Use the Wronskian to prove that the functions f(x) = ex, g(x) = cosx, and h(x) = sinx are linearly independent on the real line.

Q: Show that the function f(x) = 2x³ – 1, from f: R → R is bijective and find its inverse.

A: The function f(x)=2x3-1, from f:ℝ→ℝ

Q: Determine whether the given set of functions is linearly independent on the interval (-∞ f₁(x) = 6,…

A: A set of function is linearly independent on the given interval if its Wronskian is non zero.

Q: If f(x) ' (1 ) cos (x)

A: We have f(x)=xcosx. So,            f'(x)=xcosx'            =xsecx'            =x'secx+secx' x…

Q: Determine whether the pair of functions is linearly independent or linearly dependent on the real…

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Q: Give an interval over which the set of two functions fi(x) = x² and f2(x) = x |x| is linearly…

A: Given: The set of two function.f1(x)=x2f2(x)=xx

Q: Determine whether the functions f1(x) := x + 1; f2(x) := sin(x); f3(x) := x - 1; f4(x) := 3 are…

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Q: What does is mean for two functions f(x) and g(x) to be orthogonal

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Q: Find the harmonic conjugate of the function and find the analytic function of f(z)=u+iv as a…

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Q: Use the IVT to show that f (x) = x3 + x takes on the value 9 for some x in [1, 2].

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Q: What does it mean for the differentiability of a function if the Cauchy-Reimann equations (Ux = Vy…

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Q: find the derivative of (2³ + j)² O 23 +j O 223 + j O 625 + j6z² O 325 + j3z?

A: Derivative of (z3+j)2 is 2(z3+j)2-1(3z2+0) = 2(z3+j)×3z2 = 6z2(z3+j) = 6z5+j6z2

Q: Consider the function f(x) = 1 + 2x + 3x² + 4x³. Let s be the sum of all distinct real roots of f(x)…

A: First derivative check for monotonicity

Q: Let n be a positive integer. Define the function fn(z) by dzn (2²+2x+2)". Find the value of fn(-1 +…

A: Solution:Given: Let be a positive integer. the function is defined bywe have to find the value of

Q: Let f(2)=u+iv be an analytic function. Given that u=v² show that f(z) is a constant.

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Q: Use a formula to express w as a function of t if  w =    4s + 9 and s = et − 1.  w =

A: We have given that,  w = 4s + 9 and  s = et − 1 Then,  We will find w in terms of t?

Q: Determine whether the functions fi(x) = cos 3r, f2(x) = x, and f3(x) = cos² r independent on the…

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Q: Find the values of x for which the gradient of the tangent to the graph of y=f(x) is equal to the…

A: We have to find the values of x for which the gradient of the tangent to the graph of y=f(x) is…