Let f be the function defined by f(x) = x + ln x. what is the value of c for which the instantaneous rate of change of f at x = c is the same as the average rate of change of f over [1,4]?

The value will be "C = 2.164" for which the instantaneous rate of change is similar to the average rate.

Given:

Instantaneous rate of x=c is,

at x=c

              at x=c

              ...(2)

Now,

The average rate of change over [1, 4] will be:

→              

                  ...(3)

Since both are equal,

From equation "2" and "3", we get

→    

→

→      

→        

Thus the above answer is right.

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is differentiable for all , so by the mean value theorem there is some for which

We have derivative

so that satisfies

which can be rewritten using the change of base formula, if you wish:

geometry: triangles

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call the larger triangle abc

then the smaller triangle cde is similar to abc (all angles are equal, aaa).  

ab is the side that is 36, and we wish to find de, the side corresponding to ab

area_cde = area_abc - area_cde --> area_abc / area_cde = 2

let x=length of de

ratio of areas = (ratio of sides)^2

+2+=+%2836%2fx%29%5e2+

+2+=+1296%2fx%5e2+

++x%5e2+=+1296%2f2+=+648+

25.45584

step-by-step explanation:

i found the koficent of the linear function and i foun c acordind to the strict formula y=kx+c and then i took 2 as an x and replacet it on the formula i found and then foun the y