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# Functional Analysis

### If lines are drawn on a map, a triangle can be formed by the cities of New York City, ...

If lines are drawn on a map, a triangle can be formed by the cities of New York City, Washington, D.C., and Buffalo, New York. On the map, the distance between New York City and Washington, D.C. is 2.1 cm, New York to Buffalo is 2.5 cm, and from Buffalo to Washington D.C. it is approximately 2.85 cm. If the actual straight-line

### 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?

### Logarithmic and Exponential Functions Please see the attached file for the fully formatted problems.

Please see the attached file for the fully formatted problems. 1. If y = c^x express x in terms of y. (c is a positive constant) 2. a) Sketch the graph of y = 5^x for ?2 < x < 3, clearly showing the key features of this exponential function. (b) On the same graph, sketch the function y = log5 x. Explain the relationship

### Logarithm function - Deliverable Length: 2-4 paragraphs Details: Many different kinds of data can be modeled or measured easily using exponential and logarithmic functions. For example, consider ideal gases: Pressure versus volume give a curved plot, but a log graph of pressure versus volume provides a linear plot that is easier to interpret. Log plots are also used in the study of the rates of reactions; for many reactions, the log of remaining reactant concentration versus time is linear. For this Discussion Board, use the Library and other resources to find at least two examples of scientific data that are measured and reported using log scales. Examples can be found in such fields as chemistry, astronomy, and geology.

Deliverable Length: 2-4 paragraphs Details: Many different kinds of data can be modeled or measured easily using exponential and logarithmic functions. For example, consider ideal gases: Pressure versus volume give a curved plot, but a log graph of pressure versus volume provides a linear plot that is easier to interpret. Log

### 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

### 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

### Asymptotes

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

### 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) &#8801; &#8711;? V (3, 2, 4) 2) F (x, y) = xe2y i + y/(x + y) j. Find &#8711; ? F (4, 0) 3) F (x, y, z) = -yz i + xz j - xy k. Find curl F (1, 2, 5) = &#8711;×F ( 1, 2, 5)

### Polar Equations: Conic Section

Find an equation in x and y for the conic section with polar equation r=1/1+cos

### 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

### Polar and Rectangular Coordinates

1 Express the polar equation r^2 = 2cos2&#920; in rectangular form. 2 Find the total area enclosed by the graph of the polar equation r = 1 + cos2&#920;

### Finding Area of a Region Bounded by a Line and a Curve

Sketch the region bounded by the graph of the functions and find the area of the region 1) f(x) = - x^2+ 4x + 2, g(x) = x + 2 2) f(y) = y(2 - y), g(y) = -y 3) f(x) = 3^x, g(x) = 2x + 1 keywords: integration, integrates, integrals, integrating, double, triple, multiple

### Assume that a small tree will grow to maturity according to

Dh /dt = 2 + 0.5/t^0.5 where t is the time in years (t > 0) and h is the height of the tree in feet. How much does the tree grow between the fourth and ninth years?

### Finding Absolute Minimum

Determine the absolute minimum of the function f(x) = x^3 - 3x - 1 on the interval [0, 4]. Make sure to show all work that is involved.

### amplitude of resultant wave after superimposition

Two vibrations, x1 = 3 sin(10t + pi/6) and x2 = 2 cos(10t - pi/6) where t is in seconds, are superimposed. determine the time at which the amplitude of the resultant vibration, x1 + x2, first reaches a value of 2.

### Functional Analysis

Attachment file. Let X be a normed space and . Show that if for every bounded linear functional f on X , then .

### Rectangular Storage Unit: Effect of Changing Dimensions Upon Volume and Number of Solutions

A rectangular storage unit has dimensions 1 m by 2 m by 3 m. If each linear dimension is increased by the same amount a) what increase would create a new storage unit with a volume 10 times the original b) how many solutions are there to this problem? please explain why?

### Graph Parabolas Points

24. Graph the following parabola clearly labeling exact points for the vertx, x-intercepts(s), and y-intercept(s): y=2xsquare+4x-3

### Gradient : Find grad(f). Let f(x,y,z) = e^sinx +xy^2 -(2z + 1)^2 lnx

Find the gradient of f ( grad(f) ). Let f(x,y,z) = e^sinx +xy^2 -(2z + 1)^2 lnx

### Uniform Convergence

Please see the attached file for the fully formatted problems. We are using the book Methods of Real Analysis by Richard R. Goldberg.

### A Discussion On Rolle's Theorem

Please solve the following with as much explanation of each step as possible. Let f be continuous on the closed interval [a,b] and differentiable on the open interval (a,b). Also, suppose that f(a) = f(b) and that c is a real number in the interval such that f'(c) = 0. Find an interval for the function g over which Rolle's

### How to Use Mean Value Theorem to Find Guaranteed Points

Find the points guaranteed by the mean value theorem for F(x) = x^.5 - 2x on the closed interval [ 0 , 4 ].

### Find number of degrees...

Use the equation s=rO for the following: 1. Find the number of degrees in the central angle of a circle with radius of 20 inches if the angle subtends an arc of 13 inches. 2. Find the length of a degree on the equator considering the diameter of the equator to be 7912 miles. 3. A bicycle has a 28 inch wheel. How man

### Outer Measure Functions

Let A = union ( i from 1 to infinity) of M_i, Mi's are disjoint, show that m*(A) = sum (i from 1 to infinity) of |M_i| m*(A) is the outer measure of A, that is, m*(A) = inf sum (i from 1 to infinity) of M_i. PLEASE NOTICE THE = SIGN, A = the union, not a subset of the union.

### Intermediate value theorem

Determine if the following conforms to the intermediate value theorem. Clearly give the conditions and, if appropriate, find an appropriate value for c. f(x)=10/((x^2)+1) [0,1] k=8

### Weak Convergence

Please explain why the following sequence for otherwise is an example of a sequence in such that weakly, but not strongly. Please see the attached file for the fully formatted problem.

### Converting Polar Coordinates to Rectangular Coordinates

Find the rectangular coordinates of the point (r,degrees)= (-5,-55 degrees).