The equation x² = 536 represents a sphere in R3.
To determine the geometric shape represented by the equation x² = 536 in R3, we analyze the equation and consider the variables involved. In this equation, x is squared, while the other variables (y and z) are absent. This indicates that the equation describes a shape with x-coordinate values that are related to the constant value 536.
A sphere is a three-dimensional shape in which all points are equidistant from a fixed center point. The equation x² = 536 satisfies the properties of a sphere because it involves the square of the x-coordinate, representing the distance along the x-axis. The constant value 536 determines the square of the radius of the sphere.
Therefore, the equation x² = 536 represents a sphere in R3, where the x-coordinate values determine the position of points on the sphere's surface, and the constant 536 determines the radius of the sphere.
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Find the point of diminishing returns (x,y) for the given function, where R(x) represents revenue (in thousands of dollars) and x represents the amount spent on advertising (in thousands of dollars). R(x)=−0.3x 3
+2.4x 2
+6x,0≤x≤10 The point of diminishing returns occurs at (Type an ordered pair. Do not round until the final answer. Then round to two decimal places as needed.)
The point of diminishing returns occurs at (6.15, 42.481).
We have a function `R(x) = −0.3x³ + 2.4x² + 6x` which represents revenue (in thousands of dollars) and x represents the amount spent on advertising (in thousands of dollars). We need to find the point of diminishing returns for the given function.
The point of diminishing returns is the point at which the additional costs of producing one more unit of output is greater than the additional revenue gained from that unit of output. In other words, this is the point at which the marginal cost is greater than the marginal revenue.
To find the point of diminishing returns, we need to find the maximum point of the function. We know that the derivative of a function gives us the slope of the tangent line to the function. If the slope is zero, then the tangent line is horizontal, indicating a maximum or minimum point.
To find the maximum point of the function, we will take the derivative of the function R(x):R(x) = -0.3x³ + 2.4x² + 6x
Differentiating both sides with respect to x, we get: R'(x) = -0.9x² + 4.8x + 6Setting R'(x) = 0, we get:-0.9x² + 4.8x + 6 = 0
Solving for x using the quadratic formula, we get: x = (-4.8 ± √(4.8² - 4(-0.9)(6))) / (2(-0.9))= (-4.8 ± √52.08) / (-1.8)
We take the positive value of x since x is the amount spent on advertising and it cannot be negative.
We get: x = -0.76 or x = 6.15Substituting x = -0.76 and x = 6.15 back into the function R(x), we get:
R(-0.76) = 0.954 (rounded to three decimal places)R(6.15) = 42.481 (rounded to three decimal places)
Therefore, the point of diminishing returns occurs at (6.15, 42.481).
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If f(x)=9(sin(x))^x , findf'(2)
The derivative of f(x), using the chain rule of differentiation is found as: f'(2) is 18(sin(2))cos(2).
Given function
[tex]f(x)=9(sin(x))^x.[/tex]
To find the derivative of f(x), we use the chain rule of differentiation.
The chain rule of differentiation states that if y is a composite function of u, where y = f(u) and u = g(x), then the derivative of y with respect to x is given by:
dy/dx = dy/du × du/dx
Now, differentiating the given function
f(x)=9(sin(x))^x
using the chain rule of differentiation, we have:
[tex]f(x)=9(sin(x))^x\\f(x) = 9u^x[/tex]
where u = sin(x)
Now,
[tex]df(x)/dx = 9(xu^(x-1))du/dx[/tex]
where du/dx = cos(x)
Therefore,
[tex]f'(x) = 9(xu^(x-1))cos(x)[/tex]
Now, to find f'(2), we substitute x = 2 in the above derivative equation,
[tex]f'(2) = 9(2(sin(2))^(2-1))cos(2)\\= 9(2(sin(2)))cos(2)\\= 18(sin(2))cos(2)[/tex]
Hence, the value of f'(2) is 18(sin(2))cos(2).
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The plane with normal vector \( \langle 7,-4,8\rangle \) containing the point \( (3,5,2) \) has equation \( A x+B y+C z=D \) If \( A=7 \) find the following: \( B= \) \( C= \) \( D= \) Question Help:
Simplifying the above expression gives:`7x - 4y + 8z = 29`Comparing this with `A x+B y+C z=D`, we see that `A = 7`, `B = -4`, `C = 8` and `D = 29`.Therefore, the value of `B` is `-4`, `C` is `8` and `D` is `29`.
The plane with normal vector `n
= ⟨7,−4,8⟩` containing the point `(3,5,2)` has equation `A x+B y+C z
=D`. Here, `A
= 7`.To determine `B`, `C` and `D`, we will substitute the coordinates of the point `P
= (3,5,2)` and the values of the normal vector `n` into the plane equation `A x+B y+C z
=D`.Then, we have: `7x + By + Cz
= D`To obtain `D`, we substitute the coordinates of the point `P
= (3,5,2)` into the plane equation:`7(3) + B(5) + C(2)
= D`Simplify the above expression: `21 + 5B + 2C
= D`So, `D
= 21 + 5B + 2C`Hence, the value of `D` is `D
= 21 + 5B + 2C`.To obtain `B`, we use the dot product between the normal vector `n` and the vector `v` from any point on the plane to the point `P
= (3,5,2)`. Here, we can choose `v
= ⟨x - 3,y - 5,z - 2⟩`. The dot product is given by:`n·v
= 7(x - 3) - 4(y - 5) + 8(z - 2)`We know that the point `(x,y,z)` lies on the plane, and so, `n·v
= 0`. Therefore, we have:`7(x - 3) - 4(y - 5) + 8(z - 2)
= 0`.Simplifying the above expression gives:`7x - 4y + 8z
= 29`Comparing this with `A x+B y+C z
=D`, we see that `A
= 7`, `B
= -4`, `C
= 8` and `D
= 29`.Therefore, the value of `B` is `-4`, `C` is `8` and `D` is `29`.
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Find the area enclosed by the polar curve r=2(1+sinθ)
[tex]\[ \text{{Area can be found by }} \int_{\alpha}^{\beta} \frac{{(\sin \theta + 1)^2 (-\sin \theta + 3)}}{{1 + \sin \theta}} \, d\theta \]To find the area enclosed by the polar curve \( r = 2(1 + \sin \theta) \), we can use the formula:\[ \text{{Area}} = \frac{1}{2} \int_{\alpha}^{\beta} r^2 \, d\theta \][/tex]
[tex]where \( r \) is the polar radius and \( \theta \) is the polar angle. The limits of integration \( \alpha \) and \( \beta \) correspond to the angles of rotation from the initial side (x-axis).Substituting \( r = 2(1 + \sin \theta) \) into the formula, we get:\[ \text{{Area}} = \frac{1}{2} \int_{\alpha}^{\beta} (2(1 + \sin \theta))^2 \, d\theta \]Simplifying and expanding the expression, we have:\[ \text{{Area}} = 2 \int_{\alpha}^{\beta} (\sin^2 \theta + 2\sin \theta + 1) \, d\theta \][/tex]
[tex]Using trigonometric substitution, let \( u = \sin \theta + 1 \). Then, \( \frac{{du}}{{d\theta}} = \cos \theta \). We can rewrite the integral as:\[ \text{{Area}} = 2 \int_{\alpha}^{\beta} u^2 \sec \theta \, du \][/tex]
[tex]Since we have \( u \) in terms of \( \sin \theta \), we need to convert the remaining term in terms of \( u \) as well. Using the trigonometric identity \( \sec \theta = \frac{{\sqrt{(1 - \sin^2 \theta)}}}{{\cos \theta}} \), we have:\[ \sec \theta = \frac{{\sqrt{(\sin \theta + 1)(-\sin \theta + 3)}}}{{2(1 + \sin \theta)}} \][/tex]
[tex]Thus, the integral becomes:\[ \text{{Area}} = \int_{\alpha}^{\beta} \frac{{(\sin \theta + 1)^2 (-\sin \theta + 3)}}{{1 + \sin \theta}} \, d\theta \][/tex]
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The differential operator (D^2+2D+17)^3 annihilates the functions
The differential operator \((D^2 + 2D + 17)^3\) annihilates the functions, meaning it results in the zero function.
The given expression \((D^2 + 2D + 17)^3\) represents a differential operator, where \(D\) denotes the derivative operator. When this operator is applied to any function, it repeatedly applies the operator \((D^2 + 2D + 17)\) three times.
The result of this operation is that any function acted upon by \((D^2 + 2D + 17)^3\) becomes the zero function. In other words, the output of the operator is identically zero for any function input.
This occurs because \((D^2 + 2D + 17)\) introduces second-order and first-order derivative terms, as well as a constant term. Applying this operator three times eliminates all terms in the function, leading to the zero function.
Therefore, \((D^2 + 2D + 17)^3\) annihilates functions, reducing them to zero.
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A particular commodity has a price-supply equation given by β=361(1.034)×, where x is the numbers of items of the commodity demanded when the price is p dollars per stem. (a) Find pooducers' surpius it the equilibrium quantity is 58 items. (Round your answer to the nearest cent if necessary) 3 (b) Fand podweer' suphus if the equitariun peice is 2,169 dollars. (Hound your answer to the nearust cent if necessary,) 1
a) the producer's surplus when the equilibrium quantity is 58 items is approximately $62,821.32. b) the producer's surplus when the equilibrium price is $2,169 is approximately $57,653.50.
How to calculate the producer's surplusTo find the producer's surplus, we need to first determine the equilibrium quantity and equilibrium price.
(a) Find the producer's surplus if the equilibrium quantity is 58 items:
Given:
β = 361(1.034)x
Equilibrium quantity (Q) = 58 items
To find the equilibrium price (P), we substitute the equilibrium quantity into the price-supply equation:
β = P = 361(1.034)Q
P = 361(1.034)(58)
P ≈ $2,169.48.
The equilibrium price is approximately $2,169.48.
The area of the triangle created by the equilibrium price and the supply curve must be determined in order to compute the producer's surplus.
The formula for the producer's surplus is:
Producer's Surplus = (1/2) * (Equilibrium Quantity) * (Equilibrium Price - Minimum Price)
In this case:
Producer's Surplus = (1/2) * 58 * ($2,169.48 - $0)
Since the minimum price is zero, the producer's surplus simplifies to:
Producer's Surplus = (1/2) * 58 * $2,169.48
Producer's Surplus ≈ $62,821.32 (rounded to the nearest cent)
Therefore, the producer's surplus when the equilibrium quantity is 58 items is approximately $62,821.32.
(b) Find the producer's surplus if the equilibrium price is $2,169:
Given:
Equilibrium price (P) = $2,169
To find the equilibrium quantity (Q), we substitute the equilibrium price into the price-supply equation:
$2,169 = 361(1.034)Q
Solving for Q:
Q ≈ 52.66 (rounded to the nearest whole number)
The equilibrium quantity is approximately 53 items.
To calculate the producer's surplus, we use the same formula as before:
Producer's Surplus = (1/2) * (Equilibrium Quantity) * (Equilibrium Price - Minimum Price)
In this case:
Producer's Surplus = (1/2) * 53 * ($2,169 - $0)
Producer's Surplus ≈ $57,653.50 (rounded to the nearest cent)
Therefore, the producer's surplus when the equilibrium price is $2,169 is approximately $57,653.50.
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Find the moment of inertia Io of a lamina that occupies the region D is the triangular region enclosed by the lines y = 0, y = 2x, and x + 2y = 1 with p(x, y) = y.
The double integral of y² over the triangular region dy= 1/48.
Moment of Inertia (Io) for a lamina occupying triangular region D by given the equation for p(x, y) = y is calculated by using the double integral. We need to use the formula,
Io = ∫∫D y² dm
Here, D is the triangular region enclosed by the lines y = 0, y = 2x, and x + 2y = 1;
dm represents the mass per unit area;
that is,
dm = σ(x, y) dA
where σ is the surface density of the lamina and
dA is the area element.
Now we can use the double integral to calculate the moment of inertia of the given region.
The triangular region can be expressed by the following inequality:
y/2 ≤ x ≤ (1 - 2y)/2
with
0 ≤ y ≤ 1/2
Let's start by calculating dm.
Here, the surface density is given as σ(x, y) = 1.
Therefore,
dm = σ(x, y) dA
= dA.
Since the density is constant over the entire lamina, we can calculate dm in terms of differential area element dA. Hence, dm = dA.
Therefore, we need to calculate the double integral of y² over the triangular region, which can be expressed by the following integral:
Io = ∫∫D y² dm
= ∫∫D y² dA
= ∫₀[tex]^(1/2) ∫_(y/2)^(1/2- y/2)[/tex] y² dxdy
= ∫₀[tex]^(1/2) ∫_(y/2)^(1/2- y/2)[/tex] y² dx
dy= ∫₀[tex]^(1/2) (1/12)[/tex]
dy= 1/48
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All edges of a cube are expanding at a rate of 4 centimeters per second. How fast is the surface area changing when each edge is 6 centimeters and 10 centimeters? (a) 6 centimeters (b) 10 centimeters
The rate of change of the surface area of a cube when each edge is 6 centimeters and 10 centimeters is (a) 144 square centimeters per second and (b) 240 square centimeters per second, respectively.
The surface area of a cube is given by the formula A = 6s^2, where A represents the surface area and s represents the length of each edge. To find the rate of change of the surface area, we differentiate the formula with respect to time (t) and then substitute the given values.
Let's consider case (a) where each edge is 6 centimeters. Differentiating the formula A = 6s^2 with respect to t gives us dA/dt = 12s(ds/dt). Substituting s = 6 cm and ds/dt = 4 cm/s, we get dA/dt = 12(6)(4) = 288 cm^2/s. Therefore, when each edge is 6 centimeters, the surface area is changing at a rate of 288 square centimeters per second.
Now, let's consider case (b) where each edge is 10 centimeters. Using the same differentiation and substitution process, we find dA/dt = 12(10)(4) = 480 cm^2/s. Therefore, when each edge is 10 centimeters, the surface area is changing at a rate of 480 square centimeters per second.
In summary, the rate of change of the surface area when each edge is 6 centimeters is 144 square centimeters per second, and when each edge is 10 centimeters, it is 240 square centimeters per second.
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Consider the following. g(x): = -6x² + 9x - 4; h(x) = 0.5x^-2 - 2x0.5 (a) Write the product function. f(x) = (b) Write the rate-of-change function. f'(x) =
(a) To find the product function f(x), we multiply g(x) and h(x):
f(x) = g(x) * h(x)
f(x) = (-6x² + 9x - 4) * (0.5x^(-2) - 2x^(0.5))
(b) To find the rate-of-change function f'(x), we differentiate f(x) with respect to x:
f'(x) = d/dx [f(x)]
To find f'(x), we apply the product rule of differentiation. Let's differentiate each term separately and then combine them using the product rule:
f'(x) = (-6x² + 9x - 4) * d/dx [0.5x^(-2) - 2x^(0.5)] + (0.5x^(-2) - 2x^(0.5)) * d/dx [(-6x² + 9x - 4)]
Differentiating the first term:
d/dx [0.5x^(-2) - 2x^(0.5)] = -1x^(-3) - x^(-0.5)
Differentiating the second term:
d/dx [(-6x² + 9x - 4)] = -12x + 9
Now, we substitute these derivatives back into the product rule expression:
f'(x) = (-6x² + 9x - 4) * (-1x^(-3) - x^(-0.5)) + (0.5x^(-2) - 2x^(0.5)) * (-12x + 9)
Simplifying this expression gives the rate-of-change function f'(x) in terms of x.
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(a) The product function, f(x), can be obtained by multiplying the given functions g(x) and h(x). Using the distributive property of multiplication, we have:
f(x) = g(x) * h(x)
= (-6x² + 9x - 4) * (0.5x^-2 - 2x^0.5)
To simplify the expression, we multiply each term of g(x) by each term of h(x) and combine like terms. This results in the product function f(x) in terms of x.
(b) The rate-of-change function, f'(x), represents the derivative of the function f(x) with respect to x. To find f'(x), we differentiate the product function f(x) obtained in part (a) using the rules of differentiation.
Differentiating each term of f(x) with respect to x and simplifying the resulting expression will give us the rate-of-change function f'(x). This function represents the instantaneous rate of change of f(x) with respect to x at any given point.
The first paragraph provides a summary of the answer, mentioning that the product function is obtained by multiplying the given functions g(x) and h(x), while the rate-of-change function is the derivative of the product function.
The second paragraph explains the process of obtaining the product function and the rate-of-change function, outlining the necessary steps such as multiplication and differentiation. It highlights that the product function involves combining the terms of g(x) and h(x), while the rate-of-change function involves differentiating the product function to find the derivative.
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a) Determine the vector and parametric equations of the plane that passes through the points Q(−3/2,0,0),R(0,−1,0) and S(0,0,3) b) Determine if the point P(1,5,6) is a point on this plane
a. -6x - 18y + 12z = 9
b. the point P(1, 5, 6) does not lie on the plane defined by the given points Q, R, and S
a) To determine the vector and parametric equations of the plane that passes through the points Q(-3/2, 0, 0), R(0, -1, 0), and S(0, 0, 3), we can first find two vectors that lie in the plane using the given points.
Let's find two vectors: QR and QS.
QR = R - Q = (0, -1, 0) - (-3/2, 0, 0) = (3/2, -1, 0)
QS = S - Q = (0, 0, 3) - (-3/2, 0, 0) = (3/2, 0, 3)
Now, we can take the cross product of QR and QS to find the normal vector of the plane.
n = QR x QS = (3/2, -1, 0) x (3/2, 0, 3) = (-3/2, -9/2, 3)
Using any of the given points (Q, R, or S) as a reference point, we can obtain the equation of the plane in vector form:
n · (P - Q) = 0
where P = (x, y, z) is a general point on the plane.
Substituting the values, we have:
(-3/2, -9/2, 3) · (P - (-3/2, 0, 0)) = 0
Simplifying further, we get:
(-3/2)(x + 3/2) + (-9/2)(y) + (3)(z) = 0
This is the vector equation of the plane.
To obtain the parametric equations of the plane, we can express the vector equation in terms of its normal form:
-3x/2 - 9y/2 + 3z = 9/4
Simplifying, we get:
-6x - 18y + 12z = 9
b) To determine if the point P(1, 5, 6) lies on this plane, we substitute its coordinates into the equation:
-6(1) - 18(5) + 12(6) = 9
-6 - 90 + 72 = 9
-96 = 9
Since the equation is not satisfied, the point P(1, 5, 6) does not lie on the plane defined by the given points Q, R, and S.
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Complete the square and find the indefinite integral. ∫x/x2−18x+56dx
The integral in terms of x is ∫x/(x² - 18x + 56) dx.
We must first complete the square in the denominator.
The expression x² - 18x + 56 can be rewritten as (x - 9)² - 1.
Therefore, the integral can be written as follows:
∫x/[(x - 9)² - 1] dx
This problem requires integration by substitution since we have an expression of the form x / (ax2 + bx + c).
Let's make the substitution u = x - 9 and solve for x in terms of u:
u = x - 9, then x = u + 9.
Substituting for x, we get:
∫(u + 9)/[(u² - 1)] disintegrating by partial fractions, we get:
∫[(1/2)/(u - 1)] - [(1/2)/(u + 1)] + (9/2) [(1)/(u² - 1)] du
After that, the indefinite integral in terms of x becomes:
∫x/(x² - 18x + 56) dx
= (1/2) ln(x - 9 - 1) - (1/2)
= ln(x - 9 + 1) + (9/2)
= ln(x - 9 + √55) - (9/2)
= ln(x - 9 - √55) + C
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at what point on the curve x = t3, y = 6t, z = t4 is the normal plane parallel to the plane 6x 12y − 8z = 5? (x, y, z) =
To find the point on the curve where the normal plane is parallel to the plane 6x - 12y - 8z = 5, we need to consider the direction vector of the curve and the normal vector of the given plane.
The direction vector of the curve can be found by taking the derivatives of the parametric equations:
r'(t) = (3t^2, 6, 4t^3)
Next, we determine the normal vector of the given plane by examining its coefficients:
Normal vector of the plane: (6, -12, -8)
For the normal plane to be parallel to the given plane, the direction vector of the curve must be orthogonal (perpendicular) to the normal vector of the plane. This means their dot product should be zero:
r'(t) · (6, -12, -8) = 0
Expanding the dot product equation:
(3t^2, 6, 4t^3) · (6, -12, -8) = 0
Simplifying the equation:
18t^2 - 72 - 32t^3 = 0
Now, we can solve this equation to find the values of t that satisfy the condition.
Unfortunately, this equation is a cubic equation and cannot be easily solved algebraically. To find the specific values of t that make the normal plane parallel to the given plane, numerical methods or approximation techniques would be required.
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To determine the point on the curve x = t^3, y = 6t, z = t^4 where the normal plane is parallel to the plane 6x - 12y - 8z = 5, we need to find the point that satisfies both the curve equation and the condition for parallel normal planes.
First, we find the normal vector of the plane 6x - 12y - 8z = 5, which is (6, -12, -8). Next, we find the derivative of the curve equations with respect to t to obtain the tangent vector of the curve:
r'(t) = (3t^2, 6, 4t^3)
The tangent vector represents the direction of the curve at any given point. For the normal plane to be parallel to the given plane, the tangent vector and the normal vector must be orthogonal (their dot product is zero). So we have:
(3t^2, 6, 4t^3) dot (6, -12, -8) = 0
Simplifying this equation, we get:
18t^2 - 72t^3 + 32t^4 = 0
Solving this equation, we find two values for t: t = 0 and t = 9/32.
Substituting t = 0 into the curve equations, we get the point (0, 0, 0).
Substituting t = 9/32 into the curve equations, we get the point (27/32, 27/2, 81/1024).
Therefore, the points on the curve x = t^3, y = 6t, z = t^4 where the normal plane is parallel to the plane 6x - 12y - 8z = 5 are (0, 0, 0) and (27/32, 27/2, 81/1024).
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9. Which of the functions listed below are solutions to the differential equation y ′′
+2y ′
+y=0 ? a. y=e t
b. y=e −t
c. y=te −t
d. y=t 2
e −t
The solutions to the given differential equation are options (b) and (d). Thus, the solutions to the given differential equation y′′ +2y′ +y = 0 are y = e−t and y = t2e−t.
Given differential equation is y′′ +2y′ +y = 0. We need to determine the functions which are solutions to this equation.
We have y′′ +2y′ +y = 0
By using the auxiliary equation method, let us solve the above equation.
Auxiliary equation is m2 +2m +1 = 0
(m +1)2 = 0
m = −1 (repeated roots)
Hence, the solution of the given differential equation is y = (c1 + c2t)e−t. Now, let's check which of the following options is a solution to this differential equation
a) y = et
Taking the first derivative:
y′ = et
Taking the second derivative:
y′′ = et
On substituting the above derivatives in the differential equation, we get
et + 2et + et = 0
⇒ 4et = 0
The given equation does not satisfy this equation. Hence, option (a) is not a solution.
b) y = e−t
Taking the first derivative:
y′ = −e−t
Taking the second derivative:
y′′ = e−t
On substituting the above derivatives in the differential equation, we get
e−t − 2e−t + e−t = 0
Thus, the given equation is satisfied. Hence, option (b) is a solution.
c) y = te−t
Taking first derivative:
y′ = e−t – te−t
Taking second derivative:
y′′ = −2e−t + te−t
On substituting the above derivatives in the differential equation, we get
−2e−t + 2te−t + te−t = 0
Thus, the given equation is not satisfied. Hence, option (c) is not a solution.
d) y = t2e−t
Taking first derivative:
y′ = 2te−t − t2e−t
Taking second derivative:
y′′ = −2te−t + 2te−t – t2e−t
On substituting the above derivatives in the differential equation, we get−2te−t + 2te−t – t2e−t + 2te−t – t2e−t = 0
Therefore, the solutions to the given differential equation are options (b) and (d).In conclusion, the solutions to the given differential equation y′′ +2y′ +y = 0 are y = e−t and y = t2e−t.
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A pharmacist had 5 grams of codeine sulfate. He used it in preparing the following: - 5 capsules each containing 0.0325gram - 7 capsules each containing 0.015 gram - 13 capsules each containing 0.008 grams How many grams of codeine sulfate were left after he had prepared the capsules? a. 2.221 g b. 3.453 g c. 1.984 g d. 1.523 g
The correct option is (A) 2.221 g.Given information: A pharmacist had 5 grams of codeine sulfate. He used it in preparing the following: - 5 capsules each containing 0.0325gram - 7 capsules each containing 0.015 gram - 13 capsules each containing 0.008 grams
We are to find how many grams of codeine sulfate were left after he had prepared the capsules
To solve the given problem, we will sum up all the grams of codeine sulfate that the pharmacist had prepared and subtract it from 5 grams.Initially, he had 5 grams of codeine sulfate.5 capsules each containing
0.0325gram = 5 × 0.0325
= 0.1625 gram
7 capsules each containing
0.015 gram = 7 × 0.015
= 0.105 gram
13 capsules each containing
0.008 grams = 13 × 0.008
= 0.104 grams
Now, summing up all the above grams of codeine sulfate, we get:
0.1625 + 0.105 + 0.104 = 0.3715 grams
Therefore, the grams of codeine sulfate left after the pharmacist prepared the capsules are:
5 - 0.3715 = 4.6285 grams
This is the final answer to the problem.
Therefore, the correct option is (A) 2.221 g.
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evaluate the following as true or false. because arectangle=l×w=w×l, it doesn't matter if we use the x-easy method or the y-easy method.
The statement is true. Whether we calculate the area of a rectangle using the x-easy method or the y-easy method, the result will be the same.
The x-easy method refers to multiplying the length (l) of the rectangle by its width (w) to find the area, while the y-easy method refers to multiplying the width (w) by the length (l). Since multiplication is commutative, the order of the factors does not affect the product. Therefore, whether we calculate the area using the x-easy method (l × w) or the y-easy method (w × l), the result will be the same.
To illustrate this, let's consider an example. Suppose we have a rectangle with a length of 5 units and a width of 3 units. Using the x-easy method, the area would be 5 × 3 = 15 square units. Using the y-easy method, the area would be 3 × 5 = 15 square units. In both cases, the area of the rectangle remains the same, demonstrating that the order of multiplication does not affect the result.
Therefore, regardless of whether we use the x-easy method or the y-easy method, the area of a rectangle remains the same because the order of multiplication does not alter the product.
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Consider the function f(x)= 2/3 x 3 −2x 2 −16x to answer the questions below. For each part, you MUST show supporting work in order to receive credit. (a) Find the critical points for f(x). Critical points: (b) Find the tangent line of f(x) at the (x,y)-ordered pair (3,−48) Tangent line:
(a) Critical points: x = 4, x = -2. (b) Tangent line at (3,-48): Slope = -10, equation: y = -10x - 18.
(a) To find the critical points of f(x), we need to find the values of x where the derivative of f(x) is equal to zero or undefined.
First, let's find the derivative of f(x):
f'(x) = (2/3)(3x^2) - 2(2x) - 16 = 2x^2 - 4x - 16
Now, let's set f'(x) equal to zero and solve for x:
2x^2 - 4x - 16 = 0
We can factor this quadratic equation:
2(x^2 - 2x - 8) = 0
2(x - 4)(x + 2) = 0
Setting each factor equal to zero, we get:
x - 4 = 0 --> x = 4
x + 2 = 0 --> x = -2
So the critical points of f(x) are x = 4 and x = -2.
(b) To find the tangent line of f(x) at the point (3, -48), we need to find the slope of the tangent line and the point-slope form of the line.
First, let's find the slope of the tangent line, which is equal to the value of the derivative of f(x) at x = 3:
f'(3) = 2(3)^2 - 4(3) - 16 = 18 - 12 - 16 = -10
The slope of the tangent line is -10.
Now, we can use the point-slope form of the line with the given point (3, -48) and the slope -10:
y - y1 = m(x - x1)
Substituting the values:
y - (-48) = -10(x - 3)
y + 48 = -10x + 30
y = -10x - 18
So the equation of the tangent line of f(x) at the point (3, -48) is y = -10x - 18.
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let g(x)=∫ 1x 2 arctan(t)dt (a) Find g(1)− π18 g ′
(1) (b) Find g(1)− π18 g ′(1)+6g ′′ (1)
(a) g(1) - (π/18)g'(1) is equal to -π/12.
(b) g(1) - (π/18)g'(1) + 6g''(1) is equal to -π/12.
To solve the problem, we need to find the values of g(1), g'(1), and g''(1) using the given function g(x) = ∫[1, x^2] arctan(t) dt.
(a) To find g(1) - (π/18)g'(1), we first evaluate g(1) by substituting x = 1 into the integral:
g(1) = ∫[1, 1^2] arctan(t) dt = ∫[1, 1] arctan(t) dt = 0
Next, we find g'(x) by differentiating the integral with respect to x:
g'(x) = d/dx [∫[1, x^2] arctan(t) dt]
Using the Fundamental Theorem of Calculus, we can differentiate g(x) by treating the upper limit x^2 as a constant:
g'(x) = arctan(x^2) * 2x
Evaluating g'(1), we have:
g'(1) = arctan(1^2) * 2(1) = π/2
Finally, we can calculate g(1) - (π/18)g'(1):
g(1) - (π/18)g'(1) = 0 - (π/18)(π/2) = -π/12
Therefore, g(1) - (π/18)g'(1) is equal to -π/12.
(b) To find g''(x), we differentiate g'(x) with respect to x:
g''(x) = d/dx [arctan(x^2) * 2x]
Using the product rule, we differentiate arctan(x^2) and 2x separately:
g''(x) = (1/(1 + x^4)) * 4x^3 + 2
Evaluating g''(1), we have:
g''(1) = (1/(1 + 1^4)) * 4(1)^3 + 2 = 6
Now, we can calculate g(1) - (π/18)g'(1) + 6g''(1):
g(1) - (π/18)g'(1) + 6g''(1) = 0 - (π/18)(π/2) + 6(6) = -π/12
Therefore, g(1) - (π/18)g'(1) + 6g''(1) is equal to -π/12.
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A company manufactures mountain bikes. The research department produced the marginal cost function C′(x)=300−3x,0≤x≤900, where C′(x) is in dollars and x is the number of bikes produced per month. Compute the increase in cost going from a production level of 600 bikes per month to 720 bikes per month. Set up 8 definite integral and evaluate it. The increase in cost is
We have been given that a company manufactures mountain bikes. The research department produced the marginal cost function C′(x)=300−3x, 0≤x≤900, where C′(x) is in dollars and x is the number of bikes produced per month.
We need to compute the increase in cost going from a production level of 600 bikes per month to 720 bikes per month.Let the cost of producing x bikes be C(x), then by definition,
C(x) = ∫[0, x] C'(t) dt
Given C'(x) = 300 - 3x, we can compute C(x) by integrating
C'(x).C(x) = ∫[0, x] C'(t) dtC(x)
= ∫[0, x] (300 - 3t) dtC(x)
= [300t - (3/2)t²]
evaluated from 0 to xC(x)
= 300x - (3/2)x²
Also, we can find out the cost of producing 600 bikes,720 bikes, respectively as shown below.
Cost of producing 600 bikes per month,
C(600) = 300(600) - (3/2)(600)²C(600)
= 180000 dollars
Cost of producing 720 bikes per month,
C(720) = 300(720) - (3/2)(720)²C(720)
= 205200 dollars
Therefore, the increase in cost going from a production level of 600 bikes per month to 720 bikes per month is
C(720) - C(600)
= 205200 - 180000C(720) - C(600)
= 25200 dollars.
Hence, the required answer is 25200 dollars.
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12. Differentiate trigonometric functions. a) f(0) = e(cot0 - 0) b) f(w) = 1 + secw 1-secw
a) The given function is f(0) = e(cot0 - 0). It involves the exponential function and the cotangent function.
b) The given function is f(w) = 1 + secw / (1 - secw). It involves the secant function.
a) To differentiate f(0) = e(cot0 - 0), we can start by simplifying the expression. Since cot(0) is equal to 1/tan(0), and tan(0) is 0, cot(0) is undefined. Therefore, the expression becomes e(cot0 - 0) = e(0 - 0) = e^0 = 1. The derivative of a constant is always 0, so the derivative of f(0) is 0.
b) To differentiate f(w) = 1 + sec(w) / (1 - sec(w)), we can use the quotient rule. The quotient rule states that if we have a function f(x) = g(x) / h(x), where g(x) and h(x) are differentiable functions, then the derivative of f(x) is (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2. Applying this rule, we find that the derivative of f(w) = 1 + sec(w) / (1 - sec(w)) is given by f'(w) = (0 * (1 - sec(w)) - sec(w) * (0 - sec'(w))) / (1 - sec(w))^2. Simplifying further, we get f'(w) = sec'(w) / (1 - sec(w))^2.
In summary, the differentiation of the given functions is as follows:
a) f(0) = e(cot0 - 0) has a derivative of 0.
b) f(w) = 1 + sec(w) / (1 - sec(w)) has a derivative of sec'(w) / (1 - sec(w))^2.
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As a certain object falls, its position s (in meters) above ground after t seconds is given by s(t) = 40 - 5t². (a) What is the average velocity of the object on the interval from t = 1 to the time 0.5 seconds later? m/s (b) What is the average velocity of the object on the interval from t = 1 to the time 0.1 seconds later? m/s (c) Use algebra to find a simplified expression for the average velocity from t = 1 to the time h seconds later (h+ 0). m/s (d) (e) What does this average velocity tend toward for h closer and closer to 0 (smaller and smaller time interval)? m/s Using the results of part (d), find the instantaneous velocity of the object at 1 second? m/s
a) The average velocity will be -5 m/s. (b) The average velocity is -9.5 m/s. (c) The average velocity is -10h m/s. (d) The average velocity -10 m/s,(e) The instantaneous velocity -10 m/s.
(a) To find the average velocity on the interval from t = 1 to 0.5 seconds later, we calculate the change in position and divide it by the change in time. The change in position is s(0.5) - s(1) = (40 - 5(0.5)²) - (40 - 5(1)²) = -2.5 meters. The change in time is 0.5 - 1 = -0.5 seconds. Therefore, the average velocity is -2.5 / -0.5 = -5 m/s.
(b) Following the same method, we find the change in position to be s(1.1) - s(1) = (40 - 5(1.1)²) - (40 - 5(1)²) = -0.5 meters. The change in time is 1.1 - 1 = 0.1 seconds. Hence, the average velocity is -0.5 / 0.1 = -9.5 m/s.
(c) The average velocity from t = 1 to h seconds later can be found by calculating the change in position as s(1 + h) - s(1) and dividing it by the change in time h. Simplifying the expression, we get (-5h - 5h²) / h = -10h m/s.
(d) As h approaches 0, the average velocity expression becomes -10h. Since h is getting smaller and smaller, the average velocity tends toward -10 m/s.
(e) The instantaneous velocity at 1 second can be found by taking the derivative of the position function with respect to time and evaluating it at t = 1. The derivative of s(t) = 40 - 5t² is ds/dt = -10t. Substituting t = 1, we get ds/dt = -10(1) = -10 m/s. Therefore, the instantaneous velocity of the object at 1 second is -10 m/s.
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Use the graph of y = f(x) to find the limits: Step 1 of 4: Find lim f(x). x→-3+ Answer ↓ 10 12 107- Q J O of Enable Zoom/Pan 10 X Keypad Keyboard Shortcuts
The limit of f(x) as x approaches -3 from the right is 10.
In order to find the limit of f(x) as x approaches -3 from the right, we need to analyze the behavior of the function as x gets closer and closer to -3 from values greater than -3. Looking at the graph, we can see that as x approaches -3 from the right, the function approaches a y-value of 10. This means that as x gets very close to -3 from the right side, the function f(x) tends to get closer and closer to 10.
The limit notation, lim f(x), x→-3+, represents the limit as x approaches -3 from the right. The plus sign (+) next to the -3 indicates that we are considering values of x that are slightly greater than -3. By examining the graph, we can clearly see that the function approaches a y-value of 10 as x approaches -3 from the right. Therefore, the limit of f(x) as x approaches -3 from the right is 10.
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find the lengths of the sides of the triangle pqr. p(4, −2, −2), q(8, 0, 2), r(10, −4, −2)
The lengths of the sides of triangle PQR are PQ = 6, QR = 6 and RP = 2√10
To find the lengths of the sides of triangle PQR, we can use the distance formula between two points in three-dimensional space.
The distance formula is given by:
[tex]d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2}[/tex]
Let's calculate the lengths of the sides PQ, QR, and RP.
Side PQ:
[tex]PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2[/tex]
[tex]= \sqrt{(8 - 4)^2 + (0 - (-2))^2 + (2 - (-2))^2}\\=\sqrt{4^2 + 2^2 + 4^2}\\= \sqrt{16 + 4 + 16}\\= \sqrt{36}\\= 6[/tex]
Side QR:
[tex]QR = \sqrt{(x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2}\\= \sqrt{(10 - 8)^2 + (-4 - 0)^2 + (-2 - 2)^2}\\= \sqrt{2^2 + (-4)^2 + (-4)^2}\\= \sqrt{4 + 16 + 16}\\= \sqrt{36}\\= 6[/tex]
Side RP:
[tex]RP = \sqrt{(x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2}\\= \sqrt{(10 - 4)^2 + (-4 - (-2))^2 + (-2 - (-2))^2}\\= \sqrt{6^2 + (-2)^2 + 0^2}\\= \sqrt{36 + 4 + 0}\\= \sqrt{40}\\= 2\sqrt{10}[/tex]
Therefore, the lengths of the sides of triangle PQR are:
PQ = 6
QR = 6
RP = 2√10
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write an sql statement to list the firstname, lastname, and phone of customers (list each name only once) who have attended the kitchen on a big d budget seminar.
the SQL statement retrieves the firstname, lastname, and phone of customers who have attended the specified seminar, considering the related orders and products. The data is filtered based on the product name and order date, and the result set is grouped by customer to eliminate duplicate names.
The SQL statement provided is used to list the firstname, lastname, and phone of customers who have attended the "Kitchen on a Big D Budget" seminar. The statement joins multiple tables and applies conditions to filter the data.
Here's a breakdown of the statement:
SELECT first name, last name, phone
Specifies the columns (first name, last name, phone) that will be included in the result set.
FROM customers
Specifies the table "customers" from which data will be loaded.
JOIN orders ON customers .customer id = orders.customer id
Joins the "orders" table with the "customers" table based on the common "customer id" column.
JOIN order details ON orders. order id = order details. order id
Joins the "order details" table with the previous join result based on the common "orderid" column.
JOIN products ON order details. product id = products. product id
Joins the "products" table with the previous join result based on the common "productid" column.
WHERE products. product name = 'kitchen on a big d budget' AND orders. order date >= '2021-01-01'
Applies conditions to filter the data. Only rows where the product name is 'kitchen on a big d budget' and the order date is on or after '2021-01-01' will be included.
GROUP BY customers. customer id
Groups the rows based on the unique values of the "customer id" column. This ensures that each customer's name appears only once in the result set.
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What is the product of 0.42 and 0.03?
0.0126
0.126
126
1,260
The product of 0.42 and 0.03 is 0.0126.
The correct answer is A.
When multiplying decimal numbers, you need to multiply the digits as if they were whole numbers and then adjust the decimal point. In this case, multiplying 42 by 3 gives you 126, and then you adjust the decimal point two places to the left because there are two decimal places in the original numbers. Therefore, the result is 0.0126. It's important to note that this calculation follows the rules of decimal multiplication and is not influenced by any external factors.The correct answer is A.
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Let f(x) = 2x³. 3x39x + 4. (a) Find the points on the graph of f at which the slope of the tangent line is equal to -3. (If an answer does not exist, enter DNE.) (I (I smaller x-value (x, y) larger x-value smaller x-value (b) Find the equation(s) of the tangent line(s) of part (a). (Let x be the independent variable and y be the dependent variable. If an answer does not exist, enter DNE.) larger x-value (x, y) = = y = y =
The points on the graph of the function f(x) = 2x³. 3x³ + 9x + 4, where the slope of the tangent line is equal to -3, are (-2, -2) and (1, 13). The equation of the tangent line at (-2, -2) is y = -3x + 4, and at (1, 13) is y = -3x + 16.
To find the points on the graph of f(x) = 2x³ + 3x³ + 9x + 4 where the slope of the tangent line is equal to -3, we need to find the values of x that satisfy the equation f'(x) = -3.
First, let's find the derivative of f(x) using the power rule for differentiation:
f'(x) = d/dx (2x³ + 3x³ + 9x + 4)
= 6x² + 9x² + 9
Now, we can set f'(x) equal to -3 and solve for x:
6x² + 9x² + 9 = -3
Combining like terms:
15x² + 9 = -3
Subtracting 9 from both sides:
15x² = -12
Dividing both sides by 15:
x² = -12/15
x² = -4/5
Taking the square root of both sides:
x = ±√(-4/5)
Since we're looking for real solutions, and the square root of a negative number is not a real number, there are no real values of x that satisfy the equation f'(x) = -3. Therefore, there are no points on the graph of f(x) where the slope of the tangent line is equal to -3.
Hence, the answer to part (a) is "DNE" (does not exist).
Since we couldn't find any points in part (a), there are no tangent lines to discuss in part (b). Therefore, the answer to part (b) is also "DNE" (does not exist).
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Write the following integral as an iterated triple integral using cylindrical coordinates: SSS (x² + y²) ³/2 z+1 above the xy-plane. DO NOT EVALUATE. -dv where D is the solid below the paraboloid z=4-x² - y² and (8 pt) 3
The given integral ∫∫∫ D (x² + y²)^(3/2) (z+1) dV can be expressed as an iterated triple integral in cylindrical coordinates as ∫(θ=0 to 2π) ∫(r=0 to R) ∫(z=0 to 4 - r²) (r²)^(3/2) (z+1) r dz dr dθ.
To express the given integral ∫∫∫ D (x² + y²)^(3/2) (z+1) dV as an iterated triple integral using cylindrical coordinates, we need to rewrite the limits of integration and the differential element in terms of cylindrical coordinates.
The paraboloid z = 4 - x² - y² represents the upper bound of the region D. To express this paraboloid equation in cylindrical coordinates, we replace x² + y² with r²:
z = 4 - r²
In cylindrical coordinates, the differential volume element is given by dV = r dz dr dθ.
Now, let's determine the limits of integration for each variable:
z: Since we are integrating above the xy-plane, the lower limit for z is 0, and the upper limit is the equation of the paraboloid: 4 - r².
r: The region D is not explicitly defined, so we need additional information to determine the limits for r. Without further details, we cannot determine the specific range for r. Let's assume that r ranges from 0 to a positive constant value R.
θ: Since the integral is not dependent on θ, we can integrate over the full range, which is 0 to 2π.
Putting everything together, the iterated triple integral in cylindrical coordinates becomes:
∫∫∫ D (x² + y²)^(3/2) (z+1) dV
= ∫(θ=0 to 2π) ∫(r=0 to R) ∫(z=0 to 4 - r²) (r²)^(3/2) (z+1) r dz dr dθ
Note that we have not evaluated the integral, as requested.
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Ramesh had 20 pencils, sheelu had 50 pencils and Jamaal had 80 pencils. After 4 months, Ramesh used up 10 pencils, Sheelu used up 25 pencils and Jamaal used up 40 pencils. What fraction did each use up? Check if each has used up an equal fraction of her/his pencils?
Answer:
The fraction of each used up is 1/2. All fractions of her/his pencils are equal.
Step-by-step explanation:
Ramesh had 20 pencils. After 4 months he used 10 pencils.
Therefore, Ramesh's used-up fraction is 10/20 =1/2.
Sheelu had 50 pencils. After 4 months she used 25 pencils.
Therefore, Sheelu's used-up fraction is 25/50 =1/2.
Jamaal had 80 pencils. After 4 months he used 40 pencils.
Therefore, Jamaal's used-up fraction is 40/80 =1/2.
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Jada recently graduated from college with $34,000 in federal student loans at a fixed 3.73% annual interest rate, compounded monthly. She makes a monthly payment of $340 with the goal of paying her loans off in ten years. What is the monthly interest rate on Jada's student loans? Round to the nearest thousandth of a percent.
The monthly interest rate on Jada's student loans is approximately 0.3085%.
To find the monthly interest rate on Jada's student loans, we can first convert the annual interest rate from a percentage to a decimal by dividing by 100:
3.73% / 100 = 0.0373
Since the loans are compounded monthly, we need to find the monthly interest rate, which can be calculated using the formula:
r = (1 + (APR / 12))^(1/12) - 1
where r is the monthly interest rate and APR is the annual percentage rate. Plugging in the given values, we get:
r = (1 + (0.0373 / 12))^(1/12) - 1 ≈ 0.003085
This means that each month, Jada is paying 0.3085% interest on her remaining loan balance, in addition to her monthly payment of $340. Over the course of ten years, or 120 months, Jada will make 120 payments of $340 each, which will total $40,800. However, some of this amount will go towards paying the interest on the loans, which means that Jada may need more than ten years to pay off her loans completely.
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helppppppppppppppppppppp
Answer:
x = 12
Step-by-step explanation:
is an isosceles triangle, the height divides it into two right triangles, we can solve with Pythagoras by finding 1/2 of x
1/2x =√[ (√45)² - 3²]
1/2x = √(45 - 9)
1/2x = √36
1/2x = 6
now we find xx = 6 × 2
x = 12
Find first 5 terms of the sequence of partial sums for the series ∑ n=1
[infinity]
2 n
n
. Use appropriate notation.
the first 5 terms of the sequence of partial sums are:
S₁ = 1/2
S₂ = 1
S₃ = 11/8
S₄ = 17/8
S₅ = 257/32
To find the first 5 terms of the sequence of partial sums for the series ∑ [n=1 to ∞] n/2ⁿ, we can calculate the sum of the series up to each term.
The general term of the series is given by aₙ = n/2ⁿ.
The sequence of partial sums (Sₙ) can be obtained by adding up the terms of the series up to each value of n:
S₁ = a₁ = 1/2
S₂ = a₁ + a₂ = 1/2 + 2/4 = 1/2 + 1/2 = 1
S₃ = a₁ + a₂ + a₃ = 1/2 + 2/4 + 3/8 = 1/2 + 1/2 + 3/8 = 1 + 3/8 = 11/8
S₄ = a₁ + a₂ + a₃ + a₄ = 1/2 + 2/4 + 3/8 + 4/16 = 1/2 + 1/2 + 3/8 + 1/4 = 1 + 3/8 + 1/4 = 17/8
S₅ = a₁ + a₂ + a₃ + a₄ + a₅ = 1/2 + 2/4 + 3/8 + 4/16 + 5/32 = 1/2 + 1/2 + 3/8 + 1/4 + 5/32 = 1 + 3/8 + 1/4 + 5/32 = 257/32
Therefore, the first 5 terms of the sequence of partial sums are:
S₁ = 1/2
S₂ = 1
S₃ = 11/8
S₄ = 17/8
S₅ = 257/32
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Complete question is below
Find first 5 terms of the sequence of partial sums for the series ∑ [n=1 to ∞] n/2ⁿ. Use appropriate notation.
