Docsity
Log inSign up
Docsity
Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity

Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan

Guidelines and tips
Guidelines and tips
Sell on Docsity
Sell on Docsity
Log inSign up

Prepare for your exams

Study with the several resources on Docsity

Find documents

Prepare for your exams with the study notes shared by other students like you on Docsity

Search Store documents

The best documents sold by students who completed their studies

Search through all study resources
Docsity AINEW

Summarize your documents, ask them questions, convert them into quizzes and concept maps

Explore questions

Clear up your doubts by reading the answers to questions asked by your fellow students

Earn points to download

Earn points by helping other students or get them with a premium plan

Share documents
download point icon20 Points

For each uploaded document

Answer questions
download point icon5 Points

For each given answer (max 1 per day)

All the ways to get free points
Get points immediately

Choose a premium plan with all the points you need

Study Opportunities

Choose your next study program

Get in touch with the best universities in the world. Search through thousands of universities and official partners

Community

Ask the community

Ask the community for help and clear up your study doubts

University Rankings

Discover the best universities in your country according to Docsity users

Free resources

Our save-the-student-ebooks!

Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors

From our blog

Exams and Study
Go to the blog

MATH 241 Lecture 2: Functions and Basic Numerical Functions, Study notes of Mathematics

University of Oregon (UO)Mathematics

A portion of lecture notes from a university-level mathematics course, specifically math 241. The lecture covers the concept of functions, their types, and basic numerical functions such as polynomials, exponential and logarithmic functions, and trigonometric functions. The document also includes examples and exercises to help students understand these concepts.

Typology: Study notes

Pre 2010

Uploaded on 07/23/2009

koofers-user-jtd
koofers-user-jtd 🇺🇸

5

(1)

10 documents

1 / 2

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
MATH 241, LECTURE 2
1. Functions: the basic objects of study in calculus
Functions are means for producing one quantity (or set of quantities) from another. More formally we
have the following.
Definition 1. A function assigns to each input, from some set called the domain of the function, a unique
output, in a set called the range of the function.
Functions can be about measurements of a quantity, such as a cost function which measures for each
time the amount it costs to produce a given good. Functions can also be algebraic, such as the function
which takes a number, squares it, subtracts πfrom the answer, and then takes the reciprocal of that
quantity.
Even if you are more interested in functions which measure real-world quantities, part of the power
of calculus, and of mathematics in general, is connecting those functions with algebraic functions. For
example, both F=ma (Newton) and E=mc2(Einstein) changed the world, relating fundamental
physical quantities through simple functions. Also, with algebraic functions we can compute explicitly,
which helps develop understanding we can carry with us when we do applications.
2. Basic numerical functions
There are three kinds of basic functions we use in mathematics.
•Polynomials, ratios of polynomials, fractional powers.
For example f(x) = x2or x−1
x+1 or √x5−3.
•Exponential and logarithmic functions.
For example f(x)=2xor log10(x), which we will define later for those of you who haven’t seen
them.
•Trigonometric functions such as sin(x), which we will not be studying.
Understanding basic functions better will be an important outcome of our study of calculus.
Question 2. What is the better deal: getting one million dol lars a day for a month? getting n5dol lars
on the nth day for a month? getting one dollar the first day and then on each day getting twice what one
got the previous, for a month?
What if a month is changed to a week? or a year?
3. Making new functions from old
Adding, subtracting and multiplying functions is straightforward to both execute and understand.
Example 3. “Profit is the difference between total revenue and total cost” is translated into taking the
difference of functions.
Taking the quotient of functions can be trickier. For algebraic functions, you have to be careful because
the domain might change.
Example 4. If f(x) = √xand g(x) = x2−3x+ 2, what is the domain of the function f(x)
g(x)?
1
pf2

Partial preview of the text

Download MATH 241 Lecture 2: Functions and Basic Numerical Functions and more Study notes Mathematics in PDF only on Docsity!

MATH 241, LECTURE 2

  1. Functions: the basic objects of study in calculus Functions are means for producing one quantity (or set of quantities) from another. More formally we have the following.

Definition 1. A function assigns to each input, from some set called the domain of the function, a unique output, in a set called the range of the function.

Functions can be about measurements of a quantity, such as a cost function which measures for each time the amount it costs to produce a given good. Functions can also be algebraic, such as the function which takes a number, squares it, subtracts π from the answer, and then takes the reciprocal of that quantity. Even if you are more interested in functions which measure real-world quantities, part of the power of calculus, and of mathematics in general, is connecting those functions with algebraic functions. For example, both F = ma (Newton) and E = mc^2 (Einstein) changed the world, relating fundamental physical quantities through simple functions. Also, with algebraic functions we can compute explicitly, which helps develop understanding we can carry with us when we do applications.

  1. Basic numerical functions There are three kinds of basic functions we use in mathematics.
  • Polynomials, ratios of polynomials, fractional powers. For example f (x) = x^2 or x x−+1^1 or

x^5 − 3.

  • Exponential and logarithmic functions. For example f (x) = 2x^ or log 10 (x), which we will define later for those of you who haven’t seen them.
  • Trigonometric functions such as sin(x), which we will not be studying. Understanding basic functions better will be an important outcome of our study of calculus.

Question 2. What is the better deal: getting one million dollars a day for a month? getting n^5 dollars on the nth day for a month? getting one dollar the first day and then on each day getting twice what one got the previous, for a month?

What if a month is changed to a week? or a year?

  1. Making new functions from old Adding, subtracting and multiplying functions is straightforward to both execute and understand.

Example 3. “Profit is the difference between total revenue and total cost” is translated into taking the difference of functions.

Taking the quotient of functions can be trickier. For algebraic functions, you have to be careful because the domain might change.

Example 4. If f (x) =

x and g(x) = x^2 − 3 x + 2, what is the domain of the function f g^ ((xx))? 1

2 MATH 241, LECTURE 2

For real-world functions, which may contain error, what would seem to be a small amount of error in the denominator could lead to huge error in the final answer.

Example 5. To measure speed, we take distance travelled and divide it by time. What if we tried to measure the speed of a jet by using a stopwatch over 100 yards?

  1. Linear functions Linear functions are the simplest possible functions, so they are used often throughout mathematics. One of the main themes of calculus is the understanding of general functions through related linear functions. In short, linear functions are the basis of calculus.

Definition 6. Linear functions are functions of the form f (x) = mx + b, where m and b are constants.

The graph of a linear function y = mx + b is a line in the plane.

Question 7. Are all lines in the plane graphs of linear functions?

There are many ways to describe a given linear function. An important skill to develop is the ability to translate between the different descriptions. Key to many of these descriptions is the notion of slope.

Definition 8. The slope of the line y = mx + b is equal to m. It measures the change in y if x is increased by one. If one is not given the slope explicitly, it can be computed by m = (^) xy^22 −−yx^11 where (x 1 , y 1 ) and (x 2 , y 2 ) are any two points on the line.

Example 9. Sketch some lines with slope 1 , − 1 , 2 , − 3 , 12 and − 32.

Example 10. The points (− 1 , −1) and (3, 7) are both on the line y = 2x + 1. We can verify the formula for m in this case, and then take two other points on the line and use them to calculate the slope.

The different descriptions we will use are as follows, listed from simplest to most complicated. We translate each to the standard form, and give an example.

  • Standard, slope-intercept form: y = mx + b.
  • Linear equation form: ax + by = c.
  • Point-slope form: “the line which goes through (x 0 , y 0 ) with slope m” namely, (y−y 0 ) = m(x−x 0 ).
  • Two-point form: “the line between (x 1 , y 1 ) and (x 2 , y 2 ),” namely (y − y 1 ) = y x^22 −−yx^11 (x − x 1 ).

Example 11. Find the equations - in slope-intercept form - of the following lines:

  • The line 3 x + 4y = 7.
  • Through the points (− 1 , 4) and (2, 3).
  • Through the point (1, 2) with slope 3. Two lines are perpendicular if they intersect at ninety degree angles. In equations, it means that if the slope of one line is m, the slope of the other is −m.

Question 12. Why does this make sense?

Example 13. Find the equation of the line perpendicular to the line 2 x − 3 y = 3 at the point (3, 1).