Answer:
[tex]f(a)=1+5a^2[/tex]
[tex]f(a+h)=1+5a^2+10ah+5h^2[/tex]
[tex]\frac{f(a+h)-f(a)}{h}=10a+5h[/tex]
Step-by-step explanation:
We are given [tex]f(x)=1+5x^2[/tex].
Find [tex]f(a)[/tex]. All this means is replace [tex]x[/tex] in [tex]f(x)=1+5x^2[/tex] with [tex]a[/tex].
[tex]f(x)=1+5x^2[/tex]
[tex]f(a)=1+5a^2[/tex]
Find [tex]f(a+h)[/tex]. All this means is replace [tex](a+h)[/tex] in [tex]f(x)=1+5x^2[/tex] with [tex](a+h)[/tex].
[tex]f(x)=1+5x^2[/tex]
[tex]f(a+h)=1+5(a+h)^2[/tex]
[tex]f(a+h)=1+5(a+h)(a+h)[/tex]
[tex]f(a+h)=1+5(a^2+2ah+h^2)[/tex]
[tex]f(a+h)=1+5a^2+10ah+5h^2[/tex]
Find [tex]\frac{f(a+h)-f(a)}{h}[/tex]. So we got to put some parts together; the parts above:
[tex]\frac{f(a+h)-f(a)}{h}[/tex]
[tex]\frac{(1+5a^2+10ah+5h^2)-(1+5a^2)}{h}[/tex]
Now in the first ( ) I see 1+5a^2 and in the second ( ) I see 1+5a^2, so this means you have (1+5a^2)-(1+5a^2) which equals 0.
[tex]\frac{10ah+5h^2}{h}[/tex]
Now assuming h is not 0. we can divide top and bottom by h.
[tex]\frac{10a+5h}{1}[/tex]
[tex]10a+5h[/tex]
