Question

attempt 1: 10 attempts remaining. given the function $f(x) = 2 + 5x^2$, calculate the following values: $f(a) = $, $f(a + h) = $, $\frac{f(a + h) - f(a)}{h} = $, submit answer, next item

Explanation:

Step1: Calculate \( f(a) \)

Substitute \( x = a \) into \( f(x) = 2 + 5x^2 \).
\( f(a) = 2 + 5a^2 \)

Step2: Calculate \( f(a + h) \)

Substitute \( x = a + h \) into \( f(x) = 2 + 5x^2 \).
Using the formula \( (m + n)^2 = m^2 + 2mn + n^2 \) where \( m = a \) and \( n = h \), we have:
\( f(a + h) = 2 + 5(a + h)^2 = 2 + 5(a^2 + 2ah + h^2) = 2 + 5a^2 + 10ah + 5h^2 \)

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

Subtract \( f(a) \) from \( f(a + h) \) and divide by \( h \):
\[

$$\begin{align*} \frac{f(a + h) - f(a)}{h}&=\frac{(2 + 5a^2 + 10ah + 5h^2)-(2 + 5a^2)}{h}\\ &=\frac{2 + 5a^2 + 10ah + 5h^2 - 2 - 5a^2}{h}\\ &=\frac{10ah + 5h^2}{h}\\ &=\frac{h(10a + 5h)}{h}\\ &= 10a + 5h \quad (h eq0) \end{align*}$$

\]

Answer:

s:

• \( f(a) = \boldsymbol{2 + 5a^2} \)
• \( f(a + h) = \boldsymbol{2 + 5a^2 + 10ah + 5h^2} \)
• \( \frac{f(a + h) - f(a)}{h} = \boldsymbol{10a + 5h} \) (for \( h

eq0 \))