For the function f, construct and simplify the difference quotient .f(x) = 8x2 + 5x

1. Break Down the Problem

To construct the difference quotient for the function $f(x) = 8x^2 + 5x$, we will follow these steps:

1. Find $f(x + h)$
2. Calculate the difference $f(x + h) - f(x)$
3. Divide by $h$ to get the difference quotient
4. Simplify the expression

2. Relevant Concepts

The difference quotient is defined as: $\frac{f(x + h) - f(x)}{h}$ where $h$ is a small increment in $x$.

3. Analysis and Detail

1. Calculate $f(x + h)$: $f(x + h) = 8(x + h)^2 + 5(x + h)$ Expanding this: $f(x + h) = 8(x^2 + 2xh + h^2) + 5(x + h)$ $= 8x^2 + 16xh + 8h^2 + 5x + 5h$

2. Calculate $f(x + h) - f(x)$: Now subtract $f(x)$ from $f(x + h)$: $f(x + h) - f(x) = (8x^2 + 16xh + 8h^2 + 5x + 5h) - (8x^2 + 5x)$ This simplifies to: $= 16xh + 8h^2 + 5h$

3. Divide by $h$: $\frac{f(x+h) - f(x)}{h} = \frac{16xh + 8h^2 + 5h}{h}$ Factor $h$ out: $= \frac{h(16x + 8h + 5)}{h}$ When $h \neq 0$, the $h$ cancels out: $= 16x + 8h + 5$

4. Verify and Summarize

The difference quotient for the function $f(x) = 8x^2 + 5x$ is thus: $16x + 8h + 5$ As $h$ approaches 0, we can see that the difference quotient approaches $16x + 5$, which represents the derivative of the function.

Final Answer

The simplified difference quotient is: $16x + 8h + 5$