Begin by watching this video. As you watch, ask yourself if you are ready to study calculus. How can you better prepare yourself to be a calculus student?
History of Calculus
Before we begin, briefly review the history of calculus below to see some major historical figures who contributed to the development of calculus
Although we will encounter almost all of the historical figures above, only two of them will be considered major contributors to the invention of calculus. Which two do you think they are?
You should be able to make a good guess already. We will encounter to a great degree the contributions of Isaac Newton and Gottfried Leibniz.
Skills Required in Calculus
One important skill you will need to practice more and more as you study calculus will be to solve problems in steps. Although the final answer in a problem is important, your ability to express your ideas in words or mathematical processes is just as important. You should begin practicing this skill now.
Download the following "Completing the Square Worksheet" and practice solving the problems on scrap paper.
Then download and print out the following "Process-wise Math Problem-Solving Template" and show your working in three or more steps. Be sure to explain in words where necessary
Time to relax! Take a break and watch the following music video.
Two Problems in Calculus
There will be two major problems that calculus will try to solve which cannot be solved in any other branch of mathematics. They are:
Problem 1: The Tangent (Velocity) Problem
This problem is to find the equation of the tangent to a curve (function) at a point. In algebra, you could find the equation of a line if you knew the slope and any point on the line. However, to find the slope, you need to know two points on the line. Therefore, you couldn't find the equation of the tangent line to a curve at a point because you would only know that point. In calculus, we will try to solve this problem. Before we proceed, you will need to know the difference between a tangent line and a secant line
Check out the following line and then answer the questions that follow:
Now answer the following questions
- What is a tangent line?
- What is a secant line?
- How can you find an approximate value for the slope of the tangent line?
Let y=f(x) be the function that we want to investigate. To find an approximate value for the slope of the tangent line at the point C (c, f(c)), we can take two nearby points A and B. We can make A and B as close as possible to C. Go to the following link and investigate the slope of the tangent line to a function such as y=x^3 at the point x=1 by moving the line AB alone the curve around that point. Change the function and find estimates to slopes of tangent lines at different points
Now you know how to begin to find the slope of a tangent line. The tangent line problem is similar to a physics problem called the velocity problem. If an object moves at a constant velocity for a time t, we can easily find the velocity by dividing the total displacement by the time. Suppose the velocity is not constant, how can we find the velocity at any instant (the instantaneous velocity)? We can approach it in a similar manner with the tangent problem. We will not go into details for now.
For now, you should remember the basic method to approach the tangent problem. The tangent problem is solved in the branch of calculus called "Differential Calculus"
Problem 2: The Area Problem
We know how to find the area of defined shapes such as squares, rectangles, triangles, circles, etc. However if we are asked to find the area enclosed by the curve y=x^2, the x-axis, and the lines x=1 and x=3, how can we do this? This is one problem that calculus helps us solve. The following video explains how we can approach the area problem
We will encounter the area problem in the branch of calculus called "Integral Calculus"
To understand better how to solve these two problems, we must first understand the concept of Limits. Thus the first part of calculus (Limits) will be our next topic.
To summarize; there are three parts of calculus that depend on each other. They are Limits, differential calculus, and integral calculus. In limits, we find out what a function approaches as the independent variable approaches a certain value. In differential calculus we use limits to solve the tangent problem. In integral calculus we use limits to solve the area problem