By writing the ODEs in their normal form near the bifurcation point show that (a) ˙x = 1 + rx x cos(x) + sin(r) undergoes a saddle-node bifurcation at (r∗ , x∗ ) = (0, 0), (b) ˙x = (r2 2 r + 1)(x + x3) ) ln(1 + x) undergoes a transcritical bifurcation at (r∗ , x∗ ) = (1, 0). Please specify the order of the approximation as an error in ε > 0 and the corresponding size of r, x, e.g. |x - x∗| < ε.

Q: agrange's mea 2x in 0,

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Q: a.) If f(x)= sin(In(x* )), find f'(x). b.) If g(x) = 5x find g'(x). %3D .2 x +3

A: (a) Consider the provided function, fx=sinlnx3 Find the derivative of the provided function. Apply…

Q: *) Use the Wronskian to determine whether the functions y₁ = sin(3x) and y₂ = cos(4x) cos(4x) are…

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Q: Let f(r) = cos(4r + 4). f'(x) =-4 sin(4r -4) f'(2) = | 0

A: Solve the given problem.

Q: The function f(x) = a sin x + b cos x + c satisfies f"(x) = cos x – 2 sin x, f'(0) = f(0) = 2. Find…

A: a=2 b=-1 c=3

Q: 2- Find the third derivative for əx3 for f(x,y) = sin) )+2 sin) sin)+3cos°) 8y3sinz s()+2sin()…

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Q: Compute the following derivatives. Show all your work and indicate which differentiation rule(s) you…

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Q: Let ƒ(x) = (cos( − 4x – 5)) (sin(6x − 2)). Then ƒ'(x) =

A: f(x)=(cos(-4x-5))(sin(6x-2))

Q: If V(x)= In[sin(x)- cot(x)], find V'(x). %3D O a. cos(x)- csc-(x) sin(x) + cot(x) O b. cos(x)-…

A: Here we have, V(x)=lnsin(x)-cot(x) We have to find V'(x)=?

Q: 2) a) Let f(x) = 2* + 3x7. Find f'(x) and f" (x). b) Find the derivative f'(x) in the following…

A: a.)   Given f(x) = 2x + 3x-1/2 f'(x) , f''(x) are to be calculated. Formulas used here  ** y = ax…

Q: a) Find a function f so that f’(x) = 6x + cos(x). b) Find a function g with g’(x) = f’(x)

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Q: 2. Find the derivative of each of the following: (a) f(x)= 3" cos² (}x) %3D (b) h(x) = Vrd – 4x + 1…

A: (a) Consider the given function, fx=3xcos212x Here the objective is to find the derivative of the…

Q: If f (x) = sin 2x , find all of the stationary points of f (x) on 0<x<2n

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Q: Let f (x) =e*. sin(x²). what is the correct result of the derivative of f with respect to x?…

A: This question is related to differentiation, We will solve it using given information.

Q: Find f if f'(x) = 2 + cos(x), ƒ(0) = 10, ƒ(π/2) = 6. 2 f(x) = x² − cos(x) – (n - + - (x + 10 ) x +…

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Q: Find the derivative of ƒ (x) = cos-1(In(4x²+1)).

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Q: Compute the derivative of g(x) = cos(t²) dt. O 42 sin(z*) O cos(x*) -42 sin(z*) O 4a cos(a) O -…

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Q: Linearize f(x) = 5 cos x about x = pi over 2. f(x) = -5x

A: linearization around x=a is given by :L(x)=f(a)+f'(a)(x-a)