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) = x^{3} + 3x^{2} + 3x

c) f(x) = (5x^{2} + 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 x_{1}, x_{2}, and x_{3}** **using the matrix inversion technique.

b)Find solutions to the values of x_{1}, x_{2}, and x_{3}** **using Cramer’s rule.

**Problem Set 8**

- Partially differentiate the following equations with respect to x
_{1}and x_{2}.Note the superscript stand for a power and the subscript stands for the variable index.For example:x_{2}stands for x two; x^{2}stands for x squared, while stands for x two squared.

**Problem Set 9**

- Partially differentiate the following with respect to x
_{1}and x_{2}: - 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.

UCLA Calculus First and Second Derivatives Factorizing & Integrals Exercises