Page 1 | © 2012 by Janice L. Epstein 3.11 Differentials and Approximations Page 2 | © 2012 by Janice L. Epstein Differentials; Linear and Quadratic Approximations
(3.11)
EXAMPLE 2
Find the differential of y = t sint
3.11 Differentials and Approximations The graph of y  x 2 is shown along with the tangent line at x = 1.
4
3
2
Applications of Differentials
1
f  a  x   f  a   y  f  a   dy  f  a   f   a  dx
0
0
0.5
1
1.5
2
-1
Let y  f  x  , where f is a differentiable function. Then the
differential dx is an independent variable and the differential dy is
given by
dy  f   x  dx
(1)
EXAMPLE 3
Use differentials to find an approximate value for the following
(a) (2.02)4
EXAMPLE 1
2
If f  x   x , find dy if x = 1 and dx  0.5
(b) cos 59
Page 3 | © 2012 by Janice L. Epstein 3.11 Differentials and Approximations Page 4 | © 2012 by Janice L. Epstein 3.11 Differentials and Approximations EXAMPLE 4
The radius of a circular disk is given to be 24 cm with a maximum
error in measurement of 0.2 cm.
(a) What is the maximum error in the calculated area of the disc?
EXAMPLE 5
The circumference around the middle of a sphere is measured to be
40 cm, with a possible error of ±1 cm. Use differentials to estimate
the possible error in the volume of the sphere.
(b) Use differentials to approximate the maximum error in the area
of the disc.
The linear approximation (or linearization) of f at x = a is given by
L  x   f  a   f   a  x  a 
EXAMPLE 6
Find the linear approximation of f  x  
(c) What is the relative error in the area of the disc?
1
at x  4
x
Page 5 | © 2012 by Janice L. Epstein 3.11 Differentials and Approximations Page 6 | © 2012 by Janice L. Epstein 3.11 Differentials and Approximations EXAMPLE 7
Find the linear approximation of f  x   x at x  9 4 and use it
The quadratic approximation of f at x = a is given by
f   x 
2
Q  x   f  a   f   a  x  a  
 x  a
2
to approximate
2.
EXAMPLE 9
Find the quadratic approximation of f  x   cos x at x  0
EXAMPLE 8
Find the linear approximation of f  x   sin x at x  0 and use it to
approximate 0.1.
EXAMPLE 10
Q(x) is the quadratic approximation for f  x  
1
Find Q  
2
2
at x=1.
x