If y = 4 root(72+x^2) , find the approximate change in y when  (a) x increases from 3 to 3.01If y = 2x^2 = 3x , find the approximate percentage change in y when x decreases from 2 to 1.97....

If y = 4 root(72+x^2) , find the approximate change in y when 

(a) x increases from 3 to 3.01

If y = 2x^2 = 3x , find the approximate percentage change in y when x decreases from 2 to 1.97.

Although I know I have asked two questions, but I really need your help, the answers are not really that big that you have to spend so much time thinking about the answers. 

 

Please help! 

Asked on by johnboston

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embizze's profile pic

embizze | High School Teacher | (Level 1) Educator Emeritus

Posted on

We are given `y=root(4)(72+x^2)` and we are asked to approximate `Delta y` as x changes from 3 to 3.01.

Use `Delta y ~~ f'(x)dx` . Here `dx=.01` , `x=3`

`f(x)=(72+x^2)^(1/4)`

`f'(x)=(1/4)(72+x^2)^(-3/4)(2x)=x/(2(72+x^2)^(3/4))`

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Then `Delta y ~~ 3/(2(72+3^2)^(3/4))*.01=3/(2(81)^(3/4))*.01=3/54 *.01=.000bar(5)`

The approximation is `Delta y ~~ dy=.000bar(5)`

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Compare to the actual value of `Delta y` :

`f(3)=root(4)(72+3^2)=root(4)(81)=3`

`f(3.01)=root(4)(72+3.01^2)=root(4)(81.061)~~3.000556327`

Thus `Delta y=.000556327` to nine decimal places compared to the approximation `Delta y ~~dy=.000bar(5)`

sciencesolve's profile pic

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

a) You need to find how much y changes if x increases from 3 to 3.01, hence, you need to find `(Delta y)/(Delta x)`  such that:

`Delta y= f(x + Delta x) - f(x)`

You need to evaluate `Delta x`  such that:

`Delta x = 3.01 - 3 = 0.01`

`Delta y = f(3.01) - f(3) = 4sqrt(72 + (3.01)^2) - 4sqrt(72 + 9)`

`Delta y = 4(9.003 - 9) = 4*0.003~~ 0.012`

Evaluating the change in y yields:

`(Delta y)/(Delta x)~~ 0.012/0.01~~ 1.2`

Hence, evaluating the approximate change in y under the given conditions, yields `(Delta y)/(Delta x)~~ 1.2` .

b) Performing the same steps as above yields:

`Delta y = f(1.97) - f(2) = 2*(1.97)^2 + 3*1.97 - 8 - 6`

`Delta y = 7.7618 + 5.91 - 14 = -0.3282 `

`Delta x = 2 - 1.97 = 0.03`

`(Delta y)/(Delta x) = -0.3282/0.03 = -10.94`

Hence, evaluating the approximate change in y under the given conditions, yields `(Delta y)/(Delta x)= -10.94` .

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