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ASU / Math / MAT 265 / What is the difference between differential and linear approximation?

# What is the difference between differential and linear approximation? Description

##### Description: part of section 2.7, section 2.8 and part of 3.1
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"}.lst-kix_t31fjiodifrq-8 > li:before{content:"" counter(lst-ctn-kix_t31fjiodifrq-8,lower-roman) ". "}.lst-kix_usiffv8b1hdc-4 > li{counter-increment:lst-ctn-kix_usiffv8b1hdc-4}.lst-kix_t31fjiodifrq-7 > li:before{content:"" counter(lst-ctn-kix_t31fjiodifrq-7,lower-latin) ". "}ol.lst-kix_usiffv8b1hdc-8{list-style-type:none}ol.lst-kix_lwl07errdvwo-7.start{counter-reset:lst-ctn-kix_lwl07errdvwo-7 0}ol.lst-kix_usiffv8b1hdc-7.start{counter-reset:lst-ctn-kix_usiffv8b1hdc-7 0}ol.lst-kix_t31fjiodifrq-0.start{counter-reset:lst-ctn-kix_t31fjiodifrq-0 0}.lst-kix_t31fjiodifrq-2 > li:before{content:"" counter(lst-ctn-kix_t31fjiodifrq-2,lower-roman) ". "}.lst-kix_t31fjiodifrq-4 > li:before{content:"" counter(lst-ctn-kix_t31fjiodifrq-4,lower-latin) ". "}ol.lst-kix_usiffv8b1hdc-4{list-style-type:none}ol.lst-kix_usiffv8b1hdc-5{list-style-type:none}.lst-kix_t31fjiodifrq-6 > li{counter-increment:lst-ctn-kix_t31fjiodifrq-6}ol.lst-kix_usiffv8b1hdc-6{list-style-type:none}ol.lst-kix_usiffv8b1hdc-7{list-style-type:none}.lst-kix_t31fjiodifrq-1 > li:before{content:"" counter(lst-ctn-kix_t31fjiodifrq-1,lower-latin) ". "}.lst-kix_t31fjiodifrq-5 > li:before{content:"" counter(lst-ctn-kix_t31fjiodifrq-5,lower-roman) ". "}ol.lst-kix_usiffv8b1hdc-0{list-style-type:none}ol.lst-kix_usiffv8b1hdc-1{list-style-type:none}.lst-kix_d6n8itunri8s-7 > li{counter-increment:lst-ctn-kix_d6n8itunri8s-7}ol.lst-kix_usiffv8b1hdc-2{list-style-type:none}ol.lst-kix_usiffv8b1hdc-3{list-style-type:none}.lst-kix_t31fjiodifrq-3 > li:before{content:"" counter(lst-ctn-kix_t31fjiodifrq-3,decimal) ". 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"}ol.lst-kix_t31fjiodifrq-2{list-style-type:none}ol.lst-kix_t31fjiodifrq-1{list-style-type:none}ol.lst-kix_d6n8itunri8s-7.start{counter-reset:lst-ctn-kix_d6n8itunri8s-7 0}ol.lst-kix_t31fjiodifrq-0{list-style-type:none}.lst-kix_lwl07errdvwo-8 > li{counter-increment:lst-ctn-kix_lwl07errdvwo-8}.lst-kix_d6n8itunri8s-5 > li{counter-increment:lst-ctn-kix_d6n8itunri8s-5}ol.lst-kix_usiffv8b1hdc-8.start{counter-reset:lst-ctn-kix_usiffv8b1hdc-8 0}ol.lst-kix_lwl07errdvwo-1.start{counter-reset:lst-ctn-kix_lwl07errdvwo-1 0}ol.lst-kix_x2t4jeun8zlj-1.start{counter-reset:lst-ctn-kix_x2t4jeun8zlj-1 0}.lst-kix_x2t4jeun8zlj-3 > li:before{content:"" counter(lst-ctn-kix_x2t4jeun8zlj-3,decimal) ". "}.lst-kix_x2t4jeun8zlj-4 > li:before{content:"" counter(lst-ctn-kix_x2t4jeun8zlj-4,lower-latin) ". "}ol.lst-kix_x2t4jeun8zlj-8.start{counter-reset:lst-ctn-kix_x2t4jeun8zlj-8 0}.lst-kix_x2t4jeun8zlj-7 > li{counter-increment:lst-ctn-kix_x2t4jeun8zlj-7}.lst-kix_d6n8itunri8s-0 > li:before{content:"" counter(lst-ctn-kix_d6n8itunri8s-0,decimal) ". "}.lst-kix_x2t4jeun8zlj-5 > li:before{content:"" counter(lst-ctn-kix_x2t4jeun8zlj-5,lower-roman) ". "}.lst-kix_x2t4jeun8zlj-7 > li:before{content:"" counter(lst-ctn-kix_x2t4jeun8zlj-7,lower-latin) ". "}.lst-kix_d6n8itunri8s-2 > li:before{content:"" counter(lst-ctn-kix_d6n8itunri8s-2,lower-roman) ". "}ol.lst-kix_usiffv8b1hdc-1.start{counter-reset:lst-ctn-kix_usiffv8b1hdc-1 0}.lst-kix_x2t4jeun8zlj-6 > li:before{content:"" counter(lst-ctn-kix_x2t4jeun8zlj-6,decimal) ". "}.lst-kix_d6n8itunri8s-1 > li:before{content:"" counter(lst-ctn-kix_d6n8itunri8s-1,lower-latin) ". "}ol.lst-kix_d6n8itunri8s-6.start{counter-reset:lst-ctn-kix_d6n8itunri8s-6 0}.lst-kix_t31fjiodifrq-7 > li{counter-increment:lst-ctn-kix_t31fjiodifrq-7}.lst-kix_lwl07errdvwo-8 > li:before{content:"" counter(lst-ctn-kix_lwl07errdvwo-8,decimal) ". "}.lst-kix_d6n8itunri8s-4 > li:before{content:"" counter(lst-ctn-kix_d6n8itunri8s-4,lower-latin) ". "}.lst-kix_d6n8itunri8s-6 > li:before{content:"" counter(lst-ctn-kix_d6n8itunri8s-6,decimal) ". "}.lst-kix_lwl07errdvwo-7 > li:before{content:"" counter(lst-ctn-kix_lwl07errdvwo-7,lower-roman) ". "}.lst-kix_x2t4jeun8zlj-0 > li{counter-increment:lst-ctn-kix_x2t4jeun8zlj-0}.lst-kix_x2t4jeun8zlj-8 > li:before{content:"" counter(lst-ctn-kix_x2t4jeun8zlj-8,lower-roman) ". "}.lst-kix_d6n8itunri8s-3 > li:before{content:"" counter(lst-ctn-kix_d6n8itunri8s-3,decimal) ". "}.lst-kix_d6n8itunri8s-7 > li:before{content:"" counter(lst-ctn-kix_d6n8itunri8s-7,lower-latin) ". "}.lst-kix_lwl07errdvwo-4 > li:before{content:"" counter(lst-ctn-kix_lwl07errdvwo-4,lower-roman) ". "}.lst-kix_lwl07errdvwo-6 > li:before{content:"" counter(lst-ctn-kix_lwl07errdvwo-6,lower-latin) ". "}.lst-kix_x2t4jeun8zlj-6 > li{counter-increment:lst-ctn-kix_x2t4jeun8zlj-6}.lst-kix_t31fjiodifrq-3 > li{counter-increment:lst-ctn-kix_t31fjiodifrq-3}ol.lst-kix_t31fjiodifrq-4.start{counter-reset:lst-ctn-kix_t31fjiodifrq-4 0}.lst-kix_lwl07errdvwo-5 > li:before{content:"" counter(lst-ctn-kix_lwl07errdvwo-5,decimal) ". "}.lst-kix_lwl07errdvwo-7 > li{counter-increment:lst-ctn-kix_lwl07errdvwo-7}.lst-kix_d6n8itunri8s-5 > li:before{content:"" counter(lst-ctn-kix_d6n8itunri8s-5,lower-roman) ". "}.lst-kix_lwl07errdvwo-0 > li:before{content:"" counter(lst-ctn-kix_lwl07errdvwo-0,lower-latin) ". "}.lst-kix_lwl07errdvwo-2 > li:before{content:"" counter(lst-ctn-kix_lwl07errdvwo-2,decimal) ". "}.lst-kix_usiffv8b1hdc-5 > li:before{content:"" counter(lst-ctn-kix_usiffv8b1hdc-5,lower-roman) ". "}.lst-kix_lwl07errdvwo-1 > li{counter-increment:lst-ctn-kix_lwl07errdvwo-1}ol.lst-kix_x2t4jeun8zlj-7.start{counter-reset:lst-ctn-kix_x2t4jeun8zlj-7 0}ol.lst-kix_usiffv8b1hdc-2.start{counter-reset:lst-ctn-kix_usiffv8b1hdc-2 0}.lst-kix_lwl07errdvwo-3 > li:before{content:"" counter(lst-ctn-kix_lwl07errdvwo-3,lower-latin) ". "}ol.lst-kix_t31fjiodifrq-5.start{counter-reset:lst-ctn-kix_t31fjiodifrq-5 0}.lst-kix_usiffv8b1hdc-6 > li:before{content:"" counter(lst-ctn-kix_usiffv8b1hdc-6,decimal) ". "}ol.lst-kix_lwl07errdvwo-2.start{counter-reset:lst-ctn-kix_lwl07errdvwo-2 0}ol.lst-kix_d6n8itunri8s-5.start{counter-reset:lst-ctn-kix_d6n8itunri8s-5 0}.lst-kix_usiffv8b1hdc-3 > li{counter-increment:lst-ctn-kix_usiffv8b1hdc-3}.lst-kix_d6n8itunri8s-8 > li:before{content:"" counter(lst-ctn-kix_d6n8itunri8s-8,lower-roman) ". "}.lst-kix_usiffv8b1hdc-0 > li{counter-increment:lst-ctn-kix_usiffv8b1hdc-0}.lst-kix_usiffv8b1hdc-8 > li:before{content:"" counter(lst-ctn-kix_usiffv8b1hdc-8,lower-roman) ". "}.lst-kix_usiffv8b1hdc-7 > li:before{content:"" counter(lst-ctn-kix_usiffv8b1hdc-7,lower-latin) ". "}.lst-kix_lwl07errdvwo-1 > li:before{content:"" counter(lst-ctn-kix_lwl07errdvwo-1,lower-roman) ". "}ol.lst-kix_lwl07errdvwo-8{list-style-type:none}.lst-kix_usiffv8b1hdc-6 > li{counter-increment:lst-ctn-kix_usiffv8b1hdc-6}ol.lst-kix_lwl07errdvwo-1{list-style-type:none}ol.lst-kix_lwl07errdvwo-0{list-style-type:none}ol.lst-kix_lwl07errdvwo-3{list-style-type:none}ol.lst-kix_lwl07errdvwo-2{list-style-type:none}ol.lst-kix_lwl07errdvwo-5{list-style-type:none}ol.lst-kix_lwl07errdvwo-4{list-style-type:none}ol.lst-kix_lwl07errdvwo-8.start{counter-reset:lst-ctn-kix_lwl07errdvwo-8 0}ol.lst-kix_lwl07errdvwo-7{list-style-type:none}ol.lst-kix_lwl07errdvwo-6{list-style-type:none}.lst-kix_d6n8itunri8s-2 > li{counter-increment:lst-ctn-kix_d6n8itunri8s-2}.lst-kix_lwl07errdvwo-5 > li{counter-increment:lst-ctn-kix_lwl07errdvwo-5}ol.lst-kix_usiffv8b1hdc-6.start{counter-reset:lst-ctn-kix_usiffv8b1hdc-6 0}.lst-kix_x2t4jeun8zlj-4 > li{counter-increment:lst-ctn-kix_x2t4jeun8zlj-4}ol.lst-kix_lwl07errdvwo-6.start{counter-reset:lst-ctn-kix_lwl07errdvwo-6 0}ol.lst-kix_d6n8itunri8s-0.start{counter-reset:lst-ctn-kix_d6n8itunri8s-0 0}.lst-kix_x2t4jeun8zlj-3 > li{counter-increment:lst-ctn-kix_x2t4jeun8zlj-3}.lst-kix_t31fjiodifrq-0 > li{counter-increment:lst-ctn-kix_t31fjiodifrq-0}.lst-kix_lwl07errdvwo-4 > li{counter-increment:lst-ctn-kix_lwl07errdvwo-4}.lst-kix_usiffv8b1hdc-4 > li:before{content:"" counter(lst-ctn-kix_usiffv8b1hdc-4,lower-latin) ". "}ol.lst-kix_d6n8itunri8s-3.start{counter-reset:lst-ctn-kix_d6n8itunri8s-3 0}.lst-kix_usiffv8b1hdc-2 > li:before{content:"" counter(lst-ctn-kix_usiffv8b1hdc-2,lower-roman) ". "}.lst-kix_usiffv8b1hdc-3 > li:before{content:"" counter(lst-ctn-kix_usiffv8b1hdc-3,decimal) ". "}ol.lst-kix_x2t4jeun8zlj-0.start{counter-reset:lst-ctn-kix_x2t4jeun8zlj-0 0}.lst-kix_usiffv8b1hdc-1 > li:before{content:"" counter(lst-ctn-kix_usiffv8b1hdc-1,lower-latin) ". "}ol.lst-kix_t31fjiodifrq-7.start{counter-reset:lst-ctn-kix_t31fjiodifrq-7 0}.lst-kix_d6n8itunri8s-4 > li{counter-increment:lst-ctn-kix_d6n8itunri8s-4}ol.lst-kix_usiffv8b1hdc-0.start{counter-reset:lst-ctn-kix_usiffv8b1hdc-0 0}.lst-kix_d6n8itunri8s-1 > li{counter-increment:lst-ctn-kix_d6n8itunri8s-1}.lst-kix_usiffv8b1hdc-0 > li:before{content:"" counter(lst-ctn-kix_usiffv8b1hdc-0,decimal) ". "}ol.lst-kix_x2t4jeun8zlj-1{list-style-type:none}ol.lst-kix_x2t4jeun8zlj-2{list-style-type:none}ol.lst-kix_x2t4jeun8zlj-3{list-style-type:none}ol.lst-kix_x2t4jeun8zlj-4{list-style-type:none}ol.lst-kix_x2t4jeun8zlj-5{list-style-type:none}ol.lst-kix_lwl07errdvwo-3.start{counter-reset:lst-ctn-kix_lwl07errdvwo-3 0}ol.lst-kix_x2t4jeun8zlj-3.start{counter-reset:lst-ctn-kix_x2t4jeun8zlj-3 0}ol.lst-kix_x2t4jeun8zlj-6{list-style-type:none}.lst-kix_t31fjiodifrq-1 > li{counter-increment:lst-ctn-kix_t31fjiodifrq-1}ol.lst-kix_x2t4jeun8zlj-7{list-style-type:none}ol.lst-kix_x2t4jeun8zlj-8{list-style-type:none}ol.lst-kix_lwl07errdvwo-0.start{counter-reset:lst-ctn-kix_lwl07errdvwo-0 0}.lst-kix_x2t4jeun8zlj-1 > li{counter-increment:lst-ctn-kix_x2t4jeun8zlj-1}ol.lst-kix_usiffv8b1hdc-3.start{counter-reset:lst-ctn-kix_usiffv8b1hdc-3 0}ol.lst-kix_x2t4jeun8zlj-0{list-style-type:none}.lst-kix_x2t4jeun8zlj-2 > li:before{content:"" counter(lst-ctn-kix_x2t4jeun8zlj-2,lower-roman) ". "}ol.lst-kix_x2t4jeun8zlj-6.start{counter-reset:lst-ctn-kix_x2t4jeun8zlj-6 0}.lst-kix_x2t4jeun8zlj-1 > li:before{content:"" counter(lst-ctn-kix_x2t4jeun8zlj-1,lower-latin) ". "}.lst-kix_t31fjiodifrq-2 > li{counter-increment:lst-ctn-kix_t31fjiodifrq-2}.lst-kix_usiffv8b1hdc-2 > li{counter-increment:lst-ctn-kix_usiffv8b1hdc-2}.lst-kix_x2t4jeun8zlj-0 > li:before{content:"" counter(lst-ctn-kix_x2t4jeun8zlj-0,decimal) ". "}ol.lst-kix_t31fjiodifrq-1.start{counter-reset:lst-ctn-kix_t31fjiodifrq-1 0}ol.lst-kix_x2t4jeun8zlj-4.start{counter-reset:lst-ctn-kix_x2t4jeun8zlj-4 0}.lst-kix_usiffv8b1hdc-8 > li{counter-increment:lst-ctn-kix_usiffv8b1hdc-8}ol.lst-kix_lwl07errdvwo-4.start{counter-reset:lst-ctn-kix_lwl07errdvwo-4 0}.lst-kix_usiffv8b1hdc-7 > li{counter-increment:lst-ctn-kix_usiffv8b1hdc-7}ol.lst-kix_usiffv8b1hdc-4.start{counter-reset:lst-ctn-kix_usiffv8b1hdc-4 0}ol.lst-kix_x2t4jeun8zlj-5.start{counter-reset:lst-ctn-kix_x2t4jeun8zlj-5 0}.lst-kix_usiffv8b1hdc-1 > li{counter-increment:lst-ctn-kix_usiffv8b1hdc-1}.lst-kix_x2t4jeun8zlj-8 > li{counter-increment:lst-ctn-kix_x2t4jeun8zlj-8}.lst-kix_x2t4jeun8zlj-5 > li{counter-increment:lst-ctn-kix_x2t4jeun8zlj-5}ol.lst-kix_d6n8itunri8s-8.start{counter-reset:lst-ctn-kix_d6n8itunri8s-8 0}.lst-kix_x2t4jeun8zlj-2 > li{counter-increment:lst-ctn-kix_x2t4jeun8zlj-2}ol.lst-kix_usiffv8b1hdc-5.start{counter-reset:lst-ctn-kix_usiffv8b1hdc-5 0}ol.lst-kix_t31fjiodifrq-2.start{counter-reset:lst-ctn-kix_t31fjiodifrq-2 0}.lst-kix_lwl07errdvwo-3 > li{counter-increment:lst-ctn-kix_lwl07errdvwo-3}.lst-kix_d6n8itunri8s-0 > li{counter-increment:lst-ctn-kix_d6n8itunri8s-0}.lst-kix_d6n8itunri8s-3 > li{counter-increment:lst-ctn-kix_d6n8itunri8s-3}.lst-kix_lwl07errdvwo-6 > li{counter-increment:lst-ctn-kix_lwl07errdvwo-6}.lst-kix_d6n8itunri8s-6 > li{counter-increment:lst-ctn-kix_d6n8itunri8s-6}ol.lst-kix_lwl07errdvwo-5.start{counter-reset:lst-ctn-kix_lwl07errdvwo-5 0}.lst-kix_t31fjiodifrq-8 > li{counter-increment:lst-ctn-kix_t31fjiodifrq-8}ol.lst-kix_t31fjiodifrq-8.start{counter-reset:lst-ctn-kix_t31fjiodifrq-8 0}ol.lst-kix_d6n8itunri8s-2.start{counter-reset:lst-ctn-kix_d6n8itunri8s-2 0}

Section 2.7Don't forget about the age old question of What is morality according to Kantian Ethics?

Example:

Question 27 page 133, bookIf you want to learn more check out Why are mushrooms considered as decomposers?

If you want to learn more check out What is the Treaty of Versailles?

Grovel is being dumped from a conveyor belt at a rate of 30 ft / min, and its coarseness is such that it forms a pile on the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10 ft height?If you want to learn more check out What is meant by expanding epidemiology?

Suppose V represents the volume of the cone

Suppose r represents the radius of the cone

Suppose h represents the height of the coneWe also discuss several other topics like What year did the Civil War begin in the US?

Vcone = r2h        r = We also discuss several other topics like What are algorithms?

V = ()2h → v = ()(h)

V = h3

=h2        30 = (10)2

Section 2.8: Linear approximation and differential

Definition: if y = f(x), where f is a differentiable function then the differential (dx) is an independent variable; that is (dx) can be given the value of any real number. The differential (dy) is then defined in terms of (dx) by dy = f(x). Dx

Example:

1. Y =  = (1 + x2) ½

Dy = (1 + x2)½ (2x)dx

1. Y = (sin t)

Dy = (cos t) ()dt

Differential and linear approximation

f(a + dx) = f(a) + dy

Example 4

Use differentials to approximate  +

Define as f(x) =  +

f(1.01) = f(1 + 0.01) = f(1) + dy

f(1) =  +  = 2

Dy = f1(x)dx

Dy = (x1/2 + x-⅔)dx

Dy = +

Dy = (+)dx

= )

=

Sub 1 in 2

f(1.01 = + = 2 + =

Question 21 from book

The edge of a cube was found to be 30 cm with a possible error in measurement of olem. Used differentials to estimate the maximum possible error, relative error, and percentage error in computing (a) the volume of the cubel, (b) the surface area of the cube

Suppose x represent the edge length of the cube

Suppose v represents the volume of the cube

1. V = x3 → dv = f(x)dx         dx = error = oilem

Dv = 3x2dx

Dv = 3(30)2(0.1) = 270

Relative error =  = = 0.9

Percentage error = (relative error)(100%) = (0.01)(100)% = 1%

A = 6x2         dA = 12xdx → dA = 12(30)(0/.1)

dA = 36

Relative error =  = 0.0067

Example:

Find the domain of the following functions

Domain: largest set of real numbers that make the function definable

1.  f(x) =

Ex > 0 fx: e-3 = > 0 for all real numbers always

Domain = Df = (-,)

1.                  1 - ex = 0

Df = (-)u(0,)

1. g(x) = cos(e-x)

Dg = (-)

1. g(x) =

To find domain we make 1 - 3x > 0

1 - 3x > 0

1 > 3x

in(1) > x in (3)

xin(3) < 0

in(1) = 0

X < 0 dg = (-0)

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