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Preview of calculus
The Tangent Problem
There are two main problems in Calculus:
- Tangent Problem
- Area Problem
The word "tangent" means touch.
- Question
- What is the best way to describe the curve close to A?
- Answer
- We use a tangent line.
Why? The tangent line touches the point A and no where else in the curve. If we zoom in, they virtually merge.
One motivation comes from the study of motion.
Example 1:
- Question
- Where do the lights of a car traveling on a curvy road point?
- Answer
- The direction the lights point is a "tangent" to the road's curvature.
Example 2:
- Question
- Where would a rock released from a sling go (view from above)?
- Answer
- The rock's path is a straight line and we can measure its position.
In real life, the position is given by DATA.
We measure the "rate of change" by measuring the slope of the line from one point to the next. From the graph above, this is the change in y over the change in x.
change in y = \(64 - 36 \)
change in x = \(8 - 6 \)
slope = \( \dfrac{\text{rise}}{\text{run}} = \dfrac{64-36}{8-6} = \dfrac{28}{2} = 14 \)
Continuous Curves
The rate of change of y with respect to x seems clear for discrete data but what if the change is continuous? We can't apply this algebra because there is no "next" point. When we draw secant lines, they cut the curve at various points. Ideally, we want to choose a line that just cuts one point on the curve; the tangent line.
We see that the line AB is better than AC and the closer our next point is to A, the better. But we can't have \( B = A \) as straight lines are defined by two points. Instead, we keep making the distance between \(A\) and \(B\) smaller.
This process is called 'limit and it's the essence Calculus.
What is this about?
To finish this process, we consider what the slope of the line would be as the limit of \( B - A \) approaches zero. This is called the derivative of \(AB\) and is the slope of the tangent line.
To illustrate the notion of limits, consider the sequence of numbers. $$ 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \ldots $$ Each step, the fraction gets smaller and smaller until it approaches an infinitesimal limit.
The Area Problem
What is the area of a cardioid? and ellipse? or a circle?
You know that the area of a circle is \( a = \pi r^{2} \). But the real question is: What is the area? Do we understand it?
One thing we do know. The area of a rectangle \( a \times b\) is \( a b\). Further, triangles are the union of two rectangles. But what are the areas of curved objects?
To try to understand and compute these area, we approximate the curved areas by filling them with increasing number of rectangles or triangles and increasing the number of sides with each step. We can then calculate the area by calculating the area at each step and then taking the "limit".
The use of triangles makes the problem of determining the area more difficult than rectangles. For example, we can represent a circle using cartesian coordinates and represent it as the equation $$ y = \sqrt{r^{2} - x^{2}} $$ where \( r \) is the radius and \( x \) is a point on the x-axis.
Then
We can approximate the area under the graph through the use of rectangles, where the area under the graph \( \approx \) the sum of the areas of the rectangles.
We find the total area under the curve by adding the areas of all the rectangles. We can improve accuracy by using thinner rectangles.
This time, the width of the rectangle is $1$, then we have more rectangles to use and can reduce the error in our calculations. If we keep reducing the width to infinitesimal amounts and use an "infinite" number of rectangles, we can make the error disappear. This is what calculus is about.