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What is calculus???

And the Answer is....



Calculus is the study of change, with the basic focus being on
  • Rate of change
  • Accumulation
What is Calculus ?
In both of these branches (Differential and Integral), the concepts learned in algebra and geometry are extended using the idea of limits. Limits allow us to study what happens when points on a graph get closer and closer together until their distance is infinitesimally small (almost zero). Once the idea of limits is applied to our Calculus problem, the techniques used in algebra and geometry can be implemented.

Differential Calculus

Algebra vs Calculus

In Algebra, we are interested in finding the slope of a line.
algebra line
The slope of the line is the same everywhere. The slope is constant and is found using $$ \frac{\Delta y}{\Delta x} $$ or $$ \frac{rise}{run} $$
In Calculus, we are interested in find the slope of a curve
calculus slope of curve
The slope varies along the curve, so the slope at the red point is different from the slope at the blue point. We need Calculus to find the slope of the curve at these specific points
confused

How does calculus help with curves???



To solve the question on the Calculus side for the red point, we will use the same formula that we used in Algebra--the slope formula$$ \frac{\Delta y}{\Delta x} $$ .

However, we are going to make the blue and red points extremely close to each other. The key is, when the blue point is infinitesimally close to the red point, the curve becomes a straight line and $$ \frac{\Delta y}{\Delta x} $$ will then give us an accurate slope.
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Secant to Tangent This applet shows what happens to a curve when you 'zoom in' and look at points that are really close together.
As you can see, the more you 'zoom in' (Or the closer that the two points become), the more the curve approaches a straight line. Click here to see an applet of this concept as it relates to derivatives and slope

zoom in

Differential Calculus in the Real World


This same idea can be applied to real world situations. Consider the distance versus time graph of a slowly moving car shown below.

Average Velocity during the first 3 seconds?

Instantaneous Velocity at 6 seconds ?

Calculus is not needed.

$$velocity = \frac{\Delta distance}{\Delta time} = \frac{3-0}{3-0}= 1 \frac{m}{s} $$
Calculus is needed.

We will need to use the method described above and try to bring two points infinitesimally close to each other.
Distance vs time

Integral Calculus

Algebra vs Calculus

Find the purple region

Area
Does not require Calculus. It is simply the area of a rectangle (base)(height).
Area = 2 × 3 = 6
Find the blue region

area under curve

To find the area of blue region, we need Calculus. 

What can we do?

How does calculus help find the area under curves???

And the Answer is...

Calculus lets us break up the curved blue graph into shapes whose area we can calculate--rectangles or trapezoids. We find the area of each individual rectangle and add them all up. The key is : the more rectangles we use, the more accurate our answer becomes. When the width of each rectangle is infinitesimally small , then our answer is precise. See the example below:
10 rectangles
area under curve2
50 rectangles
area under curve 3

Integral Calculus in the Real World


This same idea can be applied to real world situations. Consider the velocity vs time graph of a person riding a bike.
Note: this is not the same graph that we looked at above. The first one that we looked at was distance vs time

Find the distance traveled during the first 3 seconds?

Find the distance traveled during the first 9 seconds?

Calculus is not needed.

Distance = (velocity)(time)
This is found, by looking at area under the velocity curve bounded by the x-axis. So we just have to find the area of the triangle from x=0 to x=3.
Calculus is needed.

We will need to use the method described above and find the area of infinitesimally small rectangles/trapezoids.
Distance vs time
In later lessons,
  • We will learn how to systemically and practically solve these problems
  • How these basic principles apply to a wide array of real worlds problems dealing with physics, chemistry, biology, business, engineering, medicine, computer science, astronomy and other everyday problems that could not have been solved without Calculus.
  • How integral and differential Calculus are connected using the Fundamental Theorem of Calculus.