given the function f(x)= -5 + 2x², calculate the following f(a)= f(a + h)= (f(a + h)-f(a))/h = question…

Answer:

1. $f(a)=-5 + 2a^{2}$
2. $f(a + h)=-5+2(a + h)^{2}=-5 + 2(a^{2}+2ah+h^{2})=-5 + 2a^{2}+4ah + 2h^{2}$
3. $\frac{f(a + h)-f(a)}{h}=\frac{(-5 + 2a^{2}+4ah + 2h^{2})-(-5 + 2a^{2})}{h}=\frac{-5 + 2a^{2}+4ah + 2h^{2}+5 - 2a^{2}}{h}=\frac{4ah+2h^{2}}{h}=4a + 2h$

Explanation:

Step1: Substitute $x = a$ into $f(x)$

$f(a)=-5+2a^{2}$

Step2: Substitute $x=a + h$ into $f(x)$

$f(a + h)=-5+2(a + h)^{2}$ Expand $(a + h)^{2}=a^{2}+2ah+h^{2}$ So $f(a + h)=-5 + 2a^{2}+4ah + 2h^{2}$

Step3: Calculate $\frac{f(a + h)-f(a)}{h}$

First find $f(a + h)-f(a)$: $(-5 + 2a^{2}+4ah + 2h^{2})-(-5 + 2a^{2})=4ah+2h^{2}$ Then $\frac{f(a + h)-f(a)}{h}=\frac{4ah + 2h^{2}}{h}=4a+2h$