Chapter 3 – Differentiation Rules
3.10 Linear Approximations and
Differentials
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3.10 Linear Approximations and
Differentials
Dr. Erickson
Differentials
If y = f(x), where f is a differentiable function,
then the differential dx is an independent
variable (can be given the value of any real
number).
The differential dy is defined in terms of dx by
dy  f '( x)dx
dy is the dependent variable that depends on
values of x and dx.
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3.10 Linear Approximations and
Differentials
Dr. Erickson
Differentials
Let P  x, f ( x) and Q  x  x, f ( x  x)  be points on
the graph of f and let dx = x. The
corresponding change in y is
y  f ( x  x)  f ( x)
Propagated
error
Measured
Value
Exact Value
Measurement
error
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3.10 Linear Approximations and
Differentials
Dr. Erickson
NOTE
The approximation y  dy becomes better as x
becomes smaller. For more complicated
functions, it may be impossible to compute y
exactly. In such cases, the approximation by
differentials is useful. The linear
approximation f ( x)  f (a)  f '(a)( x  a)
can be written as
f (a  dx)  f (a)  dy
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3.10 Linear Approximations and
Differentials
Dr. Erickson
Example 4
Find the differential of each function.
s
1. y 
1  2s
2. y  eu cos u
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3.10 Linear Approximations and
Differentials
Dr. Erickson
Example 5
y  cos x
a)
b)
Find the differential dy.
Evaluate dy for the given values of x and dx.
x
6

3
dx  0.05
3.10 Linear Approximations and
Differentials
Dr. Erickson
Example 6
Compute y and dy for the given values of x and
dx=x. Sketch a diagram showing the line
segments with lengths dx, dy, and y.
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a)
x = 1, x = 1
y x
b)
x = 0, x = 0.5
y  ex
3.10 Linear Approximations and
Differentials
Dr. Erickson
Example 7
Use a linear approximation (or differentials) to
estimate the given number.
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a)
(2.001)5
b)
e-0.015
c)
tan44o
3.10 Linear Approximations and
Differentials
Dr. Erickson
Example 8
The radius of a circular disk is given as 24 cm with a
maximum error in measurement of 0.2 cm.
a.
b.
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Use differentials to estimate the maximum error in the
calculated area of the disk.
What is the relative error? What is the percentage error?
3.10 Linear Approximations and
Differentials
Dr. Erickson
Example 9
a.
Use differentials to find a formula for the
approximate volume of a thin cylindrical shell with
height h, inner radius r, and thickness r.
b.
What is the error involved in using the formula
from part (a)?
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3.10 Linear Approximations and
Differentials
Dr. Erickson
Example 10
One side of a right triangle is known to be 20 cm
long and the opposite angle is measured as 30o,
with a possible error of 1o.
a)
b)
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Use differentials to estimate the error in
computing the length of the hypotenuse.
What is the percentage error?
3.10 Linear Approximations and
Differentials
Dr. Erickson
Example 11
If current I passes through a resistor with
resistance R, Ohm’s Law states that the voltage
drop is V=RI. If V is constant and R is
measured with a certain error, use differentials
to show that the relative error in calculating I
is approximately the same (in magnitude) as
the relative error in R.
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3.10 Linear Approximations and
Differentials
Dr. Erickson
Example 12
When blood flows along a blood vessel, the flux F (volume of blood
per unit time that flows past a given point) is proportional to the
fourth power of the radius R of the blood vessel:
F  kR 4
(this is known as Poiseuille’s Law). A partially clogged artery can
be expanded by an operation called angioplasty, in which a balloontipped catheter is inflated inside the artery in order to widen it and
restore the normal blood flow.
Show that the relative change in F is about four times the relative
change in R. How will a 5% increase in the radius affect the flow of
blood?
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3.10 Linear Approximations and
Differentials
Dr. Erickson