You are here:

# Calculus/Calculus_theory

Question
Please sir, explain: "increasing and decreasing functions" and "Approximations"

An increasing function is one that has an upward slope.
For example,
x for all x,
3x-4 for all x,
x^2 + 5 for all x >= 0, and
tan(x) for all x where tan(x) is defined.

One function can be used to approximate any other function, though most of the time it is a poor approximation.  For example, x=0 is an approximation of sin(x), but a very poor approximation.
Over all, the error is always between -1 and 1, but it fits the curve poorly.

An approximation might be x for sin(x) near x=0, for they look pretty much the same near there.
When x is taken to be a little farther from 0 than 0.8, the approximation starts to get very poor (it goes outside of 10% error).

For the function e^x, if -0.7 < x < 1.1 and the approximation function is
f(x) = 1 + x + x²/2 + x³/6, then the error is within ±3%.

The less accuracy is desired, the wider the limits could  be.
The more accuracy that is gotten, the better on function
is said to be an approximation of the other.

Increasing and decreasing are used in calculus, for when f'(x) is positive, f(x) is increasing and where f'(x) is negative, f(x) is decreasing.  When f'(x) = 0, there is a local maximum, minimum, or an inflection point.

Calculus

Volunteer

#### Scotto

##### Expertise

Any kind of calculus question you want. I also have answered some questions in Physics (mass, momentum, falling bodies), Chemistry (charge, reactions, symbols, molecules), and Biology (reproduction, insusion of chemicals into bloodstream).

##### Experience

Experience in the area: I have tutored students in all areas of mathematics since 1980. Education/Credentials: BSand MS in Mathematics from Oregon State University, where I completed sophomore course in Physics and Chemistry. I received both degrees with high honors. Awards and Honors: I have passed Actuarial tests 100, 110, and 135.

Publications
Maybe not a publication, but I have respond to well oveer 8,500 questions on the PC. Well over 2,000 of them have been in calculus.

Education/Credentials
I aquired well over 40 hours of upper division courses. This was well over the number that were required. I graduated with honors in both my BS and MS degree from Oregon State University. I was allowed to jump into a few courses at college a year early.

Awards and Honors
I have been nominated as the expert of the month several times. All of my scores right now are at least a 9.8 average (out of 10).

Past/Present Clients
My past clients have been students at OSU, students at the college in South Seattle, referals from a company, friends and aquantenances, people from my church, and people like you from all over the world.