In Taylor and fourier series, ,point out the range where the expansion provides a “good” approximation and please explain why.

Suppose that ? is the intersection of the plane ?+?+3?=4 and the surface ?^2−?^2=?^2−8 . Note that this intersection contains the point (3,4,−1) . Verify the assumptions of the implicit function theorem at this point; then if ?(?)=(?,?) φ ( x ) = ( y , z ) be the function from ℝ→ℝ2 R → R 2 verifying the conclusion of the implicit function theorem, compute ??(3) J φ ( 3 ) using the theorem. Verify your conclusion by explicitly solving for (?,?) ( y , z ) in terms of ? x and differentiating.

Solve the following initial value problem: y'''-4y''+20y'=-102e^3x, y(0)=3, y'(0)=-2, y''(0)=-2

Find the first five nonzero terms in the solution of the given initial value problem. y′′−xy′−y=0, y(0)=7, y′(0)=10

Find the first five nonzero terms in the solution of the given initial value problem.

y′′+xy′+2y=0, y(0)=8, y′(0)=9

Show an example of a system which is a quasigroup butnot a group.

Please provide explanation.

Each of the following functions has a critical point at the origin. Show that the second derivative test fails there. Determine whether the functions has a local maximum, local minimum, or saddle point at the origin by visualizing what the surface z=f(x, y) looks like. Describe your reasoning.

(a)f(x, y) =x^2y^2

(b)f(x, y) = 1−xy^2

research and  find an important mathematician or scientist whose work is especially interesting or important.?

Explain briefly why their discoveries are important for us today?

Can you find any particular analytical trait that would be advantageous for us to have?

Solve the following initial value problem using the undetermined coefficient technique:

y'' - 4y = sin(x), y(0) = 4, y'(0) = 3

Solve the given initial value problem.

y'''+2y''-13y'+10y=0

y(0)=4    y'(0)=42 y''(0)= -134

y(x)=

Let A, B, C be arbitrary sets. Prove or find a counterexample to each of the following statements:

(b) A ⊆ B ⇔ A ⊕ B ⊆ B

Let Z* denote the ring of integers with new addition and multiplication operations defined by a (+) b = a + b - 1 and a (*) b = a + b - ab. Prove Z (the integers) are isomorphic to Z*. Can someone please explain this to me? I get that f(1) = 0, f(2) = -1 but then f(-1) = -f(1) = 0 and f(2) = -f(2) = 1 but this does not make sense in order to define a function. Can someone explain why this is not right and show what it is correct?

Let f:A→B and g:B→C be maps.

(a) If f and g are both one-to-one functions, show that g ◦ f is one-to-one.

(b) If g◦f is onto, show that g is onto.

(c) If g ◦ f is one-to-one, show that f is one-to-one.

(d) If g ◦ f is one-to-one and f is onto, show that g is one-to-one.

(e) If g ◦ f is onto and g is one-to-one, show that f is onto.

1. Prove that Z[√3i]= a+b√3i : a,b∈Z is an integral domain. What are its units?