |i^2+1|[/tex]

Since the absolute value of (-1)^(i+1) is always 1, we can further simplify:

lim (i→∞) [tex]i^2 / (i^2+1)[/tex]

Now, taking the limit as i approaches infinity:

lim (i→∞) i[tex]^2 / (i^2+1) = 1[/tex]

Since the limit is equal to 1, the ratio test is inconclusive. We cannot determine the convergence of the series based on the ratio test.

b) The series [tex](i+1)! (2^n + 5)^(2i) = 7x + 9r^2 + 13x^3 + 21x + ...[/tex] can be rewritten as a power series in the form Σa_nx^n, where a_n is the coefficient of [tex]x^n.[/tex]

To find the radius of convergence for this series, we can use the formula:

R = 1 / lim (n→∞) |a_(n+1)/a_n|

In this case, the coefficient a_n is given by [tex](n+1)! (2^n + 5)^{(2n).[/tex]

Let's calculate the limit:

lim (n→∞) [tex]|(n+2)! (2^(n+1) + 5)^{(2{(n+1)})} / (n+1)! (2^n + 5)^(2n)|[/tex]

Simplifying, we have:

lim (n→∞) [tex]|(n+2)(2^{(n+1)}+ 5)^(2)| / |(2^n + 5)^{(2n)}|[/tex]

Now, taking the limit as n approaches infinity:

lim (n→∞)[tex]|(n+2)(2^{(n+1)} + 5)^(2)| / |(2^n + 5)^{(2n)}|[/tex]= ∞

Since the limit is infinity, the radius of convergence is 0. This means that the series converges only when x = 0, and diverges for all other values of x.

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To find the scalar equation of the line in two dimensions, we need to determine the slope-intercept form of the equation, which is given by:

y = mx + b

where "m" represents the slope of the line, and "b" represents the y-intercept.

Given the point (-3, 4) and the direction vector (4, -1), we can find the slope "m" using the formula:

m = (y2 - y1) / (x2 - x1)

Substituting the coordinates of the given point and the direction vector, we have:

m = (-1 - 4) / (4 - (-3))

m = -5 / 7

Now, we have the slope "m" as -5/7. To find the y-intercept "b," we can substitute the coordinates of the given point (-3, 4) into the slope-intercept form equation:

4 = (-5/7)(-3) + b

Simplifying:

4 = 15/7 + b

4 - 15/7 = b

(28 - 15) / 7 = b

13/7 = b

Thus, the y-intercept "b" is 13/7.

Now, we can write the scalar equation of the line in slope-intercept form:

y = (-5/7)x + 13/7

This is the scalar equation of the line passing through the point (-3, 4) and having a direction vector (4, -1) in two dimensions.

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In summary, for the given parametric equations, the point (x, y) on the curve where the tangent line is horizontal is (4, 16). This is found by differentiating y(t), setting the derivative equal to zero, and solving for t. Substituting the value of t back into the parametric equations gives the coordinates (x, y) of the point on the curve.

The point (x, y) on the curve where the tangent line is horizontal can be found by determining the value of t that makes the derivative of y(t) with respect to t equal to zero. For the parametric equations 4(sin(t) + cos(t)) and y = 28 — 12 cos²(t) — 24 sin(t), we can differentiate y(t) with respect to t, set the derivative equal to zero, and solve for t. Once t is known, the corresponding values of x and y can be found.

To elaborate, we differentiate y(t) = 28 — 12 cos²(t) — 24 sin(t) with respect to t, yielding dy/dt = 48 sin(t) - 24 cos(t). Setting dy/dt equal to zero, we have 48 sin(t) - 24 cos(t) = 0. Simplifying the equation, we obtain 2 sin(t) - cos(t) = 0. Solving for t, we find t = π/6. Substituting t = π/6 into the parametric equations, we get x = 4 and y = 16. Therefore, the point (x, y) where the tangent line is horizontal is (4, 16).

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Answer:

tip : (56*18)/100=10,08$

Step-by-step explanation:

The total amount of tip Linda left = $10.08

The question appears to be incomplete.

However, it could be made out from the question that you are asking for the amount of tip Linda left.

The total amount using which Linda had her dinner at restaurant = $56

Percentage of tip she left = 18% of the total amount

To calculate the amount of tip, the total amount can be multiplied to the given percentage divided by 100.

Therefore, the total amount of tip Linda left = 56*18/100

                                                                         = 1008/100

                                                                         = $10.08

Therefore, the total amount of tip Linda left = $10.08

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