Exercises 1–5, find the linearization L(x) of ƒ(x) at x = a. 1. ƒ(x) = x3 - 2x + 3, a = 2 2. ƒ(x) = sqrt(x2 + 9), a = -4 3. ƒ(x) = x + 1/ x , a = 1 4. ƒ(x) = ( x)^(1/3), a = -8 5. ƒ(x) = tan x, a =pai.

Q: 4. Find the orthogonal trajectories of the given family of curves. y = cx'.

A: Solution:-              y=cx3 →          1              Diff. w.r.t x            ⇒dydx=3cx2…

Q: Find the perticular number or orthogonal trajectories of x2+cy2=1 passing through the point (2,1)

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Q: Find the direction at which the function f(x, y, 2) = a In(e-az) increases fastest at (-3,0, 0) O…

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Q: Find the linearization of the function f(x,y) = ln(x3y6) at the point (x0,y0) = (4, 3). Leave all…

A: Given fx,y=lnx3y6     at (4, 3) The linear approximation in 2 variables is given by Lx, y=fa,…

Q: 5. Show how the origin changes stability as a changes by using the Poincaré Diagram. x - [i :]• 2 a…

A: The stability of fixed points of a system of constant coefficient linear differential equation of…

Q: Find the linearization at x = a. IT f(x) = sin³ (x), a = 4

A: Given,            The linearization at x=a.             fx=sin3x, a=π4We know that,             The…

Q: What is the formula for the linearization of a function f at the point x = a? %3D

A: Given :To provide the formula for linearization of a function f at the point x=a

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

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Q: What is the linearization L (x) of a function ƒ(x) at a point x = a? What is required of ƒ at a for…

A: linearization is finding the linear approximation to a function at a given point.It is given by,

Q: 5(a) Determine the orthogonal trajectories of the family of electric field lines described by the…

A: To find - Determine the orthogonal trajectories of the family of electric field lines described by…

Q: Find the gradient of the function at the given point. In(x2 - y) - 6, (2, 3) Z = Vz(2, 3) = %3!

A: To find out the gradient of the function at the given point..

Q: [1] Determine the orthogonal trajectories of y(1+2x?)=ax.

A: The given equation is:y1+2x2=axy=ax1+2x2              ..............(1)

Q: 8 Find the linearization L(x, y) of the function at the given point f(x, y)=1+xIn(xy-5), (2,3)

A: Given: f(x,y)=1+xln(xy−5)(2,3)L(x,y)=?

Q: Determine the orthogonal trajectories of the family of electric field lines described by the family…

A: Given    x-c3+y2-c23x-c=x3   differentiate  w.r.t.x    3x-c2+2ydydx-3c2=3x2⇒2ydydx=3x2-3x-c2+3c2…

Q: In Exercises 19–24, find the directions in which the functions increase and decrease most rapidly at…

A: Given function is Firstly, we will find gradient f at point (4,1,1)

Q: 8. Find the direction in which f(x, y, z) increases most rapidly at the point (3,2, 1). What is the…

A: We have to find direction in which function f increases most rapidly and maximum rate of increase.

Q: Make an example of an autonomous ODE, with stable equilibrium at -1 and 3, and an unstable…

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Q: Find the orthogonal trajectories of the functiony x + ce which is passing through (-2,0)

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Q: Find all equilibria of the continuous-time dynamical system given by i = cos x, x e R. Sketch the…

A: Given: Let's find all of the equilibria of the continuous-time dynamical system x˙=cosx, x∈ℝ.Draw…

Q: Find the gradient of the function at the given point. In(x² – y) - 9, (3, 8) z = Vz(3, 8) =

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Q: 5. Find the gradient of cos(xy) – 3y* + 3 at the point (1,4)

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Q: Find the linearization of z = f(x,y) at P(-3, 2) when f(-3, 2) = 2 and f-(-3, 2) = 1, fy(-3, 2) =…

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Q: Use the linearization L(3,-2) (x, y) of f(x, y) = -x²y-xy² near the point (3, -2) to approximate the…

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Q: Use the application of Calculus. TOPIC: Partial Differentiation. Solving Using the Definition az If…

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Q: Find the orthogonal trajectories of the family of circles passing through the points (1, −2) and (1,…

A: Given that

Q: Give the family of orthogonal trajectories for the equation: y2 = Cx5+5

A: To obtain the family of orthogonal trajectories for the equation: y2 = Cx5+5. Find a differential…

Q: Find the linearization of y=2^x at a=0

A: use derivative to find the Linearization of y

Q: Compute the partial derivative of h(x, z) = e4xz-3x²z. (Use symbolic notation and fractions where…

A: In this problem first we need to partially differentiate the function with respect to z and then put…

Q: 2] write the chain rule formula for the derivatives Of and E , where Z=ZCrisit).

A:

Q: Find the gradient of the function at the given point. In(x - y)- 4, (3, 8) Z = Vz(3, 8) =

A: The objective is to find the gradient function at the given point.

Q: 8. Find the orthogonal trajectories of y = cx CX

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Q: Use the Chain Rule to compute the partial derivative. би +о In (3u? + uv) = 3u + uv

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Q: 2. Show that cos(wt – B), coswt, sinwt are linearly dependent functions of t. Show Complete Solution

A: Given that: cos(wt-B), coswt, sinwt

Q: Find the orthogonal trajectories of У — сх CX

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Q: At time 0, M is deposited into Fund X, which accumulates at a force of interest δ# = 0.006t*. At…

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Q: find the partial derivatives of l(BO,B1) Σ 2 i=1 yi log ( 17 1² (B₁ + B₁ x)) + (1-y₁ ) log (1-² (B₁…

A: As per the question we are given a function l(β0 , β1) in partial summation form and from that we…

Q: 5  please answer, type the solution if you can please. or write clearly

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Q: Find the gradient of the function at the given point. In(x² - y) - 9, (2, 3) = Z Vz(2, 3) =

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Q: 3. Derive from first principles using partial differentiation the expressions satisfied by the a,…

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Q: Verify the chain rule y = t³. %3D

A: Given query is to solve the expression using chain rule of derivative.

Q: x2 - y2 + 2Cx = 0

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Q: from the initial solution (x1, X2)= (0,0) perform 2 interactive iterations of the gradient search…

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Q: 1. How do you find the linearization of f(x) at x = a? Illustrate with a concrete example

A: We have to solve

Q: So x, = c,e +c,e", and x, = 3c,e" +c,e" Sketch the trajectories in phase plane 2 -2 -2 Comment

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Q: find orthogonal trajectories of   x2+(y-c)2=c2.

A: x2+(y-c)2=c2 now differentiating both side with respect x , we get 2x+2(y-c)dydx=0 or, dydx= -x(y-c)…

Q: Find the linearization of the function z =/ at the point (-5, 1). L(1, y)

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Q: 8. Find the orthogonal trajectories of y cx

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Q: . Find the linearization of √ x at a = 49. Then use your linearization to approximate √ 56.

A: I am going to solve the given problem by using some simple calculus  get the required result.