Please write the process and draw the graphs clearly, all the questions are in the document.

Problem Set 2

• Use calculus techniques to establish the first and second derivatives of the following functions and to graph the functions below by indicating 1) points where f(x) = 0, 2) the y-intercept, 3) all relative maxima, 4) all relative minima, and 4) points of inflection, if any.Show all of your work.

a)

b) f(x) = x³ + 3x² + 3x

c) f(x) = (5x² + 6x – 2)(x – 5)

Problem Set 3

1)Solve the following equations for x:

2)Solve the equation hint: “Factor out” the term (x-1).Also, you should find three distinct solutions.

• Differentiate the following functions:

3a)

3b)

4)Graph the function indicating solutions (points where f(x) = 0), local maxima, minima, and points of inflection (if any).Use calculus techniques whenever possible.

Problem Set 4

Evaluate the following indefinite integrals:

1a)

1b)Apply integration by parts.

1c)Apply integration by parts twice.

1d)

Evaluate the following definite integrals.

Problem Set 5

Use the matrices defined below for the following set of problems.

• Find:

1a)3A

1b)A + B

1c)AB

1d)BC’

1e)CB

1f)AE

1g)FF’

1h)A+C

1I)EF

1I)2B-8E

Problem Set 6

• For the following system of equations,

a)Find solutions to the values of x₁, x₂, and x₃ using the matrix inversion technique.

b)Find solutions to the values of x₁, x₂, and x₃ using Cramer’s rule.

Problem Set 8

• Partially differentiate the following equations with respect to x₁ and x₂.Note the superscript stand for a power and the subscript stands for the variable index.For example:x₂ stands for x two; x² stands for x squared, while stands for x two squared.

Problem Set 9

• Partially differentiate the following with respect to x₁ and x₂:
• Find the total differential of the following functions:
• Find the total derivative of V with respect to X.

Z = 4 + 5X

Problem Set 10

1)Determine whether or not the following matrices are negative (semi)definite or positive (semi)definite:

a)b)

2)Determine the critical points of the following functions, and determine whether or not they correspond to (local) maxima, (local) minima, or saddle-points.Clearly distinguish the first-order conditions (FOCs) and the second-order conditions (SOCs).

a)

b)

c)

Problem Set 11

• Find the maximum of U for problems (a) and (b) below.For each problem, find the optimum using mathematical techniques.

Problem Set 12

• Given the constraint, find the stationary points for the following function and evaluate the second order conditions.Use the Lagrange technique.