1 Chapter 2 Differentiation: Tangent Lines

tangent In plane geometry, we say that a line is tangent to a circle if it intersects the circle in one point. But, for more general curves, we need a better definition. essential The concept of tangent lines is essential to your understanding of differential calculus so we must have an accurate idea of its meaning. There are many rough ideas of what tangent lines are – many of which are not only rough, but wrong. Write down what your definition of a tangent line to a curve by completing the sentence: A line is tangent to a curve _____________________

Tangent Lines There are several incorrect definitions about tangent lines that we must be careful to avoid: Definition #1: Definition #1: A line is tangent to a curveif it crosses the curve in exactly one point A line is tangent to a curve if it crosses the curve in exactly one point. NOT Its possible for a line to touch at one point and NOT be tangent. WRONG! NOT This line is NOT a tangent line.

Tangent Lines There are several misconceptions about tangent lines that we must be careful to avoid: Definition #2: Definition #2: A line is tangent to a curveif it touches the curve only once A line is tangent to a curve if it touches the curve only once. Its possible for a tangent line to touch a curve at multiple points. WRONG! This is a tangent line. But it crosses in two places

Tangent Lines There are several misconceptions about tangent lines that we must be careful to avoid: Definition #3: Definition #3: A line is tangent to a curveif it touches the curve at only one point but does not cross the curve A line is tangent to a curve if it touches the curve at only one point but does not cross the curve. NOT Its possible for a line segment to touch a curve at only one point and not cross and still NOT be a tangent. WRONG! NOT This is NOT a tangent line

Tangent Lines There are several misconceptions about tangent lines that we must be careful to avoid: Definition #4: Definition #4: A line is tangent to a curveif it grazes the curve at one point but does not cross at that point A line is tangent to a curve if it grazes the curve at one point but does not cross at that point. Its possible for a tangent to cross our curve at one point. WRONG! This is a tangent line

An Informal Definition At this point, its difficult for us to get a clear definition of a tangent line to a curve. So we need to rely on our general knowledge of a tangent line. Itll take some time before we can understand its true definition. However, since I always have students that must see the true definition, here it is: A straight line is said to be a tangent of a curve y = f ( x ) at a point x = c on the curve if the line passes through the point P( c, f ( c )) on the curve and has slope f (c) where f ' is the derivative of f… A straight line is said to be a tangent of a curve y = f ( x ) at a point x = c on the curve if the line passes through the point P( c, f ( c )) on the curve and has slope f (c) where f ' is the derivative of f… now, I know you feel much better…

Example: Try to sketch the tangent line to the curves at the indicated point:

Example:

Example:

Example:

The Key to the Tangent Line to a Curve Although it may be difficult for us to come up with a good definition for the tangent line to a curve, we get a little help from Newton and Leibniz and we just need to use their definition. limits and the slope There are two key components to the definition of the tangent line: limits and the slope ! But before we dive into studying the tangent line of a curve, we need to make sure we remember the slope. Dont forget the basic formula for slope between two points:

Practice Problems Determine the slope of the line that passes through the following two points: 1. (-4, 0) and (8, 0) 2. (-2, -5) and (3, 15) 3. (3, -2) and (-1, 1) Sketch the graph, then approximate the slope of the tangent line for each function at the given point (you may use your calculator to graph the function): 1. f (x) = (x – 3) 2 – 1 at the x = 3 2. f (x) = x 4 – x 2 at the x = f (x) = ln x at the x = e

Homework Tangent Lines Worksheet: Tangent Lines