You have given a function λ: R → R with the following properties (x ∈ R, n ∈ N): λ (n) = 0, λ (x + 1) = λ (x), λ (n+1/2) = 1 Find two functions p, q: R → R with q (x) 6= 0 for all x such that λ (x) = q (x) (p (x) + 1)

Q: let f(x)=ex for 0 < x < 1 Approximate f(0.25) using linear interpolation with x0=0  and x1=0.5 .…

A: Substitute x0=0 and x1=0.5 in fx=ex  and calculate L0(x) and L1(x), to construct the Lagrange…

Q: a2f b- For the following functions prove that ax2 ay2 1- f(x,y) = e-2y cos 2x %3D f(x,y) = In/x2 +…

A: The first step inludes the  partial differentiation of the functions individually with respect to y…

Q: Given for C[-1,1] the inner-product function (f,g)= | f(x)g(x)dx , find |F|| if f(x) =1-3x.

A: The solution to the given question is explained below.

Q: If f (x) is an invertible, differentiable function with tangent line y = 4 x + 7 at the point (0, 7)…

A: we have to swap variable for finding the inverse function

Q: 1. Show (in terms of e – 8) that a function f : R³ → R defined by f (x, y, z) = (2x + 3y + 4z) is…

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Q: 2. Let f(x) = S dt for x 2 2 In a) Find F'(x). b) Is F increasing or decreasing? What can you say…

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Q: The function g(x) = ln(x³ – 1) defined on V2, 0) is not zero unless x 2.

A: Given statement

Q: A continuous r.v. F(x1= === 2 × -IN 15 X has CDF X Co OBX C1 18X 16 17 **** 18

A: Given: Fx=0            x&lt;012x        0≤x&lt;1k            x≥1

Q: If f"(c)<0, where c is a critical value and f(x) is a continuous function over an open interval I…

A: We have to find f(c)

Q: 1 Let f(x) = 8 + In (t) dt, for x2 1. Find f'(2).

A:

Q: 1) Consider the function f(x) = In(x + 5) + x + 5 Verify that the function F(x) = x In(x + 5) is an…

A: Explanation of the answer is as follows

Q: 8. Consider the function f(x)=/25-x² . a. Find the linearization L(x) of f(x) at x=3.

A: Given expression is , fx=25-x2 to find out the linearization L(x) of f(x) at x=3.

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 Wronskian for the set of functions {e¬*, xe¯×, (x + 1)e-*}.

A:

Q: Define f : X → K by f(x) = x'(1), x € X. Again, the linear functional f is not continuous. By 6.4,…

A: Given: f:X→K by fx=x'1, x∈X To prove f is not continuous using the given corollary.

Q: A function f is approximated by a cubic spline defined by: 1+x+x² + x³ -{4+ 6(x-1) + 4(x- (4 + 6(x −…

A: Sol:- Consider the function between (0,1) ∫011+x+x2+x3dxApply the Sum Rule:…

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

A:

Q: Let f(x) = x^3 − x − 1 and x0 = 1.1, x1 = 1.2, x2 = 1.3 be approximations to a solution of f(x) = 0.…

A: By Muller's method, x3= 1.3249

Q: 1. Suppose that f : [0, 0) → R is continuous and f(x) # 0 for all x > 0. If we have (f(x))? = 2…

A:

Q: The function f(x) = x + 1/x is %3D O Decreasing on (0,1) O Increasing on (0,1) O Increasing on (-1,…

A: In this question we are given with the function and we have to find the graph of the function is…

Q: e +1 Differentiate the function f(x)= In 5x +1 Se+1 3ex+1 Sesz e3x +1 ,5x +1 e 3e3z Sesz e3x +1 5x…

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Q: 8. Given for C[0,1] the inner-product function (f,g)=[f(x)g(x)dx , find ||f|| if f(x) = x+1.

A:

Q: Find the antiderivative F of f that satisfies the given condition.  f(x) = 7 − 3(1 + x2)−1,    F(1)…

A: NOTE: Refresh your page if you can't see any equations. . take integration

Q: 1. Find grad f for the given function below. Ax, y, z) = 5xy + 3yz – 4zx

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Q: 2. Give an example of a regulated function f on [0, 1] such that f(x) > 0 for all æ e (0, 1), So…

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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: 1.4 For which values of a eR is the function ar? f(x + iy) = (a'x – axy) +i (- + ay an entire…

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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: 15. Let f(x) = x² -4x+3. What are the critical number(s)? x = 1 b. x = 3 c. x= 1 and x 3 a. d. x = 2…

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Q: Consider the regular subdivision of the interval [a, b] as a = x0 < x1 < x2 < x3 < x4 = b, with the…

A: According to Simpson's rule, if the interval a , b is divided in to n equal sub intervals then the…

Q: Given the function y=f(x), y" is found to be positive over the interval (4,8), then over this…

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Q: 9) The Wronskian of the functions f(x) = -5, g(æ) = 7x² y h(x) = e3 is given by a) W(f(x), g(x),…

A: The details calculation is given below

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: Let f(x) = e* for 0 < x<1 a. Approximate f(0.25) using linear interpolation with xo = 0 and x1 = 0.…

A:

Q: The absolute maximum of y =-x + 2x +3 on the interval [0, 3] occurs at Select one: O a. None of…

A: Simply take the derivative and equate to zero than solve for x.

Q: a'(t) = -x(t – 1), x(t) = 1 on [-1,0], is given by 1(t) = E(-1)* t – (k – 1))* k! for n-1<t <n, k=0

A: As per the question we are given the following first order time varient differential equation :…

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: For the function f(x)=sin(x)/x defined on [-1,1], there is some -1<c<1 such that f′(c)=0.f′(c)=0.…

A:

Q: 1. Suppose that f(z) is analytic for lz <1 and sa Show that there is a constant c < 1 such that…

A: To prove the estimate for }f(z)| over the disc or radius r, center 0

Q: ) Consider the function f(r) = f(1) has a critical number A %3D f"(A) = Thus we conclude that f(x)…

A: Critical point f(x) is the number at which first derivative is zero.…

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: 1 dt, for x > 4. In(t) Let F(x) 4

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Q: 1. Consider the function z = f(x, y). If fa(a, b) and fy(a, b) both exist, and they are equal, then…

A: 1. Given: z=fx,y and suppose that fxa,b and fya,b both exists and they are equal and it is…

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: 16. If f(x) = 5, (;e* G +5) dt -x, then the inflection point(s) of y = f(x) is(ar- %3D 3 A) x = 4…

A: f(x)=∫0x14t4+5dt-13x4

Q: (2) Is the function f(x) : R → R, sin a 1 ifa # πh,k & Z, ifa = Tk, k ε Z f (x) Borel measurable?

A: Consider the given function, fx=1sinxifx≠πk,k∈Z1ifx=πk,k∈Z   Thus, the given function have infinite…

Q: Given for C[-1,1] the inner-product function (f,g)= [ f(x)g(x)dx, find ||f|| if f(x) =1-3x.

A:

Q: 2. Find the principal part with respect to x for x 0 of the infinitesimal f(x) = = Vx + Vx. (That…

A: We have to use expansion of the function after some manipulations.

Q: The Gamma Function is defined by [(x) = fx-¹e-tdt, positive integer then prove that for na T(n) = (n…

A:

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.