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# Curves

A curve is representative of a line which is not straight. A curve begins and finishes similar to a line, but stops resembling a straight line as it approaches the formation of its bulge or deformation. For instance, a parabola is an example of a curve. Figure 1. This illustration is representative of a curve. As is depicted in this figure, a curve bulges outwards near the middle.

Curves are generally thought to be plane curves and are therefore two dimensional. However, in some cases curves can be three dimensional. These types of curves are referred to as space curves.

In the study of geometry, curves are particularly important to the areas of differential geometry and algebraic geometry. In differential geometry, a curve is referenced as a differentiable curve and this curve does not exist in two dimensional space.

In algebraic geometry, curves are known as algebraic curves and can be either plane curves, space curves or of a higher dimension. For example, an elliptic curve, which is studied in number theory, is an example of an algebraic curve.

### Curve Fitting Techniques

Task: Apply curve-fitting techniques and interpret the results. As such, your work will include doing scatterplots, determining the equation and graph of the curve on the scatterplot, finding the r^2 value, estimation using the curve, etc. Using a linear model, you are expected to use the data given below (University of Maryland

### Value Curve and Swing Weights in Excel

1. Generate a value curve using Excel for the following problem: Provide air movement: objective 50 mph; min 20 mph; max 100 mph. Provide electrical power: objective 30 amps; min 10 amps; max 50 amps. Provide communication: objective 120 miles; min 100 miles; max 200 miles Provide ground movement: objective 4 mph; min 1 mph

### Elliptic Curves Over a Finite Field

Let E be the elliptic curve over F5 defined by y^2 = x^3+1. 1. Write down the division polynomial (psi_3)(x) for this curve. 2. Show that the greatest common divisor of (psi_3)(x) and "x^5 -x" is "x(x-1)." 3. Use part (2) to show that the 3-torsion points in E(F5) are {Origin,(0,1),(0,-1)}." I have also attached the proble

### Curve Fitting and Input and Output Files

Problem 2 This is a three dimensional version of the two dimensional curve fitting problem associated with determining thevertical alignment of a road. In this case, the problem is that groundwater hydrologists want to map thepiezometric surface (top of the water table) in a region so they can assess the potential for developmen

### Gaussian curvature is applied.

Let f(x,y) be an infinitely differentiable function. Suppose that a. f(0,0) = 0 b. f_x(0,0) = 0 and f_y(0,0) = 0. Consider the surface z = f(x,y). Show that K(0,0) = f_xx(0,0) f_yy(0,0) - f_xy(0,0)^2.

### Gaussian Curvature of the Unit Sphere

Compute the curvature of the unit sphere in R^3. I started the problem with r = [sin u cos v sin u sin v cos v. ] From there, how do you get the principal curvatures k_1 and k_2?

### exact length of the curve

Find the exact length of the curve defined by y=2[(sqrt(x))^(3)]-1=2x^(3/2)-1 from x=0 to x=2. Set up the integral, use the substitution method( reverse chain rule), and express your answer in radical form. (ex sqrt(2), not 2.141).

### Use excel to draw an average product (AP) and marginal product (MP) curves in the same space

Please help with the following problem. See the complete description of the problem in the attached document. 1) Complete the table (Table 1) below 2) Use Excel to draw the total product (TP) curve. 3) Use Excel to draw the average product (AP) and marginal product (MP) curves in the same space 4) Use the informati

### Z scores and p values

Fill in the answers for each table below. Please report your z scores to two decimals and your p values to three decimals. If the p value is less than .001, please report p < .001. Problem 1... Problem 2 Setting µ = 60 and &#963; = 7 What is the z score of 78? What is the z score of 45? What is the probability of 59?

### Arc Length and Tangent

The equation R(t)=sint(i)+cost(j)+logsect(k) (0 less than or equal to t and t is less than pi/2) find a) element of arc length ds, along c in terms of t b) the unit tangent T c) the curvature k

### Circumference Length of an Ellipse

The first problem deals with finding the circumference length of an ellipse. This field is called differential geometry. The second problem deals with finding the equation of tangent line of a given equation. See the attached file.

### Demand Curve and Profit

Suppose the demand curve for a monopolist is QD = 500 - P, and the marginal revenue function is MR = 500 - 2Q. The monopolist has a constant marginal and average total of \$50 per unit. Calculate the monopolist 's profit.

### Curve on a Spherical Surface

Hi I have this curve C defined by x=sin(2t), y=1-cos(2t), z=2cos(t) where t lies between (or equal to) -pi and pi. How do I show that this curve lies on a spherical surface with central in origon and radius = 2?

### Characteristics of a curve

Please see the attached file for the fully formatted problems. Let a,b and w be positive constants. Let g(t) = (a cos (wt) , a sin(wt) , bt) t>0 Find explicitly the arc length parametrization h(s) of the curve Find the unit tangent and principle normal vectors at an arbitrary point h(s) Find the curvature k(s)

### Operations supply management - learning curve

A time standard was set at 0.20 hour per unit based on the 50th unit produced. If the task has a 90% learning curve what would be the expected time of the 100th, 200th and 400th unit? Please give answer and explain.

### Tangent Normal and curvature of parametric plane curves.

Find unit tangent and normal vectors at the given point: y= x^3 at (-1,1) x=t^3 y=t^2 at (-1,1) x=3sin2t y=4cos2t where t = pi/6 x=t-sint y=1-cost where t=pi/2 Find the curvature of the given plane: x=5 cosh t y=3 sinh t at t=0

### Arc length

See attachment Find the arc length of the graph on interval [1/2,2]

### Find the length of the given curve.

Find the length of the curve r(t)=[e^2, (e^t)sin(t), (e^t)cos(t)], 0 greater than or equal to t that is less than or equal to 2pi

### Find point of intersection in curves

At what point do the curves, r1(t)=(t, 1-t, 3+t^2) and r2(s)=(3-s, s-2, s^2) intersect. and find their angle of intersection

### Polar Equation Question

Find a polar equation for the curve represented by the given Cartesian equation. x^2= 4y

### Sketching a polar curve

Sketch a curve using each of the polar equations. 1.)r= 1 + cos theta 2.)r^2 theta= 1 (lituus)

### Curve Equation Values

See attached A curve has the equation... Find the period of f(x). Determine all values of x...for which f(x)=0... Find the value or values of x...for which f(x)=2

### An Elliptical Fireplace Arch

A fireplace arch is to be constructed in the shape of a semi ellipse. The opening is to have a height of 40 inches at the center and a width of 82 inches along the base. To sketch the outline of the fireplace, the contractor uses a 82-inch string tied to two thumbtacks. Where the thumbtacks should be placed. Answers 1. 21 in

### Trends, Forecasting and Curve-Fitting

Picture this- 2001----------------128 2002-----------------192 2003----------------288 2004-----------------432 2005-----------------648 How many catfish will there be in 2006? Please explain and give me a strategy for solving this problem?

### Finding a Polar Equation from a Cartesian Equation

Find a polar equation for the curve represented by the given Cartesian equation. xy = 4 r^2 =

### Find the exact length of the polar curve.

Find the exact length of the polar curve. r = e^6&#952; 0 &#8804; &#952; &#8804; 2&#960;

### Sketch the curve of a spiral polar equation.

Sketch the curve of each polar equation. See attached page for problems.

### Fcurve represented by the given Cartesian equations.

Find the polar equation for the curve represented by the given Cartesian equations. 1.) y= x+1 2.) 2xy= 1

### Tangent Curves Value Alpha

Y=alpha*x^2 + alpha*x + (1/24) x=alpha*y^2 + alpha*y + (1/24) These are two curves. For what values of alpha are these two curves tangent to each other.