Transcript What is calculus? (I still don`t know!)

What is calculus?
It is based on two BIG ideas…
BIG IDEA 1 - Differentiation
In the last few weeks, we have been leading to and discussing
this idea.
Simply put, differentiation is the process of finding the
derivative, which is defined in calculus as the slope or
steepness of a curve.
The slope of a straight line is constant.
For other shapes, the slope is the line tangent to a curve at
any point; this slope changes according to an equation.
It is based on two BIG ideas…
BIG IDEA 2 - Integration
If you signed up for calculus, you will learn all about this next
year.
Integration is the process of finding the area between a curve
and the x-axis.
Basically, calculus allows us to add up little bits of area under
the curve to get the total area. So you could say it’s just fancy
addition.
S IS THE TOTAL
AREA BETWEEN
POINTS a AND
b
You can use regular math to
figure out how much cable you’d
need for these power lines…
…but you need calculus to figure out how much cable to use for the
catenary patterns on these towers.
You can use regular math to
figure out the cost of building
and maintaining this roof…
…but you need calculus figure out cost and maintenance for this roof!
You can use regular math to figure out the proper lead for hitting the
receiver…
(and, by the way, if you miss this one, it might mean the championship…)
but you need calculus figure out the proper lead for landing on another
planet…
(and, by the way, it’s much worse if you miss this one…Feel better about
losing the Super Bowl?)
The logarithmic spiral of the Nautilus shell is a classical image used
to depict the growth and change related to calculus.
This is the standard
logistic sigmoid
function…Pretty, no?
Originally, this curve was studied in population growth. Notice how the initial
growth is approximately exponential. As saturation begins, growth slows and
eventually stops at maturity. As it turns out, the curve has applications in a
variety of fields, including The logistic function finds applications in a range of
fields, including biology, economics, chemistry, probability, statistics,
sociology, and political science.
Most importantly, we wouldn’t be able to apply it to all these fields if it weren’t
for calculus.
Calculus is used in physics and every branch of the physical
sciences:
Actuarial science (think insurance rates and pension plans)
Computer science
Statistics
Engineering
Economics
Business
Medicine
…and more!
Whenever we can describe a problem using mathematical
terms, calculus can help us find the best, or optimal solution.
Real life rarely follows a straight line.