MATH 207 The History Of Calculus Calculus in

MATH 207

The History Of Calculus

Calculus in Engineering Civil Engineering For example, in hydraulic systems, basic fluid mechanics equations require calculus. Structural Engineering Electrical Engineering For example, structural analysis relating to seismic design requires calculus. Calculations of bearing capacity and shear strength of soil are done using calculus, as is the determination of lateral earth pressure and slope stability in complex situations. Computing voltages in electronic circuits requires the use of calculus differential equations

Mechanical engineering Many examples! Calculus is used for: • Evaluating surface areas of complex objects • Computing torques and forces • Analyzing fluid flow in hydraulic systems • … Aerospace Engineering For example, • computing thrust of rockets that function in stages, • modeling gravitational effects over time and space. Many other Engineering fields: Nuclear Engineering, Material Science, etc…

Calculus is also used in many more areas: • • Architecture Acoustics Politics Optics Music Sports Arts … Calculus is everywhere!

Class website: • http: //ougouag. com • Contains ALL course material: Ø Annoucements Ø Syllabus Ø Homework Assignments Ø Lecture Notes Ø MANY more resources…! • Bookmark it!

Brightspace: brightspace. ccc. edu • How do you log in? Ø With your ccc credentials (login and password) • DO it now! (on your tablet/phone…) • Set up your profile to get my messages. • I WILL BE USING BRIGHTSPACE TO POST ALL YOUR SCORES IN TESTS AND QUIZZES.

Syllabus • Let’s go over it in detail. • Please pay attention. • I will be giving you a short quiz next time about its contents!

A couple more things…

Food in class (!) • If you must eat, please choose items that are discreet and do not disturb class (no strong smells and crinkly noises!)

About you • At the end of today’s session: turn in the handout I gave you. • Fill in as much as is comfortable for you. All info will be appreciated and will help me to get to know you better. • Let’s go around and mention our Name and Special thing

Now let’s review a few things…

Several review links are posted on the class website! For example: Pre. Calculus Tutorials MATH 2. org Just Math Tutorials

Distance Formula Example: Find distance between (-1, 4) and (-4, -2).

Midpoint Formula Example: Find the midpoint from P 1(-5, 5) to P 2(-3, 1). Answer: (-4, 3)

Equations in two variables – Example: Circle Equations y r (x, y) (h, k) x

1. 1 Functions Copyright © Cengage Learning. All rights reserved.

Definition of a Function

It’s helpful to think of a function as a machine: The input is the independent variable, The output is the dependent variable.

Four Ways to Represent a Function § verbally (by a description in words) § numerically (by a table of values) § visually (by a graph) § algebraically (by an explicit formula)

Verbally (with words) or With Diagrams: Copyright © 2011 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Numerically: using Tables -

Visually: using Graphs -

Algebraically: using Formulas – There are several Categories of Functions:

The most common method for visualizing a function is its graph. If f is a function with domain D, then its graph is the set of ordered pairs: {(x, f (x)) | x D}

Theorem: Vertical Line Test A set of points in the xy - plane is the graph of a function if and only if a vertical line intersects the graph in at most one point.

Finding the Domain of a Function

Piecewise-defined Functions: Example:

A common Piecewise Defined Function: The absolute value function: f (x) = |x| x if x 0 –x if x < 0 f(x) = |x| =

Symmetry of a Function (Odd or Even) • If a function f satisfies f (–x) = f (x) for every number x in its domain, then f is called an even function. The graph of an even function is symmetric with respect to the y-axis • If f satisfies f (–x) = –f (x) for every number x in its domain, then f is called an odd function. The graph of an odd function is symmetric about the origin An even function Figure 19

Increasing and Decreasing Functions Where is the function increasing?

Where is the function decreasing?

Where is the function constant?

Local Maxima and Minima Local Max

Local Min

Average rate of change of a Function

from 0 to 1