Theorem 6.1.7 Suppose that f:I → R and g: I → R are differentiable at c E I. Then, b) The function f +g is differentiable at c and (f +g)'(c) = f'(c) + g'(c) (Prove b)

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Q: (4-х, XS0 10) Determine if the function y(X) = 3x, is differentiable at x =0. x>0

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Q: Q3) If f(z) is analytic function and u(x, y) = x3 – 3xy find f(z)

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Q: Q2\A\ Which of the following functions are bijective 2- f: R{0}→ R\{0},f(x) = 1/x این 2- f:R→R,f(x)…

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Q: 5. Let f be defined for all real numbers and suppose f(x)-f(y)| ≤ (x - y)² for all real numbers, x…

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Q: f:[0,1] -> [0,1] such that f(x)=x^2 has two fixed points. Is this true?

A: Given that f:[0,1] to [0,1].

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Q: Assume that f: R→ R and g: R → R are infinitely differentiable functions. Prove the following…

A: We have given that ,  are both infinitely differentiable.We need to prove that , We will use…

Q: b) Prove that if f(z) and f(Z) are analytic in a domain I, then I is a constant function.

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Q: Sketch: f(x) = [x + 2]] − 1, on [−2,2]

A: We have to draw the graph

Q: 1. express the given function in the form f(z)= u+iv a) f(z)=6z-5+9i b) f(z)= z+ (1/z) c) f(z)=…

A: We know that z = x + iy We express the given function in the form f(z)= u+iv, where u = u(x,y) and v…

Q: Let g(z) = (z3 + (z² – 1)5) . Then the derivative of g is g'2) = A(2? + (z? – 1)9)* (cP + F z(z? –…

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Q: Q2) a) Find Maclaurin expression of f(x) = Cosh (x) b)Find the gradient of the function f(x, y,z) =…

A: Note: According to our guideline. We will answer the first question. As the exact one is not…

Q: 1) In this problem, let x = 2, the value Ax = dx, dy = f'(x)dx, and Ay = f(x + Ax)-f(x) to complete…

A: Note:-Since you asked multiple questions ,according to our policy we can solve only first question…

Q: ] j is a differentiable function and if f'(x

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Q: 3- Prove that (y" f(x)) = "(x)y) = fy where y E Rm and f: R" → R™ %3D %3! ax ax

A: The objective is to prove that  ∂∂xyTfx=∂∂xfTxy=fxTxy where y∈ℝm  and  f:ℝn→ℝm

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.