Calculus What is it? What good is it? What is it? What good is it?

CalculusCalculus

What is it? What good is it?

What is it? What good is it?

What is Calculus?What is Calculus?

A systematic method for solving problems about systems that change.

A language for describing dynamic systems.

A collection of ideas based on infinite processes and limits.




Systems that changeSystems that change

Precalculus Precalculus Calculus CalculusLimit

Process

With Calculus you can…With Calculus you can…

Calculate the slope of line…even when the shape of the line is always changing!

Calculate the slope of line…even when the shape of the line is always changing!

Systems that changeSystems that change Precalculus

A B 200 Miles

4 Hours

Average speed =50 mph

Precalculus

A B 200 Miles

4 Hours

Average speed =50 mph

Calculus

InstantaneousSpeed = 55 mph

Calculus

InstantaneousSpeed = 55 mph

LimitProcess


Calculate the speed of a car or a particle in space…even when the speed is always changing!

Calculate the speed of a car or a particle in space…even when the speed is always changing!

Systems that changeSystems that change Precalculus Precalculus Calculus Calculus

5

4

Area = 20

4

5

€

y = x 3 − 6x 2 + 9x +1

Area = 12

LimitProcess


Calculate the area under a curve…even when the height is always changing!

Calculate the area under a curve…even when the height is always changing!

A method for solving problems about systems that

change

A method for solving problems about systems that

change Precalculus formulas:

Slope of a lineAverage speedArea of a rectangle

Precalculus formulas:


Calculus formulas:

Slope of a curveInstantaneous

speedArea under a curve

Calculus formulas:

Slope of a curveInstantaneous

speedArea under a curve



Precalculus:


Precalculus:


Calculus:

SlopeInstantaneous

speedArea

Calculus:

SlopeInstantaneous

speedArea

Other things calculus can do:Other things calculus can do:

Explain why the angle between an observer and a rainbow is 42 degrees.

Connect designs on a laser printer to form letters of arbitrary size.

Find the best seat in the theater for viewing a movie.

Etc.

Explain why the angle between an observer and a rainbow is 42 degrees.

Connect designs on a laser printer to form letters of arbitrary size.

Find the best seat in the theater for viewing a movie.

Etc.






"Velocity is the derivative of position"

"Velocity is the derivative of position"

The relation between velocity and position is given precisely in the language of calculus.

The relation between velocity and position is given precisely in the language of calculus.€

v =ds

dt

"Force is the derivative of momentum"

"Force is the derivative of momentum"

Newton's second law of motion is accurately and precisely described in the language of calculus.

Newton's second law of motion is accurately and precisely described in the language of calculus.€

F =d(mv)

dt

The language of calculus shows that the relationship between velocity and position is the same as that between force and momentum.

The language of calculus shows that the relationship between velocity and position is the same as that between force and momentum.€

v =ds

dt F =

d(mv)

dt








Big Ideas of CalculusBig Ideas of Calculus

Limits

Differential

Calculus

Integral

Calculus

What good is calculus?What good is calculus?

…there is an opposition in all things. If not so … righteousness could not be brought to pass; neither wickedness; neither holiness nor misery; neither good nor bad. Wherefore, all things are a compound in one. II Nephi 1:81-83

…there is an opposition in all things. If not so … righteousness could not be brought to pass; neither wickedness; neither holiness nor misery; neither good nor bad. Wherefore, all things are a compound in one. II Nephi 1:81-83

Voyager II movie.Voyager II movie.QuickTime™ and a

H.264 decompressorare needed to see this picture.

Calculus was used to find:Calculus was used to find:

Trajectory of flight Effect of gravity of Saturn & Jupiter Amount of fuel needed After 12 years of flight, Voyager II

came within 41 miles of its target, 3000 miles from Neptune

Trajectory of flight Effect of gravity of Saturn & Jupiter Amount of fuel needed After 12 years of flight, Voyager II

came within 41 miles of its target, 3000 miles from Neptune

The same calculus used to:The same calculus used to:

Determine trajectories of airborne bombs and missiles

Determine trajectories of airborne bombs and missiles

The same calculus can be used:

The same calculus can be used:

To help solve the problems of resource and energy distribution that are necessary for peace on earth.

To help solve the problems of resource and energy distribution that are necessary for peace on earth.

Those who know the calculusThose who know the calculus

will be the ones who choose which problems will be solved in this generation, whether for good or for evil.

will be the ones who choose which problems will be solved in this generation, whether for good or for evil.

Remember…Remember…

To be learned is goodif they hearken to the counsels of God.

II Nephi 6:61

To be learned is goodif they hearken to the counsels of God.

II Nephi 6:61

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Calculus What is it? What good is it? What is it? What good is it?