(b) Show that the u(x, t) can be written as u(x, t) = 0(x – at) + 4(x + at), where o and are arbitrary functions.

Q: 4. Let f(x, y) = 2TY x = In 2 g and y = In s. . a. Find the equation of the tangent plane to f(x, y)…

A: Let f(x,y)= 2xy , x=t/ln2 and y= lns Here we have to find the equation of the tangent plane to…

Q: Let f(r, y, z) = ln(x² + z) + y²e#² – cos(2yz). Find fzyr- %3|

A: Consider,                                                                 f(x, y, z) = ln(x² + z) +…

Q: B: Expressand as functions of u and v both by using the Chain Rule and by expressing z directly in…

A:

Q: 2. Determine whether the functions f1(x) = e2", f2(x) = e=2#, and f3(x) = sinh 2x are linearly…

A:

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A:

Q: Let z = f(x, y) = 5x – x³ + 2xy – y². - Find and classify all the critical point(s) of this…

A: Differentiate the function fx,y=5x-x3+2xy-y2 partially with respect to x and y, to obtain fx and fy.…

Q: Solve for g'(x)?

A: Given:

Q: Prove 8xyz < (x + y)(y + z)(x + z) for every x, y, z E [0, +∞[.

A: We have to prove given identity: Concept:

Q: 2- Obtain the function x to minimize the following functional. = [ c«f + *)dt J = St: x1 = -X2 ,…

A: J=∫01x12+x22dt-------(1) x1˙=-x2-----(2) x(0)=c, x(1)=d here c,d are fixed number

Q: Find the linearization L(x) of y = e In(x) at a = 1. L(x) = %3D

A: Given a function f(x)=y=e3x ln(x) To find the linearization L(x) at a=1 The linearization L(x) is…

Q: Let z(x, y) be the function implicitly defined as the solution to 2x + y + z + sin(xyz) = 4 + that…

A:

Q: A function (f) is homogeneous of degree • if and only if : ôf + y = Af (x, y). Does the function…

A:

Q: Verify Young's Theorem y? x2 z = In(2x + y) – + e(x+2y) + - y

A:

Q: x - y -2x Prove by definition that for f(r, y) fy(x, y) = %3D x+ y' (x+ y)2

A:

Q: Find the local linearization of f(x)=x^(2) at 2

A:

Q: Find the linearization L(x) of f(x) = cot(x) at x = π/4.

A:

Q: 23. Given that f(0) = v2 cos (e-), the minimum value of o1+ v2 C. %3D COS D.-1 A. V2-1. B.2N2

A: Since you have posted multiple question in this question so I have solved first question as per our…

Q: Let f(x, y) = ln (3x + y2). Show that fxy = fyx

A:

Q: Compute the discriminant D(x, y) of the function. S(x, y) = x' + y* – 6x – 2y + 1

A:

Q: 2. Find the linearization of the function f(x, y) = In(x³y®) at the point (xo, yo) (4, 3). Leave all…

A: Consider the following function: fx,y=lnx3y6 fxx,y=ddxlnx3y6=1x3y6×3x2y6=3x…

Q: Verify Young's Theorem In(2x + y) y2 x2 + + e(x+2y)

A:

Q: [1] Show that the functions In x, In(x) are linearly dependent on (0, co)..

A:

Q: Calculate of of and for the function f(x, y) defined as : ax' dy sin (xy) 1 + y f(x, y) =

A:

Q: Let f(x, y, z) = ln(x² + z) + y²e#z – cos(2yz). Find fzya.

A:

Q: FIND THE LINEARIZATION OF f(x) = 2+ [,* dt at z= 1. %3D 1+ Lütfen birini seçin: a. L(x) = 2x – 4 b.…

A:

Q: Find the linearization L(x) of f(x) = cot x at x = 4

A:

Q: - Find the total differential df of f (x, y) = (x² + y²) 3/2.

A:

Q: Find the linearization of the function f (x)= x³ at x = 1. qb |a| sin (a) L (x) = Using this…

A: Given, fx=x5 We know that, If we have a differentiable function f defined near "a" the linearization…

Q: Find the linearization L(x) of f(x) = cot x at x = 4 %3D -

A:

Q: 9. Demonstrate that the functions Y1 = x2, y2 = sin(x), y3 = xe are linearly independent on R.

A: Recall: If y1,y2 and y3 are the given function and         Wy1,y2,y3x = y1y2y3y1'y2'y3'y1''y2''y3''…

Q: 2. Suppose that z = f (x, y) and both x and y are functions of a single variable t such that dy t2 –…

A:

Q: Find the linearization L(x) of the function at a. f(x) = sin(x), a =. 3 L(x) =

A:

Q: Find g: R R, continuous in [0. ) and position in (0, ) satisfying g1) %3D )dt

A: Here we have,  g:R→R, continuous in [0,∞) and position in (0,∞) Also given g(1)=1 and…

Q: Suppose that f(x, y) = e¯4z² - ly + 7z + 2y then find the discriminant of f.

A:

Q: Consider a regularizing term defined by where k!2* and the linear differential operator D is defined…

A: We are given the following regularizing term.                    ∫ℝN0DF(x)2 dx=∑k=0∞ak∫ℝN0DkF(x)2 dx…

Q: Show that F = (2x - 3)i - z j + (cos z)k is not conservative.

A:

Q: д. L. = a(x, y)+b(x, y) +c(x, y). – d(x, y) -

A: Proof: Consider V be any vector space of all functions of two variable x  and y whose partial…

Q: Find the linearization L(x) of the function at a. f(x) = sin(x), a = 3 L(x)

A:

Q: Let z(x, y) be the function implicitly defined as the solution to 2x + y + z + sin(xyz) = 4 + that…

A: ddxsinnx =cosnxddxnx =ncosnx

Q: (ii)If y=(x+Jx* +1. prove that y'= Vx +1

A:

Q: Find the linearization L(x) of the function at a. f(x)= sin(x), a= π\4

A:

Q: 9. Determine the analytic functions f = u+iv, for which u(x, y) = r³+rg(y), where g is a twice…

A: (9) To Find: The Analytic function f for which the real part is u(x,y)=x3+xg(y), where g is a twice…

Q: Let L(r) be the linearization of f(x) = sin x + 1 at the point x = . Find L(π/2).

A:

Q: Verify Young's Theorem z = In(2x + y) y? x2 +e(x+2y) y

A:

Q: Find the linearization L(x) of f(x) at x = a. f(x) = sin x, a = 0 O L(x) = 3x + 1 O L(x) = 0 O L(x)…

A: L(x) = f(a) + f '(a) (x - a) where, L(x) is the linear approximation of f(x) at x = a. f '(a) is…

Addition Rule of Probability

It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.

Expected Value

When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).

Probability Distributions

Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.

Basic Probability

The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.