Questions: Compute the difference quotient for the function given below. f(x) = 2x^3 + x

Solution Steps

To compute the difference quotient for the function $f(x) = 2x^3 + x$, we need to follow these steps:

1. Substitute $x + h$ into the function to get $f(x + h)$.
2. Calculate $f(x + h) - f(x)$.
3. Divide the result by $h$ to get the difference quotient.

We start with the function given by $$f(x) = 2x^3 + x.$$

Next, we substitute $x + h$ into the function: $$f(x + h) = 2(x + h)^3 + (x + h).$$ Expanding $(x + h)^3$ gives: $$(x + h)^3 = x^3 + 3x^2h + 3xh^2 + h^3.$$ Thus, $$f(x + h) = 2(x^3 + 3x^2h + 3xh^2 + h^3) + x + h = 2x^3 + 6x^2h + 6xh^2 + 2h^3 + x + h.$$

Now, we compute the difference quotient: $$\frac{f(x + h) - f(x)}{h} = \frac{(2x^3 + 6x^2h + 6xh^2 + 2h^3 + x + h) - (2x^3 + x)}{h}.$$ This simplifies to: $$\frac{6x^2h + 6xh^2 + 2h^3 + h}{h}.$$ Factoring out $h$ from the numerator gives: $$\frac{h(6x^2 + 6xh + 2h + 1)}{h} = 6x^2 + 6xh + 2h + 1.$$

Final Answer