Chapter 2. Limits and Derivatives Introduction The invention of the calculus was accomplished by the mathematician and physicist Newton (1642-1727) and the mathematician, logician, and philosopher Leibniz (1646-1716). The entire calculus is based on the concept of limit.

Limits arise when people try to find the tangent line to a given curve or the instantaneous velocity of a moving object. Tangent problem and velocity problem lead us to formulate the notion of derivative using limits. Derivative is a very powerful tool to discover the properties for a given function.

Some applications of limits. Finally, the definition of derivative. In this chapter, we will see: How the notion of limits arises when we try to find an equation of the tangent line to some curve at some point and instantaneous velocity for some moving object. Definitions of limits. How to compute limits using limit laws. We only learn how to compute limits for simple functions not for general functions. Some applications of limits. Finally, the definition of derivative.

§2.1 How limits arise Tangent line: Find an equation of the tangent line to the parabola y = f(x) = x2 at P(1,1) Challenge: An equation of tangent line at P is given by: y – 1 = m(x – 1), where m is the slope of the tangent line. We only know one point on the line. But we need two points to compute the slope m.

Instantaneous Velocity : Suppose that a ball is dropped from the top of the CN Tower, 450 m above the ground. Find the velocity V of the ball after 5 seconds. Challenge : The velocity keeps changing all the time. (1) There is no time interval given. (2) Even if a time interval is given, we can only compute an average velocity.

[5,t] [5,6] 53.9 m/s [5,5.1] 49.49 m/s [5,5.05] 49.245 m/s [5,5.01] [5,5.001] 49.0049 m/s This table suggests that the limit of average velocities is 49. So the required instantaneous velocity after 5 seconds V = 49 m/s.

Comment 1: For a long time people thought that heavier objects fall faster. The reason is that, the great Aristotle said it was so. Galileo (1564-1642) argued that all objects fall at the same rate if the effect of air resistance is ignored. Simple mathematical reasoning led Galileo to suspect Aristotle’s statement. That is, before his experiment, Galileo knew that Aristotle’s statement must be wrong. Do you know why?

Calculus, on the other side, deals with dynamic situations: Comment 2: Algebraic problems considered in high school dealt with static situations: What is the revenue when x items are sold? If the lengths of two legs in a right triangle are 3 m and 4 m, respectively, what is the length of the other side? …… Calculus, on the other side, deals with dynamic situations: How fast is a rocket going at any instant after lift-off? Air is being pumped into a spherical balloon so that its volume increasing at a rate of 100 cm3/s, how fast is the radius increasing when the diameter is 50 cm.