Write the function f (z) = z³ + z + 1 in the form f (z) = u(x, y) + iv(x, y).
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- 1. Use the definition of the derivative of complex function to show that f(z) is not differentiableat z = 0 wheref(z) = conjugate of z3/ z2 , if z is not equal to 0 and f(z) = 0, if z =0Suppose that y' = f(y) has no constant solutions, where f : R -> R is in finitely many times differentiable at all points in R. What is the behavior of solutions to y' = f(y).For the function g(x, y) = x^2 + kxy + y^2 where k is a constant, show that g(x, y) has a critical point at the origin for all values of k. Calculate the discriminant at (0, 0).
- Determine two functions, defined on the interval ( − ∞ , ∞ ) , whose Wronskian is given by W ( f 1 , f 2 ) = e 2 x . Are the functions that you found linearly independent on ( − ∞ , ∞ ) ? How do you know?Show, using the definition of the derivative, that the function f (x, y) = xy^2 isdifferentiable at every point (x0, y0) ∈ R2.Note: If you argue that f is differentiable because of continuity of its partial derivatives,you will receive zero points.Hint: Clearly the partial derivatives of f exist and are continuous, so we know from lec-tures that the derivative will exist and it will be given by A(x0, y0) = 〈∂xf (x0, y0), ∂yf (x0, y0)〉.Suppose w=x/y+y/z, wherex=e3t, y=2+sin(t)x=e3t, and z=2+cos(6t). A ) Use the chain rule to find dwdtdwdt as a function of x, y, z, and t. Do not rewrite x, y, and z in terms of t, and do not rewrite e3t as x. dw/dt= B ) Use part A to evaluate dwdtdwdt when t=0.
- Find the partial derivative of z with respect to u, if z = x^4+x(^2)y, x = s+2t-u and y=stu^2, when s=4, t=2, and u=1.1. Find the directional derivative of the function f(x, y, z) = In (3x + 6y + 9z) at the point P1(1,2,3) in the direction of P2(4, 6, 5).Given the analytical function f(z) defined by: f(z)=u(x,y)+iv(x,y) where u(x,y)=x^2−y^2 and v(x,y) is an unknown function. Using the Cauchy-Riemann equations, find v(x,y). I dont have any intitial conditions so leave the constant term as C .
- A) Find and classify the critical points of f (z, y) = x^3 + y^3 − 6y^2 − 3x + 9 using the first andsecond derivative tests(a) express dw/dt as a function of t, both by using the Chain Rule and by expressing w in terms of t and differentiating directly with respect to t. Then (b) evaluate dw/ dt at the given value of t. 1. w = x2 + y2, x = cos t + sin t, y = cos t - sin t; t=0 2. w = x/ z + y /z , x = cos^2 t, y = sin2 t, z = 1/t ; t =3