QuestionJuly 15, 2025

7. Given the function f(x)=-2+8x^2 , calculate the following. f(a)= f(a+h)= (f(a+h)-f(a))/(h)=

7. Given the function f(x)=-2+8x^2 , calculate the following. f(a)= f(a+h)= (f(a+h)-f(a))/(h)=
7. Given the function f(x)=-2+8x^2 , calculate the following. f(a)= f(a+h)= (f(a+h)-f(a))/(h)=

Solution
4.7(341 votes)

Answer

f(a) = -2 + 8a^2 ### f(a+h) = -2 + 8a^2 + 16ah + 8h^2 ### \frac{f(a+h) - f(a)}{h} = 16a + 8h Explanation 1. Calculate f(a) Substitute x = a into f(x). f(a) = -2 + 8a^2. 2. Calculate f(a+h) Substitute x = a+h into f(x). f(a+h) = -2 + 8(a+h)^2. 3. Simplify f(a+h) Expand (a+h)^2: (a+h)^2 = a^2 + 2ah + h^2. So, f(a+h) = -2 + 8(a^2 + 2ah + h^2) = -2 + 8a^2 + 16ah + 8h^2. 4. Calculate \frac{f(a+h) - f(a)}{h} Subtract f(a) from f(a+h): f(a+h) - f(a) = (-2 + 8a^2 + 16ah + 8h^2) - (-2 + 8a^2) = 16ah + 8h^2. Divide by h: \frac{16ah + 8h^2}{h} = 16a + 8h.

Explanation

1. Calculate $f(a)$<br /> Substitute $x = a$ into $f(x)$. $f(a) = -2 + 8a^2$.<br />2. Calculate $f(a+h)$<br /> Substitute $x = a+h$ into $f(x)$. $f(a+h) = -2 + 8(a+h)^2$.<br />3. Simplify $f(a+h)$<br /> Expand $(a+h)^2$: $(a+h)^2 = a^2 + 2ah + h^2$. So, $f(a+h) = -2 + 8(a^2 + 2ah + h^2) = -2 + 8a^2 + 16ah + 8h^2$.<br />4. Calculate $\frac{f(a+h) - f(a)}{h}$<br /> Subtract $f(a)$ from $f(a+h)$: $f(a+h) - f(a) = (-2 + 8a^2 + 16ah + 8h^2) - (-2 + 8a^2) = 16ah + 8h^2$. Divide by $h$: $\frac{16ah + 8h^2}{h} = 16a + 8h$.
Click to rate:

Similar Questions