1. Tangent Lines (Sections 2.1 and 3.1 )
Alex Karassev

2. Tangent line ? What is a tangent line to a curve on the plane?
Simple case: for a circle, a line that has only one common point with the circle is called tangent line to the circle
This does not work in general! ? P P

3. Idea: approximate tangent line by secant lines
Secant line intersects the curve at the point P and some other point, Px y y Px Px P P x x a x x a

4. Tangent line as the limit of secant lines
Suppose the first coordinate of the point P is a
As x → a, Px x → a, and the secant line approaches a limiting position, which we will call the tangent line y y Px Px P P x x a x x a

5. Slope of the tangent line
Since the tangent line is the limit of secant lines, slope of the tangent line is the limit of slopes of secant lines
P has coordinates (a,f(a))
Px has coordinates (x,f(x))
Secant line is the line through P and Px
Thus the slope of secant line is: m y
y=f(x) mx P Px
f(x) x x a

6. Slope of the tangent line
We define slope m of the tangent line as the limit of slopes of secant lines as x approaches a:
Thus we have: m y
y=f(x) mx P Px
f(x) x x a

7. Example
Find equation of the tangent line to curve y=x2 at the point (2,4) y P Px x 2

8. Solution
We already know that the point (2,4) is on the tangent line, so we need to find the slope of the tangent line
P has coordinates (2,4)
Px has coordinates (x,x2)
Thus the slope of secant line is: y P Px x 2

9. Solution
Now we compute the slope of the tangent line by computing the limit as x approaches 2: y P Px x 2

10. Solution y
Thus the slope of tangent line is 4 and therefore the equation of the tangent line is y – 4 = 4 (x – 2) , or equivalently y = 4x – 4 P Px x 2