# Tutor profile: Peter H.

## Questions

### Subject: Linear Algebra

Prove that if $$A+B$$ is invertible, then $$A(A+B)^{-1}B = B(A+B)^{-1}A$$

Add $$A(A+B)^{-1}A$$ to both sides and simplify. Both sides simplify to $$A$$, so the equality holds.

### Subject: Discrete Math

Prove that $$\log n! = \Theta(n \log n)$$

$$n! < n^n$$ clearly, so $$\log n! = O(n \log n)$$. For the other containment, consider $$2 \log n! = \log(n!^2)$$. Rewrite the product as $$(n \times 1)((n-1) \times 2 \times \cdots \times (1 \times n)$$. Each term in the product is greater than $$n$$. Thus $$\log (n!)^2 = \Omega(n \log n)$$

### Subject: Calculus

Evaluate $$\lim_{x\to 0} \frac{\int_0^x f(t)dt}{\int_0^{2x} f(t) dt}$$ assuming $$f$$ is continuous and $$f(0) = 1$$

For $$x$$ small the numerator is approximately the area of a rectangle of height $$f(0)$$ and width $$x$$. Similarly the denominator is the area of a rectangle twice as wide. Thus the ratio tends to $$1/2$$