Chapter 4: Partial Differentiation
Section 4.7: Approximations
Use differentials to estimate the maximum error in determining the surface area of a closed rectangular box measured (in inches) to be 15×6×21, if each measurement is accurate to .2 in.
If the length, width, and height of the box are x,y, and z respectively, the surface area of the box is given by the function fx,y,z=2y z+x z+x y. The maximal error in measuring the surface area as f15,6,21=1062, will be approximately df, where
=fx15,6,21⋅±0.2+fy15,6,21⋅± 0.2+fz15,6,21⋅± 0.2
= ± 33.6
Indeed, the actual value of the maximal error is
Maple Solution - Interactive
Define fx,y,z and its first partial derivatives
Context Panel: Assign Function
fx,y,z=2y z+x z+x y→assign as functionf
Calculus palette: Partial derivative operator
(Set the symbols fx, fy and fz as Atomic Identifiers)
f__xx,y,z=∂∂ x fx,y,z→assign as functionf__x
f__yx,y,z=∂∂ y fx,y,z→assign as functionf__y
f__zx,y,z=∂∂ z fx,y,z→assign as functionf__z
Context Panel: Evaluate and Display Inline
f__x15,6,21⋅.2+f__y15,6,21⋅.2+f__z15,6,21⋅.2 = 33.6
Compute Δf "exactly"
f15.2,6.2,21.2−f15,6,21 = 33.84
Maple Solution - Coded
f≔x,y,z→2y z+x z+x y:
Use the differential operator D to obtain the partial derivatives in df.
df≔D1f15,6,21⋅.2+D2f15,6,21⋅.2+D3f15,6,21⋅.2 = 33.6
<< Previous Example Section 4.7
Next Example >>
© Maplesoft, a division of Waterloo Maple Inc., 2023. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
Download Help Document
What kind of issue would you like to report? (Optional)