Prove, if f(x)= V1+x² –In +1+ 1 is increasing and concave on x X E (0;+∞) .

Q: Let f € L'(0,1). Assume that f(t)t2"dt = 0 for all n f = 0 a.e. 1,2,3,.... Prove that %3D

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Q: 1 Let f (x,y) =In Then f (1, – 1) =... 3x2+4y3 XX

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Q: 1. Suppose f(-2) = 3, f'(-2) =-4 g(3) = 2, g'(3) = -1 1.1. What is the linear approximation to k(x)…

A: Linear approximation formula is k(x)=k(a)+k'(a)(x-a) Formula of derivative of composition of…

Q: Let ww. f(x) = x² + Mx a) Approximate to f' with step size h = 0.1 by using central difference…

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Q: 4. Let A(x) = ſ dt. Find the local minima, local maxima, and t2+1 inflection points of A(x).

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Q: If L(x)=mx+b is the linearization of f(x)=2x³ - x +3 at x-1, then m= Using L to approximate 2(1.01)…

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Q: (a) Prove and verify that the gradient of the tangent line to the curve f(x) = V5x² + 3x – 1 at…

A: Given curve f(x)=5x2+3x-1differentiate…

Q: Let y : R →R be the real-valued function defined on the real line, which is the solution of the…

A: Ans:-  Initial value problem: y'=-xy+x,  y=0=2          solution : y'+xy=x, 40=2          is of…

Q: 1) A course repeater claims that his brother who did some Calculus at college years ago, but…

A: (1) ⇒f(x)=(2x2-1)(x2+3)x2+1⇒f'(x)=(x2+1)ddx(2x2-1)(x2+3)-(2x2-1)(x2+3)ddx(x2+1)(x2+1)2    (∵using…

Q: 2. Show that 4=x i+y°j+:'k is irotational.

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Q: 1. Show (in terms of e – 6) that a function f : [2, 7] → R be defined by f(x) = Vx² + 1 is uniformly…

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Q: Prove that for the gamma function [(x) = ftx-¹e-tdt that for n a positive integer then I(n) = (n-1)!

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Q: If the Wronskian of the three functions ex v is equal to zero, then the function y satisfies the DE…

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Q: Find the best rank 2 approximation to A from (1) with respect to ||:||F.

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Q: (a) Express the initial value problem: 1 s'(x) = -f(x), for x +1 f(0) = 1, as an integral equation…

A: The Initial value problem is given as, f'x=-1x+1fx, for  -13≤x≤12                         (1) with…

Q: 1+ By using the definition In(1+x) = / 1 -dt, show that for a certain range of values of r, t…

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Q: Find solution of an equation x X f(x) 1 1.0000 1.25 0.8000 1.50 0.6667 0.5714 1.75 2.0 0.5000 Using…

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Q: 1 1 Select all the antiderivatives of f(x)=-+ %3D x² +1

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Q: f(x) = 3 x4-4 x3-12 x2+1, has local minimum at x = %3D A O 1 .BO -2 .CO 2 .D O 0.

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Q: (a) Show whether if 0<x <1, f(x) = 2 - x if 1<x < 2 is a Dirichlet function.

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Q: 2. Show that g(x)= *** has a unique fixed point in [0,1] and find an approximation to the point that…

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Q: If Σ An diverges then > can diverges O only if c>0 O only if c< 0 O only if c+0 O for any real value…

A: Given, the series ∑an diverges, then we have to find the values of c for which the series ∑can…

Q: 1. Let flxx) = x³ + y3 - 3x2-3y2 - 9y. (a) Find all of the critical points of f.

A: Part (a): (i) Find derivative of x and y. (ii) Equate to zero, to get critical points.

Q: 39. Suppose c and d are not both 0, and let ax + b f(x) : cx + d Show that fhas no critical numbers…

A: Please see the explanation below

Q: In x 2. Use a local linear approximation of f(x) = 1+x² to estimate In (0.9) 1.81

A: Given - fx=lnx1+x2 To estimate ln0.91.81 using a local linear approximation of fx. Here we will use…

Q: 1. Evaluate the limits. e' cos (t – 4) dt - 4 (а) lim x2 x2 – 4 e2 + 2 (b) lim x00 e2x + 100

A: Solve the following

Q: Find the degree n=4 Taylor polynomial for f(x) = ln (x + 1) at c=1.

A: The degree n=4 Taylor polynomial for f(x) at c is given by f(x) = f(c) + f(1)(c)1!(x-c) +…

Q: Suppose that y = f(x) is differentiable at x = a and that g(x) = m(x - a) +c is a linear function in…

A: You have correctly answered the first part. I will straightway go to the second part.

Q: et flk) = - ** = **(- at k+1 dt k+1 1 Let f(k) %3D 1 Prove that f(k) satisfies 0<f(k)s÷ k2

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Q: t²x"" +2tx" + x' +2022x 0 for t> 0. If their Wronskian satisfies W(2) = 20, then find W(t) for all t…

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Q: 2 Consider the function f(x) - A. 6x + 4x8. f(z) has a critical point at z = Find the value of A: A…

A: The given function is f(x)=-6x2+4x-8 .............................. (1)  Differentiate with respect…

Q: Let f(x) = 7x2(x – 11) + 12. Find the critical points c that correspond to local minima.

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Q: Suppose the function f(x)=3x^4+ax^3+bx^2 has a critical point at x=2, for which f(2)= −80. Determine…

A: Suppose the function f(x)=3x4+ax3+bx2 has a critical point at x=2, for whichf(2)= −80. Determine a,…

Q: The critical Points of f(x, y) - x' -3x +3y2-6y are: None of these O (1,1) and (-1,1). O (01) and…

A: We will find the first order partial derivatives of f and equate them to 0 to find the critical…

Q: Define (a) P.T. f(x) 0. (b) P.T. f(x) = x+1 [*** sin(1²) dt x 2x f(x) = cos(x²) - cos[(x + 1)²]…

A: Note: As company policy, we solve first three sub-parts only. Please resubmit the remaining…

Q: 3.2 Let f(x) = Approximate the integral x4 x L f(x) dx using the Trapezoidal rule with n = 4. -2

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Q: -Find the least squares approxtmation polynomial of degree two for the continuous function f(x), (x)…

A: Least square method to approximate polynomial: Least square method to approximate the two degree…

Q: 4. Decompose into f(x) and g(x) such that h(x) = f(g(x)). %3D (a) h(x) = (3x + 2)² f(x) =_ g(x) =,…

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Q: Construct a suitable Liapunov function of the form ax? + cy², where a and c are to be determined.…

A: To Show - Construct a suitable Liapunov function of the form ax2 + cy2, where a and c are to be…

Q: The critical Points of f (x, y) = x³ - 3x +3y2 -6y are: O (-1,1) and (1,1) O (1,1) and (1,1) O None…

A: It is given that, f(x, y)=x3-3x+3y2-6y   Critical points of 2 variable function are the point where…

Q: 7] The Wronskian of the functions 1, x, and e -2x is (а) е' -2x (2x+1) (b) 4e-2x (c) -e(2x +1) (d)…

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Q: If the functions f(x², y²) and g(x, y) are homogeneous of degree 2 and 4, f (x, y) g(x, y) 1…

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Q: 4. Expand f (x) = 1 around a = 0, to get linear, quadratic and - 1-x cubic approximations.

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Q: 1 Let f (x,y)=In . Then f „(1, – 1) =...... 3x²+4y3 XX

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Q: (c) Apply the golden ratio search method to locate the minimum of the function f(x) = 4x³ – 20x +…

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Q: Suppose that we don't have a formula for g(x) but we know that g(1) = -4 and g'(x) = V x2 + 15 for…

A: g(1)=-4, g'(x)=x2+15 Finding the value of g '(1) g '(1)=12+15=1+15=16=4

Q: If f(x) - g(x) = 1, then f and g have the same antiderivative. %3D O True False

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Q: If f" (c) > 0 where c is a critical number of f' (x), then (c, f (c)) is a relative maximum of of f…

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Q: 1. Determine the discrete least squares trigonometric polynomial S2 (x) on the interval [-T, T] for…

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