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Apply the chain rule to a partial derivative

Use the chain rule to find dw/dt as a function of x, y, z, and t. Do not rewrite x, y, and z in terms of t, and do not rewrite e5t as x. dw/dt = Suppose: w = x/y + y/z ... (Please see the attached file for the fully formatted problem.) Note: Use exp() for the exponential function. Your answer should be an expression in

Cauchy-Riemann Equations : First-Order Partial Derivatives

Please see the attached file for full problem description. 3. Use Cauchy-Riemann equations and the given theorem to show that the function _ f (z) = e^z is not analytic anywhere. Theorem: Suppose that f (z) = u (x, y) + i v (x, y) and that f'(z) exists at a point z0 = x0 + i y0. Then the first-

Verifying Entire Functions

Apply the given theorem to verify that each of these functions is entire: (a) f (z) = 3x + y + i (3y - x) (b) f (z) = sin x cosh y + i cos x sinh y (c) f (z) = e-y sin x - i e-y cos x (d) f (z) = (z2 - 2) e-x e-iy. See attached file for proper formatting.

Change of Coordinates Lagrangian

Consider a Lagrangian system, with configuration space R^n, given by (x^1, ... x^n); and Lagrangian L(x', ..., x^n; v^1, ... v^n). Now consider a new system of coordinates, (y^1,... ^n), for this same system, so the y's are functions of the x's; and, inverting, the x's are also functions of the y's. Find the Lagrangian in the y-

Lennard-Jones Potential : First, Second, and Third Derivatives

Please see the attachment. I am trying to take the first, second, and third derivatives of this equation, so I can utilize them to determine relationship to force and modulus, and thermal expansion coefficient. Please help me take the derivatives properly. Thanks!

Word Problems : Derivatives and Rate of Change

41) Suppose that the average yearly cost per item for producing x items of a business product is C(x)=10+(100/x) . if the current production is x=10 and production is increasing at a rate of 2 items per year, find the rate of change of the average cost. 45) Suppose a 6ft tall person is 12 ft away from a 18-ft tall lamppost. i

Derivatives and Tangent Lines (8 Problems)

Find the Derivative y'(x) implicitly. 1) 3xy³-4x = 10yy² 2) sinxy= x² - 3 3) 3x+y³-4y = 10x² 4) xe^(power y) -3ysinx = 1 5) cos y - y² = 8 6) e^x² - 3y = x² + 1 Find the Equation of the Tangent Line at the Given Point. 7) x³ - 4y² = 4 at ( 2,1) 8) x³y² = -3xy at (-1

Derivatives : Composite Function

Let f(x) = sin(2x + 1) and g(x) = x3 + 3 for all real x. Which of the following is equal to the derivative of the composite function f[g(x)]?

Derivatives : Rate of Flow of Water

The volume (in gallons) of water in a tank after t hours is given by f(t) = 600 sin^2(Pi*t/12) for 0 <= t <= 6. What is the rate of flow of water into the tank, in gallons per hour?

Directional Derivative and Tangent Vector

Give a simple proof or counterexample to disprove: If tangent vector p E Rn is such that the directional derivative of f by k vanished for every function f then k =0. If function f on Rn is such that the directional derivative of f by every tangent vector at every point vanishes, then f is constant. The directional deri

Max, min, inflection pts

If f(x) = x^4 - 4x^3 + 10 find the relative extrema of the function and the points of inflection of its graph. Also, sketch the graph of the function. a. The x-value(s) of the relative minima of function f: ________________ b. The x-value(s) of the relative maxima of function f: _______________ c. The x-value(s) of poi

Directional Differentiation

Let (see equation in attached file). - find all the directional derivatives of f at 0 - is f continuous at 0? - Is f differentiable at 0?

Mathematics/derivatives and curve sketching

Find the derivative of f(x)= (3-4x^2)/(x^2-x-6) and then determine intervals of increase/decrease,local max and min values and points of concavity and inflection. Also, please tell me that if the derivative of the function is (4x^2+42x+3)/(x^2-x-6) then do I use the denominator or numerator to determine whether the function is

Gompertz Equation

2. The Gompertz equation y'(t) = y[a-b*ln(y)] is an important model for avascular tumor growth. In the avascular growth phase, tumor cells obtain nutrients directly from the surrounding tissue. (The transition from avascular to vascular growth is marked by the onset of angiogenesis, the formation of blood vessels, which are