Transcript Tangent lines

Tangent Lines
(Sections 2.1 and 3.1 )
Alex Karassev
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
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
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
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:
f ( x)  f (a)
mx 
xa
m
y
y=f(x)
Px
P
mx
f(x)
x
x
a
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:
m  lim mx
xa

m
y
y=f(x)
Thus we have:
f ( x)  f (a)
m  lim
xa
xa
Px
P
mx
f(x)
x
x
a
Example

Find equation of the tangent line to curve y=x2 at the
point (2,4)
y
P
Px
x
2
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
y
has coordinates (2,4)
Px
has coordinates (x,x2)
Thus
P
Px
x
2
the slope of secant line is:
f ( x)  f (a)
mx 

xa
x 2  a 2 x 2  22 x 2  4


xa
x2
x2
Solution

Now we compute the slope of the tangent line by
computing the limit as x approaches 2:
y
P
Px
x
2
f ( x)  f (a)
m  lim mx  lim

x 2
x 2
xa
x2  4
( x  2)( x  2)
lim
 lim

x 2 x  2
x 2
x2
lim ( x  2)  2  2  4
x 2
Solution
y

P
Px
x
2
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