# Tutor profile: Alex L.

## Questions

### Subject: Organic Chemistry

Explain how to achieve a final product of t-butyl alcohol with chloroethanal as the starting ingredient. List the steps (i.e., reagents) in the order necessary to achieve the final product.

1. Add the following Grignard reagent: methyl magnesium bromide (MeMgBr). As a result of the Grignard reaction, the oxygen atom becomes more negatively charged and a methyl group binds to the central carbon atom. The double bond between oxygen and the central carbon atom becomes a single bond. 2. Add hydrogen chloride (HCl, specifically the chloride ion Cl-) The chlorine atom leaves and forms a chlorine molecule with the additional chloride ion and the double bond reforms. 3. Add MeMgBr again. An additional methyl group binds to the central carbon atom and the single bond between the oxygen and the central carbon atom reforms 4. Add acid (specifically the hydronium ion H3O+). The oxygen atom accepts a hydrogen ion (H+), thus forming t-butyl alcohol.

### Subject: Linear Algebra

Find the eigendecomposition of A, for A = [-4 2] [ 3 1 ]

Assuming that A is diagonizable, define A such that A = X*D*inv(X) 1. Find D: A - lambda*I = [-4 - lambda 2 ] [ 3 1 - lambda] (-4 +3lambda + lambda^2) - 6 = 0 --> lambda^2 + 3lambda - 10 = 0 By solving this quadratic equation, lambda1 = 2 and lambda2 = -5 D = [lambda1 0 ] --> D = [2 0] [ 0 lambda2] [0 -5] Solve for x1 and x2, for Ax1 = 0 and Ax2 = 0 Know that the identity matrix I = [1 0] [0 1] 2. Find X, for X = [x1 x2] A -2I = [-6 2] --> x1 = [1] [ 3 -1] [3] A +5I = [1 2] --> x2 = [ 1 ] --> [ 2] [3 6] [-1/2] [-1] 3. Find inv(X), for X*inv(X) = I Let inv(X) = [x1 x3] [x2 x4] [1 2] * [x1 x3] = [x1 + 2x2 x3 + 2x4] = [ 1 0 ] [3 -1] [x2 x4] [3x1 - x2 3x3 - x4 ] [ 0 1 ] x1 + 2x2 = 1 ---> 7x1 = 1, for x2 = 3x1 ---> x1 = 1/7 ---> x2 = 3/7 3x1 - x2 = 0 x3 + 2x4 = 0 ---> -7x4 = 1, for x3 = -2x4 ---> x4 = -1/7 ---> x3 = 2/7 3x3 - x4 = 1 inv(X) = [1/7 2/7] = (-1/7) * [-1 -2] [3/7 -1/7] [-3 1] Therefore, A = (1/7) * [1 2] [2 0] [-1 -2] [3 -1] [0 -5] [-3 1]

### Subject: Calculus

Find the following integral: int[a, b] (x*cos(3x^2))dx

Remember the chain rule: If F(x) = f((g(x)), then F'(x) = f'(x)(g(x)*g'(x) Remember basic integration: int[a,b](f'(x))dx = f(x)[a,b] Here, let f'(x) = cos(x) -> f(x) = sin(x) Let g(x) = 3x^2 -> g'(x) = 6x. In the case of this problem, there should be a coefficient of 1/6. Thus, int[a, b] (x*cos(3x^2))dx = (1/6)*(sin(3x^2))[a,b]

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