Answer:
S(20) = [tex]\sqrt{20+2}[/tex] = 4.69 ; value of total sales after 20 months is 4.69 million dollars
S'(20) = [tex]\dfrac{1}{\sqrt{20+2} }[/tex] = 0.21 ; increase in the total sales after 20 months is 0.21 millions of dollars
S(24) = [tex]\sqrt{24+2}[/tex] = 5.10; 5.10 millions of dollars
S(25) = [tex]\sqrt{25+2}[/tex] = 5.20 ; 5.20 millions of dollars
Step-by-step explanation:
[tex]S(t) = \sqrt{t+2}[/tex]
[tex]S(t+h) = \sqrt{t+2+h}[/tex]
[tex]S(t+h) - S(t)= \sqrt{t+2+h }- \sqrt{t+2}[/tex]
[tex]\dfrac{S(t+h) - S(t)}{h}= \dfrac{\sqrt{t+2+h }- \sqrt{t+2}}{h}[/tex]
By rationalization:
[tex]\dfrac{\sqrt{t+2+h }- \sqrt{t+2}}{h} * \dfrac{ {\sqrt{t+2+h }- \sqrt{t+2}} }{ {\sqrt{t+2+h }- \sqrt{t+2}}}[/tex]
[tex]\\ \\ \dfrac{t+2+h-t-2}{h \sqrt{t+2+h }+ \sqrt{t+2} } = \dfrac {h}{\sqrt{t+2+h }+\sqrt{t+2} }[/tex]
= [tex]\dfrac {1}{\sqrt{t+2+h }+\sqrt{t+2} }[/tex]
[tex]\lim_{k \to \infty} \dfrac {1}{\sqrt{t+2+h }+\sqrt{t+2} } = \dfrac{1}{t+2}[/tex]
[tex]S' (t) = \dfrac{1}{\sqrt{t+2} }[/tex]
S(20) = [tex]\sqrt{20+2}[/tex] = 4.69 ; value of total sales after 20 months is 4.69 million dollars
S'(20) = [tex]\dfrac{1}{\sqrt{20+2} }[/tex] = 0.21 ; increase in the total sales after 20 months is 0.21 millions of dollars
S(24) = [tex]\sqrt{24+2}[/tex] = 5.10; 5.10 millions of dollars
S(25) = [tex]\sqrt{25+2}[/tex] = 5.20 ; 5.20 millions of dollars
