Question

for the given functions f(x), let x0=1, x1=1.25, x2=1.6. Construct interpolation polynomials of degree at most one and at most two to approximate f(1.4), and find the absolute error. a. f(x)=sin (pi x)

Answer #1

therefore the absolute error is 0.1394 and 0.03283

Let x0=1, x1=1.25, and x2=1.6. Using data at these xi, construct
interpolating polynomial of degrees at one and two and use them to
approximate f(1.4). Find the absolute errors
d) Fx=ln(10x)

1 Approximation of functions by polynomials
Let the function f(x) be given by the following:
f(x) = 1/ 1 + x^2
Use polyfit to approximate f(x) by polynomials of degree k = 2,
4, and 6. Plot the approximating polynomials and f(x) on the same
plot over an appropriate domain. Also, plot the approximation error
for each case. Note that you also will need polyval to evaluate the
approximating polynomial.
Submit your code and both plots. Make sure each of...

Let P2 denote the vector space of polynomials in x with real
coefficients having degree at most 2. Let W be a subspace of P2
given by the span of {x2−x+6,−x2+2x−1,x+5}. Show that W is a proper
subspace of P2.

Let F (x1, x2) = ln(1 + 4x1 + 7x2 + 6x1x2), x = (x1, x2) ∈ R
.
→−
(a) Find the linearization of F at 0 .
Show F is continuously differentiable, that is, C , at 0 .

Let
f(x)=sin(x)+x^3-2. Use the secant method to find a root of f(x)
using initial guesses x0=1 and x1=4. Continue until two consecutive
x values agree in the first 2 decimal places.

The second-order Taylor polynomial fort he functions f(x)=√1+x
about X0= is P2=1+(x/2)-(x^2/2) using the given Taylor polynomial
approximate f(0.05) with 2 digits rounding and the find the
relative error of the obtained value (Note f(0.05=1.0247). write
down the answer and all the calculations steps in the text
filed.

Let X =( X1,
X2, X3 ) have the joint pdf
f(x1, x2,
x3)=60x1x22, where
x1 + x2 + x3=1 and
xi >0 for i = 1,2,3. find the
distribution of X1 ? Find
E(X1).

Let W⊂ C1 be the subspace spanned by the two polynomials x1(t) =
1 and x2(t) =t^2. For the given function y(t)=1−t^2 decide whether
or not y(t) is an element of W. Furthermore, if y(t)∈W, determine
whether the set {y(t), x2(t)} is a spanning set for W.

: Consider f(x) = 3 sin(x2) − x.
1. Use Newton’s Method and initial value x0 = −2 to approximate
a negative root of f(x) up to 4 decimal places.
2. Consider the region bounded by f(x) and the x-axis over the
the interval [r, 0] where r is the answer in the previous part.
Find the volume of the solid obtain by rotating the region about
the y-axis. Round to 4 decimal places.

The second-order Taylor polynomial fort he functions f(x)=xlnx
about X0= 1 is P2= -1+(x-1)^2/2 using the given Taylor polynomial
approximate f(1.05) with 2 digits rounding and the find the
relative error of the obtained value (Note f(1.05=0.0512). write
down the answer and all the calculations steps in the text
filed.

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