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Functional Analysis

Weather balloon rising

A weather balloon that is rising vertically is observed from a point on the ground 300ft from the spot directly beneath the balloon. At what rate is the balloon rising when the angle between the ground and the observer's line of sight is 45 degrees and is increasing at 1 degree per second?

Mean Value Theorem

10. Does the function , , satisfy the hypotheses of the Mean Value Theorem? Find such that is the slope of the secant line passing through , and . Please see the attached file for the fully formatted problems.

Compact Subset of R^m with Convergent Sequences

Please help with the following problem. Let A be a proper subset of R^m. A is compact, x in A, (x_n) sequence in A, every convergent subsequence of (x_n) converges to x. (a) Prove the sequence (x_n) converges. Is this because all the subsequences converge to the same limit? (b) If A is not compact, show that (a

Maximum Value on an Interval

Find the absolute maximum value for f(x) = x^2 - 4x - 32 on the interval [-3, 9]. 8 11 13 14 18 24 30 none of these.

Interval

Where on the interval [0, 2π] is the function f (x) = x - sin x concave up? (π=pi)

Intervals

Please explain the steps and solution, thank you: Where on the interval [0, 2π] is the function f (x) = x - sin x concave up? (π=pi)

Transfer Function Analysis and Models

Please see the attached file for the fully formatted problems. Two identical stirred tanks with a recycle stream are connected as shown in the diagram below: Fr CAi,Fi CA1

Density and Center of Mass

1) A plate is bounded by the curves y = -x^2, y = x^2, and x = 1. It has density d(x,y)=x. Make a sketch and find its center of mass. 2) A solid occupies a half-cylinder W: x^2 + y^2 is less than or equal to 4, y is greater than or equal to 0, z is greater than or equal to -3 and less than or equal to 3. It has constant

Mobius Transformations: Example Problem

Suppose T is a Mobius transformation such that the image of the real axis under T is the real axis. Prove that T may be written in the form T(z) = (az+b)/(cz+d) with a, b, c, and d real.

Turns of a Helix

^ ^ ^ ^ For the helix r = a cos t i + a sin t j + ct k find c ( c > 0) so that the helix will make one complete turn in a distance of 3 units measured along the z - axis.

Asymptotes

Find all vertical and horizontal asymptotes of f(x) = (x^2+ 4)/(x^2- 4x-12) Please show steps & graph if applicable!

Singularities and Poles

The function f(z) = zsin(pi/z)/[(z-1)(z-2)^2] has isolated singularities only. Determine the singularities of f(z) and classify each of them as removable, a pole, or an essential singularity. If z0 is a removable singularity, find the value f(zo) that makes f(z) analytic at z0. If z0 is a pole. find the singular part of f(z) at

Asymptotes

Explain and contrast the types of asymptotes considered for rational functions.

Vectors : Divergence, Gradients and Curls

1) V(x, y, z) = (x + y + z)2 i + (x + y)2 j + x2 k. Find div V(3, 2, 4) ≡ ∇? V (3, 2, 4) 2) F (x, y) = xe2y i + y/(x + y) j. Find ∇ ? F (4, 0) 3) F (x, y, z) = -yz i + xz j - xy k. Find curl F (1, 2, 5) = ∇×F ( 1, 2, 5)

Parametric Equations

Find an equation in x and y for the curve given by the parametric equations x=e-t,y=-2t,t in R.

Equation of an Ellipse

25*y^2 + (10/sqrt2)*y*z + 4*z^2 - 50 = 0 I know when compared with the standard 2nd degree conic equation that this is an ellipse rotated 16.24 degrees. My question is , How do I put this equation in the standard form of an ellipse equation? ie: y^2/a^2 + z^2/b^2 = 1

Ellipses and Parametric Equations

7. Find the equation of the ellipse whose center is (-3,1); vertex (-3,3) and focus (-3,0). 8. Graph by hand the curve whose parametric equations are given and show its orientation. x=t+3 y=2t^2 0<=t<=5 9. Find the two different parametric equations for the rectangular equation y=x^2 +2

Holomorphic Map Functions

Suppose z= phi(&) and w=psi(&) are one-to-one analytic maps from the unit disc D(0,1) onto the regions G_1 and G_2. Set phi(0)=z_0 and psi(0)=w_0. Let 0<r<1 and omega_1(r)=phi(D(0,r)), omega_2(r)=psi(D(0,r)). Assume f: G_1->G-2 be holomorphic map with f(z_0)=w_0. Show that f(omega_1(r)) is contained in omega_2(r)

Green's Theorem Enclosed Curves

Use Green's Thereom to find the area enclosed by the curve: {abs(x)}^(1/2) + {abs(y)}^(1/2) = 1. It is important that you solve this problem using Green's Method because that is how my professor prefers it. I know that you have to make the above statement true and switch to a different set of coordinates. Something al

Equation of ellipse and hyperbola

1 Find an equation of the ellipse with the center (0,0) , vertical major axis 14 and minor axis 10. 2 Find an equation for the hyperbola with the focus (11,12) and asymptotes 4x-3y=18 and 4x+3y=30. 3 Find the arc length of the curve given by x = sin t - cos t, y = sin t + cos t, pi/1 <= t <= 3pi/4