Name:
Mods:
Date:
2.2b Notes on Tangent/Normal Lines
Tangent Lines:
Definition: the tangent line at a point (𝑎, 𝑓(𝑎)) has the same slope as the instantaneous slope of
𝑓(𝑥)at a.
Example 1:
Find the equation for the line tangent to 𝑓(𝑥) = 𝑥 2 at 𝑥 = 3.
1. First find the slope
Δ𝑦
𝑓′(3) =
=
Δ𝑥
2. Then find the x-y coordinate when x = 3
3. Last, create a line using the point-slope form (then graph above!)
Point-Slope Form:
y  y1  m( x  x1 )
Example 2:
1
Find the slope of the tangent line to the function 𝑓(𝑥) = 𝑥 at 𝑥 = 1.
Now create the tangent line.
Normal Lines:
Definition: the normal line at a point (𝑎, 𝑓(𝑎)) intersects the tangent line and is perpendicular to it.
Example 3:
First, find the slope like you did in the last example of 𝑓(𝑥) = 𝑥 2 − 3𝑥 at (−2,10).
What is the perpendicular slope to this:
Now write the equation for the normal line to f (x).
Graph each.
Example 4:
Find the normal line to f ( x)  3x 2  2 at x = 1
Exercises:
1. Find the tangent line to f ( x)  x3  2 x 2  2 at x = 2
3
2. Find the tangent line to f ( x)  2
at x = -4
x  25
3. Find the normal line to f ( x)  x3  x 2  2 at x = 1
1
4. Find the normal line to f ( x) 
at x = 5
x4
Answers: 1. f ( x)  4 x  6
2. f ( x) 
Homework: 2.2b Worksheet parts a-c
8
23
x
27
27
3. f ( x)   x  1
4. f ( x)  x  4