Consider the function f(x)=6√x+4 on the interval [4,6]. Find the average or mean slope of the function on this interval.

Question

Consider the function f(x)=6√x+4 on the interval [4,6]. Find the average or mean slope of the function on this interval.

 

By the Mean Value Theorem, we know there exists a c in the open interval (4,6) such that f'(c) is equal to this mean slope. For this problem, there is only one c that works. Find it.

 

Answer 1

Hi,

f(x)=6√x+4 [4,6].  a = 4, b = 6

1) f(4) = 6 √4 + 4 = 16 AND f(6) = 6 √6 + 4

2)we know f'(c) = [ f(a) - f(b) ] / (a - b ) -------> f'( c ) = ( 16 - 6 √6 - 4 ) / ( 4 - 6 )

f'( c ) = ( 12 - 6√6 ) / (-2) = -6 + 3√6

f'(x) = 6 ( 1/2√x)

f'(c) = 6( 1 / 2√c ) = -6 + 3√6 now solve this equation for c

1/2√c = (-6 + 3√6 ) / 6 reciprocal both sides

2√c = ( 6/ (-6 + 3√6 )) both side to the power 2

4 c = 36/ ( 36 -36√6 + 54) both sides dived by 4

c = 9/ ( 90 - 36√6) = 4.95 acceptable because between given interval

I hope it is useful,

Minoo

c = ( 100 -60√6 + 54 ) / 36 = ( 154 -60√6) / 36 = 1.39