9/30/2011
Linear Approximations
The idea is that it might be easy to
calculate a value f(a) of a function, but
difficult (or even impossible) to compute
nearby values of f.
Linear Approximation
AP Calculus BC
Radnor High School
2
y
As you can see, graph of a differentiable
function and its tangent line at the point of
tangency are pretty much the same.
The tangent line is said to approximate the
function for values near the point of
tangency.
xx






3
The equation of the tangent line to the
graph of f(x) at the point (x0, f(x0)) is
4
If x  x  x0 , then
f ( x0  x )  f ( x0 ) x  f ( x0 )
New y value  f ( x0 ) x  old y value
y  f ( x0 )( x  x0 )  f ( x0 )
When x is close to x0 the tangent line
closely approximates the graph of f(x).
f ( x )  f ( x0 )( x  x0 )  f ( x0 )
This is called local linear approximation of
f(x) at x0.
5
6
1
9/30/2011
Differentials
Example:
Find the local linear approximation of
dy
denotes the derivative of y w.r.t. x
dx
f ( x)  1  x at x0  0.
It also represents the ratio of dy and dx
called differentials.
U iit to approximate
Use
i
0.9 & 1.1 .
dx is the differential of x.
dy is the differential of y.
7
If x is a fixed value, then we can have
If y = f(x) is differentiable at x = x0, then
we define the differential of f at x0, to be
the function of dx given by the formula
dy  f ( x)  dx
f (x)
(x) is the slope of the tangent line to the
graph of f at (x, f(x)), dy and dx can be
viewed as the rise and the run of this
tangent line.
dy  f ( x0 )  dx
If dx ≠ 0, then we can have
8
dy
 f ( x0 ).
dx
9
10
There is a distinction between y and dy.
y represents the change in y that occurs
when we travel along the curve starting at x
and end at x + x. y = f(x + x) – f(x).
Where as, dy is the change in y when we
travel along the tangent line.
dy  f ( x)  dx
11
12
2
9/30/2011
Example:
y
Let y = 1/x. Find dy and y at x = 1 with
dx = x = -0.5.
yy
dy
dx=x
x
x
x+x
13
14
Error in Applications
Although y and dy are different, dy is
actually a good approximation for y
provided that dx = x is close to 0.
Let xa be the actual value of x and x0 be the
measured value of x and dx = x0 – xa.
y
x  0 x
y
f ( x) 
 y  f ( x)x  f ( x)dx  dy
x
f ( x)  lim
If y = f(x), then f(xa) is the actual value of y
and f(x0) is the measured value of y.
y = f(x0) – f(xa)  dy = f (x0)dx, this is
called the propagated error in the computed
value of y.
15
16
Example:
Definition:
The ratio of the error to the true value of the
quantity is called the relative error.
Suppose the side of a cube is measured to
be 25cm with a possible error of ±1cm.
The relative error expressed as a percentage
is called the percentage error.
a)) Use
U diff
differentials
ti l tto estimate
ti t th
the error iin
the calculated volume.
b) Estimate the percentage error in the
side and volume.
17
18
3
9/30/2011
Example:
The radius of a sphere is measured with a
percentage error within ±0.004%.
g error in the
Estimate the ppercentage
calculated volume of the sphere.
19
4