What is the difference between natural and clamped Cubic Splines?
Solve the following problems with a clear explanation.
[1] A natural cubic spline S on [0,2] is defined by
S(x) = { S0(x) = 1 + 2*x - x^3 , if 0 <= x <= 1
S(x) = { S1(x) = 2 + b*(x-1) + c*(x-1)^2 + d*(x-1)^3 , if 1 <= x <=

Answer the following questions for a nonperiodic B-splinecurve of order 3 defined by the control points P0,P1,P2,P3.
a- What are the knot values?
b- How many different curves is theB-splinecurve composed of?
c- Expand theB-splinecurveequation to get the separate equations of thecurves in b.

Dear TA,
Please be kind to show me how to factor following equation. Find solution for x. The answer is x=2. I try to factor out an x but that doesn't work and I don't think using the quadratic equation would work either. Also, I try factoring out in pairs but that doesn't work either. Thank you in advance.
Y=X+6
Y=X^3
X

(i) Use the tabulated data and Newton's method of divided differences to (a) calculate the degree three interpolating polynomial based on x_0, x_1, x_2, x_3 and (b) calculate the degree four interpolating polynomial based on x_0, x_1, x_2, x_3, x_4.
x_k || 0 | -1 | 1 | 3 | 2
y_k || 5 | 15 | 3 | 47 | 9
(Use exact arithmeti

Can you show me how to work this? Is there only one way and what is most acceptable?
Find the values of the complex conjugate roots fortheequation x^3 + 3X^2 +2 = 0.

7. This problem generalizes the factorial function, as in n!=n(n-1)(n-2)...(2)(1), to more general arguments than just the positive integers.
(a) Use integration by parts to show that for any positive integer n, the integral with respect to x from 0 to infinity of xne-x is n!
(b) Make a clear case that the integral exists

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An equation of theform ax2 + bx + c, where a 0, is called a equation.
Solve theequation.
(x - 1)(x - 3) = 0
x = (smaller value)
x = (larger value)
Solve theequation.
x2 + 38x = 0
x = (smaller value)
x = (larger value)
Solve theequation.
x2 - 23x = 0
x = (smaller value)
x =

Question: Each of the three dimensions of a cube with a volume of x^3 cubic centimeters is increased by a whole number of centimeters. If the new volume is x^3 + 10x^2 + 31x + 30 cubic centimeters and the new height is x + 2 centimeters, then what are the new length and width?

Maria is using a cylindrical oxygen tank while scubadiving. It holds 80 cubic inches of air. How many cubic feet of air is she using? Round to the nearest hundredth if necessary.