Calculus

Calculus is the study of calculations that include infinitesimals. Calculus comes in two flavors, Differentiation and Integration. Calculus is not derived from Algebra, although when you do calculus calculations, there’s a lot of algebra involved. Calculus earns our respect by how it works with curved functions. If you are new to calculus you might like Special Calculus where all the functions are straight lines.

Differentiation

Differentiation calculates the slope of a function at a point. If you have studied slopes before, you know a slope is calculated from two points. There’s some magic involved to calculate a point slope. For our example below, the function is a straight line and so all the point slopes are equal to the slope of the line.

A small jogging mammal plots her position as a function of time. She starts at the origin and jogs in the positive x direction.

  • 1 second, x=2 ft
  • 2 seconds, x=4 ft
  • 3 seconds, x=6 ft

The function for this is

x=2t

The slope of this line is 2. This is true for every point in the function f(t)=2t. Her velocity is 2 ft/second. For a graph with feet values for the y-axis and time in seconds for the x-axis, the derivative at any point is the velocity at that point.

The derivative is the plot of all the point slopes. For this story, f'(t)=2. The apostrophe indicates that it is a first derivative.

Velocity is the first derivative of Distance.

Integration

Integration is the calculation of area under a function.

Assume you drive 100 km/hr for 4 hours. Plot this on a graph. You have a straight line at y=100. As you can see in the illustration below, we are measuring the area of a rectangle with sides of 100 and 4. It should make sense that after driving four hours at 100 kilometers per hour the distance traveled would be 400 km.

The next story may be more fun. At the surface of earth, gravity provides an acceleration of 9.8 m/s^2. If we are on a spaceship and we use this acceleration, it will feel like we are on Earth. We also have the option to increase the acceleration just a little bit to make it 10.0 m/s^2 and that will make math calculations easier.

We can make a graph that plots acceleration as a function of time. Since we saw the acceleration never changes, its going to be a graph that’s a horizontal line at y=10 or a(t)=10.

Assume we are interested in our velocity after one minute and after one hour.

v(t) = 10t

v(60) = 600 m/s

v(3600) = 36,000 m/s

But you are probably wondering, how far have we traveled?

To answer this, we will give you one definite integral:

v_f - v_i = \dfrac {1}{2} a t^2

Revelation

A student did some work using f(x)=x^2