7. if $f(x)=3x^{2}+9$, find and simplify $\frac{f(a + h)-f(a)}{h}$, where $h\neq0$.\n$3a + 6h$\n$3h +…
7. if $f(x)=3x^{2}+9$, find and simplify $\frac{f(a + h)-f(a)}{h}$, where $h\neq0$.\n$3a + 6h$\n$3h + 12a$\n$6a+3h$\n$3h + 3a$\n$9h + 6a$
Answer
Explanation:
Step1: Find $f(a + h)$
Substitute $x=a + h$ into $f(x)=3x^{2}+9$. $f(a + h)=3(a + h)^{2}+9=3(a^{2}+2ah+h^{2})+9=3a^{2}+6ah + 3h^{2}+9$
Step2: Find $f(a)$
Substitute $x = a$ into $f(x)=3x^{2}+9$. $f(a)=3a^{2}+9$
Step3: Calculate $f(a + h)-f(a)$
$f(a + h)-f(a)=(3a^{2}+6ah + 3h^{2}+9)-(3a^{2}+9)=6ah+3h^{2}$
Step4: Simplify $\frac{f(a + h)-f(a)}{h}$
$\frac{f(a + h)-f(a)}{h}=\frac{6ah + 3h^{2}}{h}=\frac{h(6a + 3h)}{h}=6a+3h$
Answer:
$6a + 3h$