In: Advanced Math

Expand f(x) = 10 + 2x + x^4 in terms of the Legendre polynomials, showing the first 5 coefficients.

Write f(x)=x^4+2x^3+2x+1 as a product of irreducible
polynomials, considered as a polynomial in Z3[x], Z5[x], and Z7[x],
respectively.
1. 2. Let f(x) be as in the previous exercise. Choose D among
the polynomial rings in that exercise, so that the factor ring
D/〈f(├ x)〉┤i becomes a field. Find the inverse of x+〈f├ (x)〉┤i in
this field.

Expand the function, f(x) = x, defined over the interval 0 <x
<2, in terms of:
A Fourier sine series, using an odd extension of f(x)
and A Fourier cosine series, using an even extension of f(x)

What is the leading coefficient of the polynomial function f(x) = 2x+x³+4?

Calculate the all of the second order derivatives of f(x,y)=x^4
y- 〖3x〗^4 y^3+5y-10
Let be f(x,y,x)=2x^5 yz^3-3x^(-4) y^3 calculate the f_xyz=?
If f(x,y,z)〖=5e〗^(-xyz^6 ) +lnx calculate the f_yxz=?
Let f(x,y)=cos(x^3 y) ise df/dx=? df/dy=?
z=2x+y , x=sin(3t ) and y=cos(3t) using the chain rule calculate
the dz/dt where t=π/2.
Find the critical points of the f(x,y)=〖6x〗^2+12y^2+12x-24y and
provide the information about the critical points obtained (if it
is max, min or saddle point).

Please find f(x') and f(x") for all:
1) f(x) = 3(x2 -2x)3/4(7 -
5x2)6
2) f(x) = (y - x)4(y2 - x)3
3) f(x) = x7/3 + 16x3 + x
4) = f(x) = (x2 - x) / (y2 - y)

f(x)=(x-3)^4/(x^2+2x) Find the values of x at which the curve y
= f(x) has a horizontal tangent line

F(x) = 0 + 2x + (4* x^2)/2! + (3*x^3)/3! + .....
This is a taylors series for a function and I'm assuming there
is an inverse function with an inverse taylors series, I am trying
to find as much of the taylors series of the inverse function
(f^-1) as I can

f(x) = (2x − 3)(x 2 − 6)
(a) Write formulas for f '(x) and f ''(x).
(b) Find all x-intercepts of f(x). (Exact answers, no
decimals.)
(c) Find all critical points of f(x). (x-values only; y-values
not needed.) Classify them using the 1st or 2nd derivative
test.
(d) Find all inflection points of f(x). (x-values only; y-values
not needed.)

Using Matlab, consider the function f(x) =
x^3 – 2x + 4 on the interval [-2, 2] with h
= 0.25. Write the MATLAB function file to find the
first derivatives in the entire interval by all three methods i.e.,
forward, backward, and centered finite difference
approximations.
Could you please add the copiable Matlab code and the associated
screenshots? Thank you!

Differentiate the following:
1) f(x) = √2x-4. (all under square root)
2) f(x) = x/5-x
3) y=cos(4x^3)
4) f(x)=tan(x^2)
5) f(x)= 3e^2x cos(2x)
6) y= sin2x/cosx
7) y= √sin(cosx) (all under the square root)

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