Question

Find
1.f(a+h)
2,Difference Quotient
(f(a+h)-f(a))/h
3.The instantaneous rate of change of f when a=5.

f(x)=x^2+x

Answer 1

just plug x=a+h inf(x)=(x)^2+x
\[diff. Quotient \frac{ d }{dx } f(x)=\lim_{h \rightarrow 0}\frac{ f \left( a+h \right)-f \left( a\right) }{h }\]
\[\to find instantaneous rate plug a=5 \in \frac{ d }{dx}f(x)\]

Answer 2

so what is the answer for question 3

Answer 3

@surjithayer

Answer 4

\[f \left( a+h \right)=\left( a+h \right)^{2}+\left( a+h \right)=a ^{2}+2ah+h ^{2}+a+h\]
\[f \left( a \right)=a ^{2}+a\]

Answer 5

\[f \prime \left( a \right)=\frac{ d }{dx }f \left( x \right) at x=a is =\lim_{h \rightarrow 0}\frac{ f \left( a+h \right)-f \left( a \right) }{h }\]
\[=\lim_{h \rightarrow 0} \frac{ f \left( a+h \right)-f \left( a \right) }{ h }\]
\[=\lim_{h \rightarrow 0}\frac{ a ^{2}+2ah+h ^{2}+a+h-a ^{2}-a }{ h }\]
\[=\lim_{h \rightarrow 0}\frac{ a ^{2}+2ah+h ^{2}+a+h-a ^{2}-a }{ h }\]
\[=\lim_{h \rightarrow 0}\frac{ 2ah+h ^{2}+h }{h}=\lim_{h \rightarrow 0}\frac{ h \left( 2a+h+1 \right) }{ h }\]
\[\lim_{h \rightarrow 0}\left( 2a+h+1 \right)=2a+0+1=2a+1\]
\[f \prime \left( 5 \right)=2*5+1=11\]

Answer 6

perfect