Suppose water is leaking from a tank through a circular hole of area An at its bottom. When water leaks through a hole, friction and contraction of the stream near the hole reduce the volume of water leaving the tank per second to cApv 2gh, where c (0

Q: 3. Find the maximum and minimum values of a 1 āz - on |z| ≤ 1 (where a with la| < 1 is fixed)

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Q: 1. Find the orthogonal trajectories of the family of curves x? + y2 = cy.

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Q: 1. Find the orthogonal trajectories of the given family of curves: a. y = Ce-mx; with m held fixed

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Q: Use Taylor's formula for f(x,y) at the origin to find quadratic and cubic approximations of f(x,y) =…

A: The function fx,y=11-3x-y Then the taylor's formula for fx,y is, fxx,y=31-3x-y2fyx,y=11-3x-y2 Then…

Q: 5Find the Orthogonal trajectories of the family of r=aeº

A: Question:- Find the orthogorial trajectories of the family of r=aeθ ?Solution :- We have r=aeθ -(i)…

Q: Assume that after  months, butterflies’ populace  in 1000 is calculated as: f(w) = 12w + cos(16w) +…

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Q: Show that f(x, y) = x^2 + y^2 − 2xy + 4 has infinite number of critical points and that D = 0 at…

A: Given that      f(x,y) = x2+y2 -2xy+4 has infinite number of critical points and D= 0 at each one.…

Q: 1. Find the orthogonal trajectories of the given family of curves: a. y= Ce¯mx; with m held fixed b.…

A: This question is related to family of curves, We will solve it using given information.

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A: Given, cylinder has diameter (D) 4 cm and height (H) 8 cm error in diameter and height=dD=dH=0.5

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Q: 5. If c is a critical point of F(x), when the is a local maximum or minimum at c? Explain

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Q: (6) Find the orthogonal trajectories of the 3C cos 0, where C family of curves r being parameter.

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Q: 1.Test the function def ined by f(x.y)=xlnx- ylnx for its homogeneity. If it is homogeneous, give…

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Q: Assume that after months, butterflies’ populace in 1000 is calculated as: f(w) = 12w + cos(16w)…

A: Given f(w) = 12w + cos(16w) +n * 100

Q: Find the orthogonal trajectories of the family of curves xy = a where a ≠ 0is an arbitrary constant.

A: Consider the equation: xy=Cy=Cxy'=-Cx2 Substitute the value of C in the first derivative:…

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A: The boundary conditions are y(0,t) = y(l,t) = 0 and initial conditions are y(x,0) = μ0sinπxl and…

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Q: (3) A study on optimizing revenue from a website considered dividing cus- tomers into two groups…

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Q: 4 ース -where do es t have relahive minimums ? the relahi ve minimum is which occuRs at X= where does…

A: topic- relative maximum and minimum of function  relative minimum is the value of function where…

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Q: Q1: Use ewton- Raphson method to estimate the Yoot of fcx) = x -x -1 , emp loying an initial %3D…

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Q: Test the function def ined by f (x,y) = - x- 4y for its homogeneity. If it is homogeneous, give its…

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Q: 4. Find the orthogonal trajectories of the family of parabolas with vertices at the origin and foci…

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Q: Assume that after  months, butterflies’ populace  in 1000 is calculated as:…

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Q: 3) Consider a particle of mass m in a one-dimensional box of width L. Apply the variational…

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Q: Find the orthogonal trajectories to the family of curves: r=c- (in y -1).

A: Orthogonal trajectories can be find in the following way: (i) differentiate given family of curves,…

Q: 2. A firm uses capital, K, and labor, L, to produce output P according to P = KaL, a, 3 are positive…

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Q: 5) Construct a linear approximation of the function f(x ) = lacX+s) about the point x,= 1. Compare…

A: Tangent line approximation or linear approximation is very significant in differential calculus.…

Q: Suppose water is leaking from a tank through a circular hole of area A, at its bottom. When water…

A: Mathematical modeling is the real-life application of the differential equation. One such important…

Q: 5. Find the orthogonal trajectories for a family of circles tangent to the at y-axis.

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Q: Given f(x)= x' – rx + k where r E r > 0 and r # 1, and k to be any constant. Prove tha if 0 1, fhas…

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Q: Given the function y"=-6X(x²-ax+4) (x+a)³ (x -1)³ 3. with (X+2)(x-) - analysiS Enclude all maximums…

A: given that y=-x3x+2x-2find the maxima and minima for the given function

Q: 1. Find the orthogonal trajectories of the family of curve z² + 3y which is passing through (10,…

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Q: let fie) - 71* Ihen f has a stotionary point at z- The stationary point is CNo arswer given Oa local…

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Q: Suppose water is leaking from a tank through a circular hole of area A, at its bottom. When water…

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Q: Show that (x, y, z) = xy + y² + 22 has no local extrema by showing that its unique critical point is…

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Q: Suppose water is leaking from a tank through a circular hole of area A, at its bottom. When water…

A: Given: Water is leaking from a circular hole(in the bottom) whose area is Ah. When water leaks…

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Q: Sketch, by hand (without using any graphing device), several trajectories in the phase…

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Q: A clamped cubic spline for a function f on 0,3 is given by [1+ax² -2x', S(x) = . [1-2(x-1)+b(x-1)² +…

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Q: Q6) A string of infinite length is stretched from -00 to +o0 and subjected to the following initial…

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Q: 1 Find skewness and kurtosis for the RV X with PDF f(x) = 2T -∞ < x < x.

A: PDF of X,  fX(x) = 12πe-x22, -∞&lt;x&lt;∞ So, μ = E(X) = ∫-∞∞x f(x) dx=∫-∞∞x 12πe-x22 dx=0 [as…

Q: Once again, a particle travels at velocity t – 1 at time t. Divide the interval 0, 2 into 2k…

A: A particle travels at the velocity t-1 at time t.  We now divide the interval 0, 2 into 2k intervals…