S=6x^2 is the surface area of a cube. Find the exact change in surface area when x is increased from 6 cm to 6.1 cm .

Question

Please Help.
It makes no sense.

Answer 1

I assume x is a side length.

dS/dx=12x
dS=12x dx

find dS, you are given x as 6cm, dx as .1

S(6)=6*36
S(6.1)=6*37.21 EXACT change 6*1.21cm^2=7.38
Now, using linear approximation.
dS=12*6*.1=72*.1=7.2

Now notice the exact change is 7.38, the incremental change is 7.2

This indicates a bias in an increasing function. If you had used the midpoint value initially (6.05) dx=.1
dS=12*6.05*.1=7.26

So why is it off slightly? The area equation is NOT linear, yet we used the linear approximation model (origvalue+ slope*changex)