1. Section 2.7

2. Tangent Line Problem
Problem of finding the tangent line at a given point boils down to finding the slope of the tangent line at that point
Slope secant line

3. Def: Tangent line with slope m
If function f is defined on an open interval containing c and the
exists, then the line passing through (c, f(c)) with slope m is called the “tangent line.”

4. Examples of finding the derivative using the limit process
1. Find the slope of the tangent line to the graph of at the point (-1,2).

5. Ex. 1 ctd.
The formula of 2x will find the slope of any tangent line to the graph of
AT THE POINT (-1,2)
m = 2(-1) , so the slope is -2
AT THE POINT (0,1)
The slope would be 2(0) = 0

6. Ex.2
Write the equation of the line that is tangent to at (4,2).

7. Equation of tangent line
Ex. 2 ctd.
Since x=4
Check on your calculator!
Equation of tangent line

8. Notation for Derivatives
= the derivative of y with respect to x
gives us a formula for finding the slope of the tangent line at the point (x, f(x)).

9. The “Alternative Form”
If you only need to find f’(x) for one point, this form will be shorter.
Start with:
Let x = c +∆x
x-c = ∆x

10. Example 1
Find f '(1)

11. Example 2
Find f '(2)