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Rate of Change : Exponential Growth

4.) Ever wonder why bacteria are so hard to contain ? A certain coloney of bacteria has an initinal population of 10,000. After t hours, the coloney has grown to a number P(t) given by P(t) = 10000(1+.86t + t^2). a.) Find the growth rate (rate of change) of the population P with respect to time t. [FInd P'(t)] b.) Fi

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-

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-

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


What is the derivative of f(x) = 4x(x2 + 1)3 ?

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)]?


If f(x) = xe^x/sin(x) for 0<x<PI, then f' (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

Chain Rule

Please See Attachment. Suppose f: R &#61664; R is differentiable and let Show that

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?

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

Derivatives and Rate of Change : Calculate the rate a shadow is moving up a wall.

I need to determine how fast a shadow is moving up a wall. Given the heigth of the wall the height if the object that cast the shadow. The length of the wire the object moves on, and the height of the light that casts the shadow. I have worked out the first sections in an Excel 2000 spreadsheet but I need a push in the right di


The morse potential is D*[1-exp(-ax)]^2 , where D, a are positive constants. Show that x=0 is a stable equilibrium, and that the period of small oscillations about it is... (See attachment for full question)

Lagrangian multipliers

1) Who can explain me Lagrangian multipliers with drawings scheme etc... 1)I just can't imagine what is happening in space with Lagrangian multipliers. 2) I did this problem but here also I can't understand it, because I can't understand what is happening in space! could you explain it with drawings and schemes please : the

Partial differentiation

When I write (Wx)y it means the partial derivative of W according to x with y constant ! Supposethat g(x,y)=c a constant and W=f(x,y,z) . Which of the following makes sense as the derivative Wx ? : a) (Wx)x b) (Wx)y c) (Wx)z 2) suppose that cos(x-y)=5u and W=x^2*y*u. Find (Wu)x. 3) Consider the cu

Derivative and a Tangent Line

Compute the derivative of the given function and find the equation of the line that is tangent to its graph for the specified value of x0. Equation: f(x)= x^3-x; x0=-2

Solve: Computing a Derivative

Complete the following: Solve for the derivative of the given function, finding the slope of the line that is tangent to its graph for the specified value of the independent variable f(x) = x^2-1; x=-1

Find the Gradient of the function

Find the gradient of the function: (3&#8730;&#952;^3) / 2sin2&#952; I have a number of these questions to complete could you please explain each step involved to get the correct answer