A differential equation is an equation that relates a function and its derivatives, describing how the function changes over time or space.the general solution of the given differential equation is[tex]= C_1 e^{3x} + C_2 e^{-3x} + \dfrac{9}{2} x - \dfrac{2}{9} + C e^{2x}[/tex]
Given differential equation is[tex]\dfrac{d^2 y}{dx^2} - 9 y &= x + e^{2x} \\[/tex] Here, the auxiliary equation is m² - 9 = 0 which gives m = ±3 From the characteristic roots, the complementary solution will be given by [tex]y_c = C_1 e^{3x} + C_2[/tex] e^(-3x)
Now we must use the method of uncertain coefficients to find the solution of a differential equation. For the particular solution, assume y_p = Ax + B + Ce^(2x)
Substituting this in the differential equation, we get:
[tex]\dfrac{d^2 y_p}{dx^2} - 9 y_p &= x + e^{2x} \\\\A e^{2x} + 4C e^{2x} - 9(Ax + B + Ce^{2x}) &= x + e^{2x}[/tex]
On compare the coefficient, we get:
A - 9C = 0 => A
9C4C - 9B = 0
=> B = 4C/9
Therefore, the particular solution is:
[tex]y_p = \dfrac{9}{2} x - \dfrac{2}{9} + C e^{2x}[/tex]
Hence, the general solution of the given differential equation is:
[tex]y &= y_c + y_p \\\\&= C_1 e^{3x} + C_2 e^{-3x} + \dfrac{9}{2} x - \dfrac{2}{9} + C e^{2x}[/tex]
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Suppose you take out a loan for 180 days in the amount of $13,500 at 11% ordinary interest. After 50 days, you make a partial payment of $1,000. What is the final amount due on the loan? (Round to the nearest cent)
The final amount due on the loan after the partial payment is approximately $13,070.41 (rounded to the nearest cent).
To calculate the final amount due on the loan, we need to consider the principal amount, the interest accrued, and the partial payment made.
Given information:
Principal amount: $13,500
Interest rate: 11% (per year)
Loan period: 180 days
Partial payment: $1,000
Partial payment date: 50 days
First, let's calculate the interest accrued on the loan from the loan start date to the partial payment date:
Interest accrued = Principal amount * Interest rate * (Number of days / 365)
Interest accrued = $13,500 * 11% * (50 / 365)
Interest accrued ≈ $201.37
Next, let's calculate the remaining principal balance after the partial payment:
Remaining principal balance = Principal amount - Partial payment
Remaining principal balance = $13,500 - $1,000
Remaining principal balance = $12,500
Now, let's calculate the interest accrued on the remaining principal balance for the remaining loan period (180 - 50 days):
Interest accrued = Remaining principal balance * Interest rate * (Number of days / 365)
Interest accrued = $12,500 * 11% * (130 / 365)
Interest accrued ≈ $570.41
Finally, we can calculate the final amount due on the loan by adding the remaining principal balance and the interest accrued:
Final amount due = Remaining principal balance + Interest accrued
Final amount due = $12,500 + $570.41
Final amount due ≈ $13,070.41
Therefore, the final amount due on the loan after the partial payment is approximately $13,070.41 (rounded to the nearest cent).
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Problem 4. Consider the plant with the following state-space representation. 0 *---**** _x+u; U; = y = [1 0]x
(a) Design a state feedback controller without integral control to yield a 5% overshoot and 2 sec settling time. Evaluate the steady-state error for a unit step input.
(b) Redesign the state feedback controller with integral control; evaluate the steady-state error for a unit step input. Required Steps:
(i) Obtain the gain matrix of K by means of coefficient matching method or Ackermann's formula by hand. You may validate your results with the "acker" or "place" function in MATLAB.
(ii) Use the following equation to determine the steady-state error for a unit step input, ess=1+ C(A - BK)-¹B
(iii) When ee-designing the state feedback controller with integral control, obtain the new gain matrix of K = [k₁ k₂] and ke
State feedback controllers with integral control are useful for reducing or eliminating steady-state errors in a system. The following is a step-by-step process for designing a state feedback controller with integral control:Problem 4 Consider the plant with the following state-space representation.
0⎡⎣x˙x⎤⎦=[0−4.4−20.6]⎡⎣xu⎤⎦y=[10]Part (a)To get a 5% overshoot and 2-second settling time, we design a state feedback controller without integral control. The first step is to check the controllability and observability of the system.The rank of the controllability matrix is 2, which is equal to the number of states, indicating that the system is controllable. The system is also observable since the rank of the observability matrix is 2.
The poles of the closed-loop system can now be placed using Ackermann's formula or the coefficient matching method. Ackermann's formula is used in this example. The poles are located at -5 ± 4.83i.K = acker(A,B,[-5-4.83j,-5+4.83j])The gain matrix is calculated as:K = [4.4000 10.6000]The steady-state error for a unit step input is calculated using the following equation:ess=1+ C(A - BK)-¹Bwhere C = [1 0] and D = 0. The steady-state error for a unit step input is found to be 0.Part (b)To reduce the steady-state error to zero, integral control is added to the system. The augmented system's state vector is [x xₐ]
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Use interval notation to indicate where
f(x)= x−6 / (x−1)(x+4) is continuous.
Answer: x∈
Note: Input U, infinity, and -infinity for union, [infinity], and −[infinity], respectively.
The function f(x) = (x - 6) / ((x - 1)(x + 4)) is continuous for certain intervals of x. The intervals where f(x) is continuous can be expressed using interval notation.
To determine where f(x) is continuous, we need to consider the values of x that make the denominator of the function non-zero. Since the denominator is (x - 1)(x + 4), the function is not defined for x = 1 and x = -4.
Therefore, to express the intervals where f(x) is continuous, we exclude these values from the real number line. In interval notation, we indicate this as:
x ∈ (-∞, -4) U (-4, 1) U (1, ∞).
This notation represents the set of all x-values where the function f(x) is defined and continuous. It indicates that x can take any value less than -4, between -4 and 1 (excluding -4 and 1), or greater than 1. In these intervals, the function f(x) is continuous and can be evaluated without any discontinuities or breaks.
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A mass of 100 grams of a particular radioactive substance decays according to the function m(t)=100e−ᵗ/⁶⁵⁰, where t>0 measures time in years. When does the mass reach 25 grams?
In the given radioactive decay function, t represents time in years, and m(t) represents the mass of the radioactive substance at time t. The mass of the substance reaches 25 grams at approximately t = 899.595 years.
To solve for t, we can set the mass function equal to 25 grams and solve for t:
25 = 100[tex]e^(-t/650)[/tex].
To isolate [tex]e^(-t/650)[/tex], we divide both sides by 100:
25/100 = [tex]e^(-t/650)[/tex].
Simplifying further:
1/4 = [tex]e^(-t/650)[/tex].
To eliminate the exponential function, we can take the natural logarithm (ln) of both sides:
ln(1/4) = ln([tex]e^(-t/650)[/tex]).
Using the property of logarithms, ln([tex]e^x[/tex]) = x, we can simplify the equation:
ln(1/4) = -t/650.
Now, we can solve for t by multiplying both sides by -650:
-650 * ln(1/4) = t.
Using a calculator to evaluate ln(1/4) ≈ -1.3863 and performing the multiplication:
t ≈ -650 * (-1.3863)
t ≈ 899.595.
Therefore, the mass of the substance reaches 25 grams at approximately t = 899.595 years.
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Given the vector valued function: r(t) = <4t^3,tsin(t^2),1/1+t^2>, compute the following:
a) r′(t) = ______
b) ∫r(t)dt = ______
a) The derivative of the vector-valued function r(t) = <4t^3, tsin(t^2), 1/(1+t^2)> is r'(t) = <12t^2, sin(t^2) + 2t^2cos(t^2), -2t/(1+t^2)^2>.
To compute the derivative of the vector-valued function r(t), we differentiate each component of the vector separately.
For the x-component, we use the power rule to differentiate 4t^3, which gives us 12t^2.
For the y-component, we differentiate tsin(t^2) using the product rule. The derivative of t is 1, and the derivative of sin(t^2) is cos(t^2) multiplied by the chain rule, which is 2t. Therefore, the derivative of tsin(t^2) is sin(t^2) + 2t^2cos(t^2).
For the z-component, we differentiate 1/(1+t^2) using the quotient rule. The derivative of 1 is 0, and the derivative of (1+t^2) is 2t. Applying the quotient rule, we get -2t/(1+t^2)^2.
The derivative of the vector-valued function r(t) is r'(t) = <12t^2, sin(t^2) + 2t^2cos(t^2), -2t/(1+t^2)^2>.
Regarding the integral of r(t) with respect to t, without specified limits, we can compute the indefinite integral. Each component of the vector r(t) can be integrated separately. The indefinite integral of 4t^3 is (4/4)t^4 + C1 = t^4 + C1. The indefinite integral of tsin(t^2) is -(1/2)cos(t^2) + C2. The indefinite integral of 1/(1+t^2) is arctan(t) + C3.
Therefore, the indefinite integral of r(t) with respect to t is ∫r(t)dt = <t^4 + C1, -(1/2)cos(t^2) + C2, arctan(t) + C3>, where C1, C2, and C3 are integration constants.
Note that if specific limits are given for the integral, the answer would involve evaluating the definite integral within those limits, resulting in numerical values rather than symbolic expressions.
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Find the maximum value of f(x,y,z)=21x+16y+23z on the sphere x2+y2+z2=324.
the maximum value of f(x, y, z) = 21x + 16y + 23z on the sphere [tex]x^2 + y^2 + z^2[/tex] = 324 is 414.
To find the maximum value of the function f(x, y, z) = 21x + 16y + 23z on the sphere [tex]x^2 + y^2 + z^2 = 324[/tex], we can use the method of Lagrange multipliers. The idea is to find the critical points of the function subject to the constraint equation. In this case, the constraint equation is [tex]x^2 + y^2 + z^2 = 324[/tex].
First, we define the Lagrangian function L(x, y, z, λ) as follows:
L(x, y, z, λ) = f(x, y, z) - λ(g(x, y, z) - c)
Where g(x, y, z) is the constraint equation [tex]x^2 + y^2 + z^2[/tex] and c is a constant. In this case, c = 324.
So, our Lagrangian function becomes:
L(x, y, z, λ) = 21x + 16y + 23z - λ([tex]x^2 + y^2 + z^2 - 324[/tex])
To find the critical points, we take the partial derivatives of L(x, y, z, λ) with respect to x, y, z, and λ, and set them equal to zero:
∂L/∂x = 21 - 2λx
= 0 ...(1)
∂L/∂y = 16 - 2λy
= 0 ...(2)
∂L/∂z = 23 - 2λz
= 0 ...(3)
∂L/∂λ = -([tex]x^2 + y^2 + z^2 - 324[/tex])
= 0 ...(4)
From equation (1), we have:
21 = 2λx
x = 21/(2λ)
Similarly, from equations (2) and (3), we have:
y = 16/(2λ) = 8/λ
z = 23/(2λ)
Substituting these values of x, y, and z into equation (4), we get:
-([tex]x^2 + y^2 + z^2 - 324[/tex]) = 0
-(x^2 + (8/λ)^2 + (23/(2λ))^2 - 324) = 0
-(x^2 + 64/λ^2 + 529/(4λ^2) - 324) = 0
-(441/4λ^2 - x^2 - 260) = 0
x^2 = 441/4λ^2 - 260
Substituting the value of x = 21/(2λ), we get:
(21/(2λ))^2 = 441/4λ^2 - 260
441/4λ^2 = 441/4λ^2 - 260
0 = -260
This leads to an inconsistency, which means there are no critical points satisfying the conditions. However, the function f(x, y, z) is continuous on a closed and bounded surface [tex]x^2 + y^2 + z^2 = 324[/tex], so it will attain its maximum value somewhere on this surface.
To find the maximum value, we can evaluate the function f(x, y, z) at the endpoints of the surface, which are the points on the sphere [tex]x^2 + y^2 + z^2 = 324[/tex].
The maximum value of f(x, y, z) will be the largest value among these endpoints and any critical points on the surface. But since we have already established that there are no critical points, we only
need to evaluate f(x, y, z) at the endpoints.
The endpoints of the surface [tex]x^2 + y^2 + z^2 = 324[/tex] are given by:
(±18, 0, 0), (0, ±18, 0), and (0, 0, ±18).
Evaluating f(x, y, z) at these points, we have:
f(18, 0, 0) = 21(18) + 16(0) + 23(0)
= 378
f(-18, 0, 0) = 21(-18) + 16(0) + 23(0)
= -378
f(0, 18, 0) = 21(0) + 16(18) + 23(0)
= 288
f(0, -18, 0) = 21(0) + 16(-18) + 23(0)
= -288
f(0, 0, 18) = 21(0) + 16(0) + 23(18)
= 414
f(0, 0, -18) = 21(0) + 16(0) + 23(-18)
= -414
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Michael and Sara like ice cream. At a price of 0 Swiss Francs per scoop, Michael would eat 7 scoops per week, while Sara would eat 12 scoops per week at a price of 0 Swiss Francs per scoop. Each time the price per scoop increases by 1 Swiss Francs, Michael would ask 1 scoop per week less and Sara would ask 4 scoops per week less. (Assume that the individual demands are linear functions.) What is the market demand function in this 2-person economy? x denotes the number of scoops per week and p the price per scoop. Please provide thorough calculation and explanation.
The market demand function for ice cream in this 2-person economy is x = 19 - 5p, where x represents the total quantity of ice cream demanded and p represents the price per scoop.
In the given problem, we are asked to determine the market demand function for ice cream in a 2-person economy, where Michael and Sara have individual demand functions that are linear. We are given their consumption quantities at two different price levels and the rate at which their consumption changes with price. The market demand function represents the total quantity of ice cream demanded by both individuals at different price levels.
Let's denote the price per scoop as p and the quantity demanded by Michael and Sara as xM and xS, respectively. We are given the following information:
At p = 0, xM = 7 and xS = 12.
For every 1 Swiss Franc increase in price, xM decreases by 1 and xS decreases by 4.
Based on this information, we can write the demand functions for Michael and Sara as follows:
xM = 7 - p
xS = 12 - 4p
To find the market demand function, we need to sum up the individual demands:
xM + xS = (7 - p) + (12 - 4p)
= 7 + 12 - p - 4p
= 19 - 5p
Therefore, the market demand function for ice cream in this 2-person economy is:
x = 19 - 5p
This equation represents the total quantity of ice cream demanded by both Michael and Sara at different price levels. As the price per scoop increases, the total quantity demanded decreases linearly at a rate of 5 scoops per 1 Swiss Franc increase in price.
In conclusion, the market demand function for ice cream in this 2-person economy is x = 19 - 5p, where x represents the total quantity of ice cream demanded and p represents the price per scoop.
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help 4. Analysis and Making Production Decisions a) On Monday, you have a single request: Order A for 15,000 units. It must be fulfilled by a single factory. To which factory do you send the order? Explain your decision. Support your argument with numbers. b) On Tuesday, you have two orders. You may send each order to a separate factory OR both to the same factory. If they are both sent to be fulfilled by a single factory, you must use the total of the two orders to find that factory’s cost per unit for production on this day. Remember that the goal is to end the day with the lowest cost per unit to produce the company’s products. Order B is 7,000 units, and Order C is 30,000 units. c) Compare the two options. Decide how you will send the orders out, and document your decision by completing the daily production report below.
A) we would send Order A to Factory 3.
B) we would send both Order B and Order C to Factory 3.
B 7,000 Factory 3
C 30,000 Factory 3
Total number of units produced for the company today: 37,000
Average cost per unit for all production today: $9.00
To make decisions about which factory to send the orders to on Monday and Tuesday, we need to compare the costs per unit for each factory and consider the total number of units to be produced. Let's go through each day's scenario and make the production decisions.
a) Monday: Order A for 15,000 units
To decide which factory to send the order to, we compare the costs per unit for each factory. We select the factory with the lowest cost per unit to minimize the average cost per unit for the company.
Let's assume the costs per unit for each factory are as follows:
Factory 1: $10 per unit
Factory 2: $12 per unit
Factory 3: $9 per unit
To calculate the total cost for each factory, we multiply the cost per unit by the number of units:
Factory 1: $10 * 15,000 = $150,000
Factory 2: $12 * 15,000 = $180,000
Factory 3: $9 * 15,000 = $135,000
Based on the calculations, Factory 3 has the lowest total cost for producing 15,000 units, with a total cost of $135,000. Therefore, we would send Order A to Factory 3.
b) Tuesday: Order B for 7,000 units and Order C for 30,000 units
We have two options: sending each order to a separate factory or sending both orders to the same factory. We need to compare the average cost per unit for each option and select the one that results in the lowest average cost per unit.
Let's assume the costs per unit for each factory remain the same as in the previous example. We will calculate the average cost per unit for each option:
Option 1: Sending orders to separate factories
For Order B (7,000 units):
Average cost per unit = ($10 * 7,000) / 7,000 = $10
For Order C (30,000 units):
Average cost per unit = ($9 * 30,000) / 30,000 = $9
Total number of units produced for the company today = 7,000 + 30,000 = 37,000
Average cost per unit for all production today = ($10 * 7,000 + $9 * 30,000) / 37,000 = $9.43 (rounded to two decimal places)
Option 2: Sending both orders to the same factory (Factory 3)
For Orders B and C (37,000 units):
Average cost per unit = ($9 * 37,000) / 37,000 = $9
Comparing the two options, we see that both options have the same average cost per unit of $9. However, sending both orders to Factory 3 simplifies the production process by consolidating the orders in one factory. Therefore, we would send both Order B and Order C to Factory 3.
Production Report for Tuesday:
Order # of Units Factory
B 7,000 Factory 3
C 30,000 Factory 3
Total number of units produced for the company today: 37,000
Average cost per unit for all production today: $9.00
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Owners of a boat rental company that charges customers between $125 and $325 per day have determined that the number of boats rented per day n can be modeled by the linear function n(p)=1300-4p. where p is the daily rental charge. How much should the company charge each customer per day to maximize revenue? Do not include units or a dollar sign in your answer.
The company should charge $162.5 to each customer per day to maximize revenue.
The revenue function can be represented by [tex]R(p) = p * n(p)[/tex]. Substituting n(p) with 1300-4p, [tex]R(p) = p * (1300-4p)[/tex]. On expanding, [tex]R(p) = 1300p - 4p²[/tex]. For maximum revenue, finding the value of p that gives the maximum value of R(p). Using differentiation,[tex]R'(p) = 1300 - 8p[/tex]. Equating R'(p) to 0, [tex]1300 - 8p = 08p = 1300p = 162.5[/tex] Therefore, the company should charge $162.5 to each customer per day to maximize revenue.
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Find the third derivative of the given function. f(x)=2x5−2x4+5x2−5x+5 f′′′(x)=___
Therefore, the third derivative of f(x) is [tex]f'''(x) = 120x^2 - 48x.[/tex]
To find the third derivative of the function [tex]f(x) = 2x^5 - 2x^4 + 5x^2 - 5x + 5,[/tex]we need to take the derivative of the second derivative.
First, let's find the first derivative:
[tex]f'(x) = d/dx (2x^5 - 2x^4 + 5x^2 - 5x + 5)[/tex]
[tex]= 10x^4 - 8x^3 + 10x - 5[/tex]
Next, let's find the second derivative:
[tex]f''(x) = d/dx (10x^4 - 8x^3 + 10x - 5)\\= 40x^3 - 24x^2 + 10[/tex]
Finally, let's find the third derivative:
[tex]f'''(x) = d/dx (40x^3 - 24x^2 + 10)\\= 120x^2 - 48x[/tex]
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Evaluate
d/dx (x^6e^x) = f(x)e^x , then f(1) = ______
Let f(x) = e^x tanx , Find f’(0) = _____
The values of f’(0) = 1 and of f(1) = 2.446.
The problem requires us to find the value of f(1) and f’(0).
Given,
d/dx(x6 e^x) = f(x) e^x
Let us find the first derivative of the given function as follows:
d/dx(x^6 e^x) = d/dx(x^6) * e^x + d/dx(e^x) * x^6 [Product Rule]
= 6x^5 e^x + x^6 e^x [d/dx(e^x) = e^x]
= x^5 e^x(6+x)
We are given that,
f(x) = e^x tan x
f(1) = e^1 * tan 1
f(1) = e * tan 1
f(1) = 2.446
To find f’(0), we need to find the first derivative of f(x) as follows:
f’(x) = e^x sec^2 x + e^x tan x [Using Product Rule]
f’(0) = e^0 sec^2 0 + e^0 tan 0 [When x = 0]
f’(0) = 1 + 0
f’(0) = 1
Therefore, f’(0) = 1.
Thus, we get f’(0) = 1 and f(1) = 2.446.
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Suppose f(x)=−8x2+2. Evaluate the following limit.
limh→0 f(−1+h)−f(−1) / h =
Note: Input DNE, infinity, -infinity for does not exist, [infinity], and −[infinity], respectively.
The limit of the given expression can be evaluated by substituting the values into the function and simplifying. The result will be a finite number.
To evaluate the limit, we substitute the values into the expression:
limh→0 f(-1+h) - f(-1) / h
Substituting -1+h into the function f(x), we get:
f(-1+h) = -8(-1+h)^2 + 2
Expanding and simplifying:
f(-1+h) = -8(1 - 2h + h^2) + 2
= -8 + 16h - 8h^2 + 2
= -8h^2 + 16h - 6
Substituting -1 into the function f(x):
f(-1) = -8(-1)^2 + 2
= -8 + 2
= -6
Now, we can rewrite the limit expression as:
limh→0 (-8h^2 + 16h - 6 - (-6)) / h
Simplifying further:
limh→0 (-8h^2 + 16h) / h
= -8h + 16
Finally, taking the limit as h approaches 0, we have:
limh→0 (-8h + 16) = 16
Therefore, the limit of the given expression is 16
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[By hand] Sketch the root locus for positive K for the unity feedback system with open loop transfer function L(s) = K - s+1 s²+4s-5 Show each necessary step of the sketching procedure AND for any step that is not needed, explain why it is not needed. Further, answer the following questions: A. Is this system stable if operated without feedback? B. Under unity feedback, what range of gains, K, stabilize the closed-loop system? C. Assuming the gain stabilizes the closed-loop system, how much steady-state error do you expect the system to exhibit in response to a unit step change in the reference signal? D. If K = 6, do you expect the dominant pole approximation to hold for this system? If so, estimate the 1% settling time of the system's step response. If not, explain why not. Aside from evaluating a square root, this entire problem can (and should) be done by hand (no calculator; no Matlab).
To sketch the root locus for the given unity feedback system, we follow the steps of the root locus construction:
1. Identify the open-loop transfer function: L(s) = K - s + 1 / (s^2 + 4s - 5)
2. Determine the poles and zeros of the open-loop transfer function. The poles are obtained by setting the denominator of L(s) equal to zero, which gives s^2 + 4s - 5 = 0.
3. Determine the branches of the root locus. Since there are two poles, there will be two branches starting from the poles. The branches will move towards the zeros and/or to infinity.
4. Determine the angles of departure and arrival for the branches. The angle of departure from a pole is given by the sum of the angles of the open-loop transfer function at that pole.
5. Determine the real-axis segments. The real-axis segments of the root locus occur between the real-axis intersections of the branches. In this case, there are two real-axis segments.
6. Determine the breakaway and break-in points. These are the points where the branches of the root locus either originate or terminate. The breakaway points occur when the derivative of the characteristic equation with respect to s is zero.
Based on the sketch of the root locus, we can answer the following questions:
A. The system without feedback is not stable because the poles of the open-loop transfer function have positive real parts.
B. Under unity feedback, the closed-loop system will be stable if the gain, K, lies to the left of the root locus branches and does not encircle any poles of the open-loop transfer function.
C. Assuming stability, the steady-state error for a unit step change in the reference signal will be zero because there is a pole at the origin (zero steady-state error for unity feedback).
D. With K = 6, the dominant pole approximation may hold since the other poles are further away. To estimate the 1% settling time, we can calculate the settling time of the dominant pole, which is the pole closest to the imaginary axis.
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If z = (x+y)e^y, x = 5t, y = 5 – t^2, find dz/dt using the chain rule.
Assume the variables are restricted to domains on which the functions are defined.
dz/dt = ______
dz/dt = (5 - 2t)e^(5 - t^2). To find dz/dt using the chain rule, we can differentiate z = (x + y)e^y with respect to t by considering x and y as functions of t.
Given x = 5t and y = 5 - t^2, we can substitute these expressions into z. By substituting x and y, we have z = (5t + 5 - t^2)e^(5 - t^2). To find dz/dt, we apply the chain rule. The chain rule states that if z = f(g(t)), where f(u) and g(t) are differentiable functions, then dz/dt = f'(g(t)) * g'(t). In this case, f(u) = u * e^(5 - t^2) and g(t) = 5t + 5 - t^2. Taking the derivatives, we find f'(u) = e^(5 - t^2) and g'(t) = 5 - 2t. Applying the chain rule, we multiply the derivatives: dz/dt = f'(g(t)) * g'(t) = (e^(5 - t^2)) * (5 - 2t). Therefore, dz/dt = (5 - 2t)e^(5 - t^2).
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Ayana has saved $200 and spends $25 each week. Michelle just started saving $15 per week. in how many weeks will Ayana and Michelle have the same amound of money saved?
Answer:
In 5 weeks, Ayana and Michelle have the same amount of money saved
(Namely $75)
Step-by-step explanation:
Ayana has $200 and spends $25 per week.
Michelle has $0 and saves $15 per week.
So, after one week,
Ayana has $200 - $25 = $175
Michelle has $0 + $ 15 = $15
After 2 weeks,
Ayana has $175 - $25 = $150
Michelle has $15 + $15 = $30
After 4 weeks,
Ayana has $150 - $50 = $100
Michelle has $30 + $30 = $60
After 5 weeks,
Ayana has $100 - $25 = $75
Michelle has $60 + $15 = $75
So, in 5 weeks, Ayana and Michelle have the same amount of money saved
Ayana and Michelle will have the same amount of money saved in 5 weeks.
To calculate the number of weeks Ayana and Michelle will take to have the same ammount of money, we have to make use of assumption. The reason for this is, as the number of weeks are yet to be found, so the value can only be found by substituting that particular entity into a variable.
Let's assume that number of weeks Ayana and Michelle will take to have the same ammount of money is "x".
So, Amount saved by Ayana after x weeks will be $200 - $25*x,
Amount saved by Michelle in x weeks will be $15 * x.
In the question, we have been told that Ayana and Michelle have the same amount of money saved, So we need to equate to above two equations to find the value of "x".
$200 - $25*x = $15 * x
$200 = $15 * x + $25*x
$200 = $40*x
$200 / $40 = x
x = 5
Therefore, Ayana and Michelle will take 5 weeks to have the same amound of money saved.
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a) Find the first four nonzero terms of the Taylor series for the given function centered at a.
b) Write the power series using summation notation.
f(x)=e^x , a=ln(10)
a) The first four nonzero terms of the Taylor series for [tex]f(x) = e^x[/tex]centered at a = ln(10) are:
10, 10(x - ln(10)), [tex]\dfrac{5(x - ln(10))^2}{2}[/tex], [tex]\dfrac{(x - ln(10))^3}{3!}[/tex]
b) The power series using summation notation is:
[tex]\sum_{n=0}^{\infty} \dfrac{(10 (x - ln(10))^n)}{ n!}[/tex]
a)
To find the first four nonzero terms of the Taylor series for the function [tex]f(x) = e^x[/tex] centered at a = ln(10), we can use the formula for the Taylor series expansion:
[tex]f(x) = f(a) + \dfrac{f'(a)(x - a)}{1!} + \dfrac{f''(a)(x - a)^2}{2!} + \dfrac{f'''(a)(x - a)^3}{3!} + ...[/tex]
First, let's calculate the derivatives of [tex]f(x) = e^x[/tex]:
[tex]f(x) = e^x\\f'(x) = e^x\\f''(x) = e^x\\f'''(x) = e^x[/tex]
Now, let's evaluate these derivatives at a = ln(10):
[tex]f(a) = e^{(ln(10))}\ = 10\\f'(a) =e^{(ln(10))}\ = 10\\f''(a) =e^{(ln(10))}\ = 10\\f'''(a) = e^(ln(10)) = 10[/tex]
Plugging these values into the Taylor series formula:
[tex]f(x) = 10 + 10\dfrac{(x - ln(10))}{1!} + \dfrac{10(x - ln(10))^2}{2!} + \dfrac{10(x - ln(10))^3}{3!}[/tex]
Simplifying the terms:
[tex]f(x) = 10 + 10(x - ln(10)) + \dfrac{10(x - ln(10))^2}{2} + \dfrac{10(x - ln(10))^3}{3!}[/tex]
Therefore, the first four nonzero terms of the Taylor series for [tex]f(x) = e^x[/tex]centered at a = ln(10) are:
10, 10(x - ln(10)), [tex]\dfrac{5(x - ln(10))^2}{2}[/tex], [tex]\dfrac{(x - ln(10))^3}{3!}[/tex]
b) To write the power series using summation notation, we can rewrite the Taylor series as:
[tex]\sum_{n=0}^{\infty} \dfrac{(10 (x - ln(10))^n)}{ n!}[/tex]
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Consider the following linear trend models estimated from 10 years of quarterly data with and without seasonal dummy variables d . \( d_{2} \), and \( d_{3} \). Here, \( d_{1}=1 \) for quarter 1,0 oth
The linear trend models estimated from 10 years of quarterly data can be enhanced by incorporating seasonal dummy variables [tex]d_{2}[/tex] and [tex]d_{3}[/tex], where d₁ =1 for quarter 1 and 0 for all other quarters. These dummy variables help capture the seasonal patterns and improve the accuracy of the trend model.
In time series analysis, it is common to observe seasonal patterns in data, where certain quarters or months exhibit consistent variations over time. By including seasonal dummy variables in the linear trend model, we can account for these patterns and obtain a more accurate representation of the data.
In this case, the seasonal dummy variables [tex]d_{2}[/tex] and [tex]d_{3}[/tex] are introduced to capture the seasonal effects in quarters 2 and 3, respectively. The dummy variable [tex]d_{1}[/tex] is set to 1 for quarter 1, indicating the reference period for comparison.
Including these dummy variables in the trend model allows for a more detailed analysis of the seasonal variations and their impact on the overall trend. By estimating the model with and without these dummy variables, we can assess the significance and contribution of the seasonal effects to the overall trend.
In conclusion, incorporating seasonal dummy variables in the linear trend model enhances its ability to capture the seasonal patterns present in the data. This allows for a more comprehensive analysis of the data, taking into account both the overall trend and the seasonal variations.
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You would like to develop a variable control chart with
three-sigma control limits. If your 10 samples each contain 20
observations, what value of D4 should you use for your R-
Chart?
To develop a variable control chart with three-sigma control limits for 10 samples, each containing 20 observations, the value of D4 that should be used for the R-Chart is approximately 2.282.
The value of D4 is a constant used in the calculation of control limits for the R-Chart, which monitors the variability or range within each sample. The control limits for the R-Chart are typically set at three times the average range (R-bar) of the samples.
The value of D4 depends on the sample size and is found in statistical tables or can be calculated using mathematical formulas. For a sample size of 10, the value of D4 is approximately 2.282. This value ensures that the control limits are set at three times the average range, providing an appropriate measure of variability and indicating when a process is out of control.
By using the value of D4 = 2.282 in the R-Chart calculation, you can establish three-sigma control limits that effectively monitor the variability in the process and help identify any unusual or out-of-control variation.
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Carry out the following arithmetic operations. (Enter your answers to the correct number of significant figures.) the sum of the measured values 521, 142, 0.90, and 9.0 (b) the product 0.0052 x 4207 (c) the product 17.10
We need to carry out the arithmetic operations for the following :
(a) The sum of the measured values 521, 142, 0.90, and 9.0 is: 521 + 142 + 0.90 + 9.0 = 672.90
(b) The product of 0.0052 and 4207 is: 0.0052 x 4207 = 21.8464
(c) The product of 17.10 is simply 17.10.
In summary, the values obtained after carrying out the arithmetic operation are:
(a) The sum is 672.90.
(b) The product is 21.8464.
(c) The product is 17.10.
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Find the derivatives. Please do not simplify your answers.
a. y = xe^4x
b. F(t)= ln(t−1)/ √t
The derivatives of the given functions are as follows:
a. y' = (1 + 4x)e^(4x)
b. F'(t) = (1/(t-1)) * (1/2√t) - ln(t-1)/(2t^(3/2))
a. To find the derivative of y = xe^(4x), we use the product rule. Let's differentiate each term separately:
y = x * e^(4x)
y' = x * (d(e^(4x))/dx) + (d(x)/dx) * e^(4x)
= x * (4e^(4x)) + 1 * e^(4x)
= (4x + 1) * e^(4x)
b. To find the derivative of F(t) = ln(t-1)/√t, we use the quotient rule. Differentiate the numerator and denominator separately:
F(t) = ln(t-1)/√t
F'(t) = (d(ln(t-1))/dt * √t - ln(t-1) * d(√t)/dt) / (√t)^2
= (1/(t-1) * √t - ln(t-1) * (1/2√t)) / t
= (1/(t-1)) * (1/2√t) - ln(t-1)/(2t^(3/2))
Therefore, the derivatives of the given functions are y' = (4x + 1) * e^(4x) for part (a), and F'(t) = (1/(t-1)) * (1/2√t) - ln(t-1)/(2t^(3/2)) for part (b).
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Find the Laplace transform of the given function: f(t)={0,(t−6)4,t<6t≥6 L{f(t)}= ___where s> ___
The Laplace transform of the given function is [tex]L{f(t)} = 4!/s^5[/tex], where s > 0.
For t < 6, f(t) = 0, which means the function is zero for this interval.
For t ≥ 6, [tex]f(t) = (t - 6)^4.[/tex]
To find the Laplace transform, we use the definition:
L{f(t)} = ∫[0,∞[tex]] e^(-st) f(t) dt.[/tex]
Since f(t) = 0 for t < 6, the integral becomes:
L{f(t)} = ∫[6,∞] [tex]e^(-st) (t - 6)^4 dt.[/tex]
To evaluate this integral, we can use integration by parts multiple times or look up the Laplace transform table. The Laplace transform of (t - 6)^4 can be found as follows:
[tex]L{(t - 6)^4} = 4! / s^5.[/tex]
Therefore, the Laplace transform of the given function is:
[tex]L{f(t)} = 4! / s^5, for s > 0.[/tex]
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A (7,4) linear coding has the following generator matrix.
G = 1 0 0 0 1 1 0
0 1 0 0 0 1 1
0 0 1 0 1 1 1
0 0 0 1 1 0 1
(a) If message to be encoded is (1 1 1 1), derive the corresponding code word?
(b) If receiver receive the same codeword for (a), calculate the syndrome
(c) Write equations for output code for the below
(d) What is the code rate of (c)
a. The corresponding codeword for the message [1 1 1 1] is [0 0 0 0 0 0 0].
b. The syndrome for the received codeword [0 0 0 0 0 0 0] is [0 0 0].
c. [c1 + c4 c2 + c4 c3 + c4 (c1 + c3 + c4) (c1 + c2 + c3 + c4) (c2 + c3 + c4) (c1 + c2 + c4)]
d. the code rate is 4/7
(a) To derive the corresponding codeword using the generator matrix G, we multiply the message vector by the generator matrix:
Message vector: m = [1 1 1 1]
Codeword = m * G
= [1 1 1 1] * G
= [1 1 1 1] * [1 0 0 0 1 1 0; 0 1 0 0 0 1 1; 0 0 1 0 1 1 1; 0 0 0 1 1 0 1]
= [1 0 0 0 1 1 0] + [1 1 1 1 0 1 1] + [0 0 0 1 1 0 1]
= [2 2 2 2 2 2 2]
= [0 0 0 0 0 0 0] (mod 2)
Therefore, the corresponding codeword for the message [1 1 1 1] is [0 0 0 0 0 0 0].
(b) To calculate the syndrome for the received codeword, we need to multiply the received codeword by the parity check matrix H:
Received codeword: r = [0 0 0 0 0 0 0]
Syndrome = r * H
= [0 0 0 0 0 0 0] * [1 1 1 0 1 0 1; 1 1 0 1 0 1 0; 1 0 1 1 0 1 1]
= [0 0 0] (mod 2)
Therefore, the syndrome for the received codeword [0 0 0 0 0 0 0] is [0 0 0].
(c) To write equations for the output code, we can use the generator matrix G. The output code can be represented as:
Output code = Input code * G
Let's represent the input code as a vector c = [c1 c2 c3 c4], where ci represents the ith bit of the input code. Then, the output code can be written as:
Output code = c * G
= [c1 c2 c3 c4] * [1 0 0 0 1 1 0; 0 1 0 0 0 1 1; 0 0 1 0 1 1 1; 0 0 0 1 1 0 1]
= [c1 + c4 c2 + c4 c3 + c4 c1 + c3 + c4 c1 + c2 + c3 + c4 c1 + c2 + c3 + c4 c2 + c3 + c4 c1 + c2 + c4]
= [c1 + c4 c2 + c4 c3 + c4 (c1 + c3 + c4) (c1 + c2 + c3 + c4) (c2 + c3 + c4) (c1 + c2 + c4)]
(d) The code rate represents the ratio of the number of message bits to the number of transmitted bits. In this case, the generator matrix G has 4 columns representing the message bits and 7 columns representing the transmitted bits. Therefore, the code rate is 4/7.
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The number of visitors P to a website in a given week over a 1-year period is given by P(t) = 123 + (t-84) e^0.02t, where t is the week and 1≤t≤52.
a) Over what interval of time during the 1-year period is the number of visitors decreasing?
b) Over what interval of time during the 1-year period is the number of visitors increasing?
c) Find the critical point, and interpret its meaning.
a) The number of visitors is decreasing over the interval ________ (Simplify your answer. Type integers or decimals rounded to three decimal places as needed. Type your answer in interval notation.)
b) The number of visitors is increasing over the interval ____ (Simplify your answer. Type integers or decimals rounded to three decimal places as needed. Type your answer in interval notation.)
c) The critical point is __________ (Type an ordered pair. Type integers or decimals rounded to three decimal places as needed.) Interpret what the critical point means. The critical point means that the number of visitors was (Round to the nearest integer as needed.)
a) The number of visitors is decreasing over the interval (52.804, 84]
b) The number of visitors is increasing over the interval [1, 52.804)
c) The critical point is (52.804, 3171.148).
Solution:
The given function is: P(t) = 123 + (t-84) e^0.02t
We need to find the intervals of time during the 1-year period is the number of visitors increasing or decreasing.
To find the intervals of increase or decrease of the function, we need to find the derivative of the function, i.e., P'(t).
Differentiating P(t), we get:
P'(t) = 0.02 e^0.02t + (t-84) (0.02 e^0.02t) + e^0.02t
On simplifying, we get:
P'(t) = (t-83) e^0.02t + 0.02 e^0.02t
We need to find the critical points of the function P(t).
Let P'(t) = 0 for critical points.
(t-83) e^0.02t + 0.02
e^0.02t = 0
e^0.02t (t - 83.5)
= 0
Either e^0.02t = 0, which is not possible or(t - 83.5) = 0
Thus, t = 83.5 is the critical point.
We can check if the critical point is maximum or minimum by finding the value of P''(t),
i.e., the second derivative of P(t).
On differentiating P'(t), we get:
P''(t) = e^0.02t (t-83+0.02) = e^0.02t (t-83.02)
We can see that P''(83.5) = e^0.02(83.5) (83.5 - 83.02) = 3.144 > 0
Thus, t = 83.5 is the point of local minimum and P(83.5) is the maximum number of visitors to the website over the 1-year period.
(a) We need to find the interval(s) of time during the 1-year period when the number of visitors is decreasing.
P'(t) < 0 for decreasing intervals.
P'(t) < 0(t-83)
e^0.02t < -0.02
e^0.02t(t - 83) < -0.02 (We can cancel e^0.02t as it's positive for all t)
Thus, t > 52.804
This means the number of visitors is decreasing over the interval (52.804, 84].
(b) We need to find the interval(s) of time during the 1-year period when the number of visitors is increasing.
P'(t) > 0 for increasing intervals.
P'(t) > 0(t-83)
e^0.02t > -0.02
e^0.02t(t - 83) > -0.02
Thus, t < 52.804This means the number of visitors is increasing over the interval [1, 52.804).
(c) We need to find the critical point of the function and its interpretation.
The critical point is (83.5, 3171.148).This means that the maximum number of visitors to the website over the 1-year period was 3171.148 (rounded to the nearest integer).
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Find the indefinite integral. ∫x5−5x/x4 dx ∫x5−5x/x4 dx=___
The indefinite integral of ∫(x^5 - 5x) / x^4 dx can be found by splitting it into two separate integrals and applying the power rule and the constant multiple rule of integration.
∫(x^5 - 5x) / x^4 dx = ∫(x^5 / x^4) dx - ∫(5x / x^4) dx
Simplifying the integrals:
∫(x^5 / x^4) dx = ∫x dx = (1/2)x^2 + C1, where C1 is the constant of integration.
∫(5x / x^4) dx = 5 ∫(1 / x^3) dx = 5 * (-1/2x^2) + C2, where C2 is another constant of integration.
Combining the results:
∫(x^5 - 5x) / x^4 dx = (1/2)x^2 - 5/(2x^2) + C
Therefore, the indefinite integral of ∫(x^5 - 5x) / x^4 dx is (1/2)x^2 - 5/(2x^2) + C, where C represents the constant of integration.
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∫√5+4x−x²dx
Hint: Complete the square and make a substitution to create a quantity of the form a²−u². Remember that x²+bx+c=(x+b/2)²+c−(b/2)²
By completing the square and creating a quantity in the given form, the result is ∫√(5+4x-x²)dx = (2/3)(5+4x-x²)^(3/2) - (8/3)arcsin((2x-1)/√6) + C, where C is the constant of integration.
To evaluate the integral ∫√(5+4x-x²)dx, we can complete the square in the expression 5+4x-x². We can rewrite it as (-x²+4x+5) = (-(x²-4x) + 5) = (-(x²-4x+4) + 9) = -(x-2)² + 9.
Now we have the expression √(5+4x-x²) = √(-(x-2)² + 9). We can make a substitution to create a quantity of the form a²-u². Let u = x-2, then du = dx.
Substituting these values into the integral, we get ∫√(5+4x-x²)dx = ∫√(-(x-2)² + 9)dx = ∫√(9 - (x-2)²)dx.
Next, we can apply the formula for the integral of √(a²-u²)du, which is (2/3)(a²-u²)^(3/2) - (2/3)u√(a²-u²) + C. In our case, a = 3 and u = x-2.
Substituting back, we have ∫√(5+4x-x²)dx = (2/3)(5+4x-x²)^(3/2) - (2/3)(x-2)√(5+4x-x²) + C.
Simplifying further, we get ∫√(5+4x-x²)dx = (2/3)(5+4x-x²)^(3/2) - (8/3)(x-2)√(5+4x-x²) + C.
Finally, we can rewrite (x-2) as (2x-1)/√6 and simplify the expression to obtain the final answer: ∫√(5+4x-x²)dx = (2/3)(5+4x-x²)^(3/2) - (8/3)arcsin((2x-1)/√6) + C, where C is the constant of integration.
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Find an equation of the tangert tine to the given nirve at the speafied point.
y= x² + 1/x²+x+1, (1,0)
y =
The equation of the tangent line to the curve y = x^2 + 1/(x^2 + x + 1) at the point (1, 0) is y = 2x - 2.
To find the equation of the tangent line, we need to determine the slope of the tangent line at the given point and then use the point-slope form of a linear equation.
First, let's find the derivative of the given function y = x^2 + 1/(x^2 + x + 1). Using the power rule and the quotient rule, we find that the derivative is y' = 2x - (2x + 1)/(x^2 + x + 1)^2.
Next, we substitute x = 1 into the derivative to find the slope of the tangent line at the point (1, 0). Plugging in x = 1 into the derivative, we get y' = 2(1) - (2(1) + 1)/(1^2 + 1 + 1)^2 = 1/3.
Now we have the slope of the tangent line, which is 1/3. Using the point-slope form of a linear equation, we can write the equation of the tangent line as y - 0 = (1/3)(x - 1), which simplifies to y = 2x - 2.
Therefore, the equation of the tangent line to the curve y = x^2 + 1/(x^2 + x + 1) at the point (1, 0) is y = 2x - 2.
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A satellite is 13,200 miles from the horizon of Earth. Earth's radius is about 4,000 miles. Find the approximate distance the satellite is from the Earth's surface.
The satellite is approximately 9,200 miles from the Earth's surface.
To find the approximate distance the satellite is from the Earth's surface, we can subtract the Earth's radius from the distance between the satellite and the horizon. The distance from the satellite to the horizon is the sum of the Earth's radius and the distance from the satellite to the Earth's surface.
Given that the satellite is 13,200 miles from the horizon and the Earth's radius is about 4,000 miles, we subtract the Earth's radius from the distance to the horizon:
13,200 miles - 4,000 miles = 9,200 miles.
Therefore, the approximate distance of the satellite from the Earth's surface is around 9,200 miles.
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Give an equation for the sphere that passes through the point (6,−2,3) and has center (−1,2,1), and describe the intersection of this sphere with the yz-plane.
The equation of the sphere passing through the point (6, -2, 3) with center (-1, 2, 1) is[tex](x + 1)^2 + (y - 2)^2 + (z - 1)^2[/tex] = 70. The intersection of this sphere with the yz-plane is a circle centered at (0, 2, 1) with a radius of √69.
To find the equation of the sphere, we can use the general equation of a sphere: [tex](x - h)^2 + (y - k)^2 + (z - l)^2 = r^2[/tex], where (h, k, l) is the center of the sphere and r is its radius. Given that the center of the sphere is (-1, 2, 1), we have[tex](x + 1)^2 + (y - 2)^2 + (z - 1)^2 = r^2[/tex]. To determine r, we substitute the coordinates of the given point (6, -2, 3) into the equation: [tex](6 + 1)^2 + (-2 - 2)^2 + (3 - 1)^2 = r^2[/tex]. Simplifying, we get 49 + 16 + 4 = [tex]r^2[/tex], which gives us [tex]r^2[/tex] = 69. Therefore, the equation of the sphere is[tex](x + 1)^2 + (y - 2)^2 + (z - 1)^2[/tex] = 70.
To find the intersection of the sphere with the yz-plane, we set x = 0 in the equation of the sphere. This simplifies to [tex](0 + 1)^2 + (y - 2)^2 + (z - 1)^2[/tex] = 70, which further simplifies to [tex](y - 2)^2 + (z - 1)^2[/tex] = 69. Since x is fixed at 0, we obtain a circle in the yz-plane centered at (0, 2, 1) with a radius of √69. The circle lies entirely in the yz-plane and has a two-dimensional shape with no variation along the x-axis.
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Find the partial derative f(x) for the function f(x, y) = √ (l6x+y^3)
The partial derivative ∂f/∂x of the function f(x, y) = √(16x + y^3) with respect to x is given by: ∂f/∂x = 8 / √(16x + y^3)
To find the partial derivative of f(x, y) with respect to x, denoted as ∂f/∂x, we treat y as a constant and differentiate f(x, y) with respect to x.
f(x, y) = √(16x + y^3)
To find ∂f/∂x, we differentiate f(x, y) with respect to x while treating y as a constant.
∂f/∂x = ∂/∂x (√(16x + y^3))
To differentiate the square root function, we can use the chain rule. Let u = 16x + y^3, then f(x, y) = √u.
∂f/∂x = ∂/∂x (√u) = (1/2) * (u^(-1/2)) * ∂u/∂x
Now, we need to find ∂u/∂x:
∂u/∂x = ∂/∂x (16x + y^3) = 16
Plugging this back into the expression for ∂f/∂x:
∂f/∂x = (1/2) * (u^(-1/2)) * ∂u/∂x
= (1/2) * ((16x + y^3)^(-1/2)) * 16
= 8 / √(16x + y^3)
Therefore, the partial derivative ∂f/∂x of the function f(x, y) = √(16x + y^3) with respect to x is given by:
∂f/∂x = 8 / √(16x + y^3)
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Prove the quotient rule by using the product rule and chain rule
Quotient Law: f(x)=h(x)g(x),f′(x)=[h(x)]2g′(x)⋅h(x)−h′(x)⋅g(x)
Product law: f(x)=g(x)⋅h(x),f′(x)=g′(x)⋅h(x)+h′(x)⋅g(x)
Chain rule: f(x)=g[h(x)],f′(x)=g′[h(x)]⋅h′(x)
Hint: f(x)=h(x)g(x)=g(x)⋅[h(x)]−1
To prove the quotient rule using the product rule and chain rule, we can express the quotient as a product with the reciprocal of the denominator. By applying the product rule and chain rule to this expression, we can derive the quotient rule.
Let's consider the function f(x) = h(x)/g(x), where g(x) ≠ 0.
We can rewrite f(x) as f(x) = h(x)⋅[g(x)]^(-1).
Now, using the product rule, we differentiate f(x) with respect to x:
f'(x) = [h(x)⋅[g(x)]^(-1)]' = h(x)⋅[g(x)]^(-1)' + [h(x)]'⋅[g(x)]^(-1).
The derivative of [g(x)]^(-1) can be found using the chain rule:
[g(x)]^(-1)' = -[g(x)]^(-2)⋅[g(x)]'.
Substituting this into the previous expression, we have:
f'(x) = h(x)⋅(-[g(x)]^(-2)⋅[g(x)]') + [h(x)]'⋅[g(x)]^(-1).
Simplifying further, we obtain:
f'(x) = -h(x)⋅[g(x)]^(-2)⋅[g(x)]' + [h(x)]'⋅[g(x)]^(-1).
To express the derivative in terms of the original function, we multiply by g(x)/g(x):
f'(x) = -h(x)⋅[g(x)]^(-2)⋅[g(x)]'⋅g(x)/g(x) + [h(x)]'⋅[g(x)]^(-1)⋅g(x)/g(x).
Simplifying further, we have:
f'(x) = [-h(x)⋅[g(x)]'⋅g(x) + [h(x)]'⋅g(x)]/[g(x)]^2.
Finally, noticing that -h(x)⋅[g(x)]'⋅g(x) + [h(x)]'⋅g(x) can be expressed as [h(x)]'⋅g(x) - h(x)⋅[g(x)]' (by rearranging terms), we obtain the quotient rule:
f'(x) = [h(x)]'⋅g(x) - h(x)⋅[g(x)]'/[g(x)]^2.
Therefore, we have proven the quotient rule using the product rule and chain rule.
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