## Midterm 2

Throughout this exam, you may use any theorems from class, class assignments, or in the class textbook Chapters 1-9, for which a proof has been provided.

### Question 1: Fourier Series

Consider the function, f : [−π, π] → R, given by f(x) = 1 −

1.Show that the Fourier Series of f(x) is given by:

2.From 1., prove:

3.From 1., prove:

### Question 2: Multivariate Difffferentiation  数学原理代考

Consider the function : R2 → R2:

(x, y) = (x2 − y2, 2xy).

1.Relative to the standard basis on R2, show using only the defifinition of the derivative that:

2.Assume a function : U →V,U,V⊂Rn open,admits an inverse  :V→U.

If is difffferentiable at ∈ U, and is difffferentiable at ()∈V,prove that ′ () is invertible.

3.Returning to the function (x, y) = (x2 − y2, 2xy), we defifine E ⊂ R2:

E := {(x, y) ∈ R2 ∈ admits alocal C1 inverse in a neighborhood of (x, y)}.

Find the set E ⊂ R2 (Hint: use 2.). Showis not one-to-one on E.

### Question 3: Multivariate Difffferentiation (cont.)  数学原理代考

Consider the function f : R3 → R:

f(x, y, z) = x3z + ex − y2.

1.Show that f(0, 1, 2) = 0, (0, 1, 2)≠0,and that there therefore exists a difffferentiable function g in some neighborhood of (1,2) ∈R2, such that g(1, 2)=0, and

f(g(y, z), y, z) = 0.

2.Find that f(1, 2), (0, 1, 2)≠0,and that there therefore exists a difffferentiable function g in some neighborhood of (1,2) ∈R2, such that g(1, 2)=0, and

f(g(y, z), y, z) = 0.

2.Find(1, 2), (1, 2).