Question

Determine c ∈ (a, b) dado pelo Teorema do Valor Médio:
h(x)=2^x ; a=0,b=2

Answer 1

h(x) = 2^x

a = 0
b = 2

h'(x) = (h(b) - h(a))/(b - a)
h'(c) = (h(2) - h(0))/(2 - 0)
h'(c) = (h(2) - h(0))/2

h(2) = 2^2 = 4
h(0) = 2^0 = 1

h'(c) = (4 - 1)/2
h'(c) = 3/2

h'(x) = (2^x)'
h'(x) = ln 2 . 2^x

ln 2 . 2^c = 3/2
2^c = 3/2/ln2
2^c = 3/2.ln2

ln 2^c = ln(3/2.ln2)
c . ln2 = ln 3 - ln(2.ln2)
c = (ln 3 - ln(2.ln2))/ln2