Chapter Two: Section One The Derivative and the Tangent Line Problem

Chapter Two: Section One One of the problems that Newton was trying to solve was the problem of finding the rate at which the Earth was traveling along its elliptical orbit around the Sun at any instant. We are familiar with the idea of rate of change of a line, but we are not familiar with the idea of instantaneous rate of change for a curve. By the way, what word would we use to describe the rate of change along a linear path?

Chapter Two: Section One That word, of course, is slope. I hope that you were thinking of this idea… So, how might we find the slope of a non-linear function? Since these curves are always changing direction and do not have a constant slope, Newton needed to find a way to calculate the slope of a curve at a single point.

Chapter Two: Section One But wait, we cannot calculate slope at a single point, can we? This is where our idea of a limit, the central idea of Chapter One, comes into play. Try to remember your days in Geometry and think about the terms tangent line and secant line.

Chapter Two: Section One A secant line to a curve is a line that intersects a curve twice. A tangent line to a curve is a line that intersects a curve at only one point. Newton came up with a way, using limits, to start with a secant line, so that we can find slope between two points, and converge to a tangent line whose slope is calculated at the point of tangency.

Chapter Two: Section One Follow the link below to see an animation of a secant line converging to a tangent line to a curve. gSecantLine/MovingSecantLine.MOV gSecantLine/MovingSecantLine.MOV

Chapter Two: Section One What is happening in the animation is a representation of one of the most important definitions in Calculus (it’s coming soon). There is a fixed point, call it ( c, f(c)) and a movable point. The coordinates of the movable point are (c +  x, f(c +  x)). As the quantity  x approaches zero, the movable point collapses onto the fixed point and our secant line through two points on the curve is now reduced to a tangent line through one point on the curve.

Chapter Two: Section One Here’s the definition of the derivative, which is the function that tells us the slope of a curve at a point:

Chapter Two: Section One Know this definition!!!

Chapter Two: Section One The way we read the left side of the formula below as:  f prime of c The complex looking formula on the right hand side of the formula will give us a numerical value. This numerical value is the value of the slope of a line tangent to a curve at the point ( c, f(c)).

Chapter Two: Section One This numerical value is also the slope of the curve at that instant. Welcome to Calculus!