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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. 

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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).

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

Area under a Curve

Calculate the area under the curve y=1/(x^2) above the x-axis on the interval [1, positive infinity]. keywords: finding, find, calculate, calculating, determine, determining, verify, verifying

Finding Curvature

Compute the curvature k(t) of the curve r(t) = 2t i + 4sint j +4cost k

Area under the curve

Find the area of the surface obtained when the graph of y=x^2 , 0<= x <= 1, is rotated around the y axis.

Eliminating the Parameter and Sketching a Curve

A.) Eliminate the parameter to find a Cartesian equation of the curve b.) Sketch the curve and indicate with an arrow the direction in which the cuve is traced as the parameter increases x= 4 cos t, y= 5 sin t, -pi/2 (</=) t (</=) pi/2.

Higher-Order System : Unit-step Response Curve

Consider a higher order system defined by (see attached file for equation): a) Using MATLAB, plot the unit-step response curve of this system. b) Obtain the rise time, peak time, maximum overshoot, and settling time of the system. Please type it in Microsoft Word using Equation Editor. It makes it easier to understand.


I am asked to find the curvature at the point where t=1 of the curve given by: vector r(t) = t i + 1/2 t^2 j + 1/3t^3 k How do I do this problem and what is the final answer?

Curvature of a Curve in Space

The curvature of a curve in space r(t) is given by k(t) = | r'(t) à? r''(t) | / | r'(t) |^3 . Consider now the curve r(u) = r(sigma(u)), given by the reparametrization t = sigma(u) of the initial curve. Show that the curvature k of the curve r is given by k(u) = k(sigma(u)), where k is the curvature of the initia

Calculating Volumes of Bounded Regions (Curve, Origin and Interval)

1. Sketch the region bounded between the given curves and then find the area of each region a) {see attachment} b) Find the area of the region than contains the origin and is bounded by the line {see attachment} 2. Sketch the given region and then find the volume of the solid whose base is the given region and which has the