Question

According to data from a General Motors study, the average speed of your trip A (in miles per hour) is related to the number of stops per mile you make on the trip x by the equation

A =26.5/x^0.45 .†

(a) Compute dA/dx for x = 0.27. (Round your answer to two decimal places.)
(b) Compute dA/dx for x = 2. (Round your answer to two decimal places.)

Answer 1

`(a)\hspace{3}-79.61mi/h^2`

`(b)\hspace{3}-4.36mi/h^2`

Explanation:

Let's rewrite the equation as:

`A=26.5*(x^(-0.45 ))`

Now, let's find its derivate:

`(dA)/(dx) =(-0.45)*(26.5)*x^(-0.45-1) =-11.925*x^(-1.45) =-(11.925)/(x^(1.45) )`

Let's evaluate x=0.27 and x=2:

`(dA)/(dx) \left \{  {x=0.27}} \right. =-(11.925)/(0.27^(1.45) )  =-79.61mi/h^2`

`(dA)/(dx) \left \{  {x=2}} \right. =-(11.925)/(2^(1.45) )  =-4.36mi/h^2`

Keep in mind that when we derivate A(average speed) we find the average acceleration, thats why the result is given in mi/h^2, also it explains the minus sign, because for every stop you make on the trip you are decelerating.