Questions: Find the difference quotient of f; that is, find (f(x+h)-f(x))/h, h ≠ 0, for the following function. Be sure to simplify. f(x) = 2x² + x + 3

△ Find the difference quotient \(\frac{f(x+h)-f(x)}{h}\) for \(f(x) = 2x^2 + x + 3\). ○ Calculate \(f(x+h)\) ☼ \(f(x+h) = 2(x+h)^2 + (x+h) + 3\). Expanding \((x+h)^2\) gives \(x^2 + 2xh + h^2\). Thus, \(f(x+h) = 2(x^2 + 2xh + h^2) + x + h + 3 = 2x^2 + 4xh + 2h^2 + x + h + 3\). ○ Calculate \(f(x+h) - f(x)\) ☼ \(f(x+h) - f(x) = (2x^2 + 4xh + 2h^2 + x + h + 3) - (2x^2 + x + 3)\). Simplifying, the terms \(2x^2\), \(x\), and \(3\) cancel out, leaving \(4xh + 2h^2 + h\). ○ Calculate \(\frac{f(x+h)-f(x)}{h}\) ☼ \(\frac{f(x+h)-f(x)}{h} = \frac{4xh + 2h^2 + h}{h}\). Factoring out \(h\) gives \(\frac{h(4x + 2h + 1)}{h}\). Since \(h \neq 0\), cancel \(h\) to get \(4x + 2h + 1\). ✧ The difference quotient is \(4x + 2h + 1\). ☺ The difference quotient for $f(x)=2 x^{2}+x+3$ is $4x + 2h + 1$.