# UCLA Calculus First and Second Derivatives Factorizing & Integrals Exercises

### Question Description

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. 4.6/5

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