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