Problem 61 - Algebra and Trigonometry

Question

If \(f(x)=3 x^{2}-x+5,\) find \(\frac{f(x+h)-f(x)}{h}, h \neq 0,\) and simplify.

Step-by-Step Solution

Answer

The simplified form of \(\frac{f(x + h) - f(x)}{h}\) for \(f(x) = 3x^2 - x + 5\) is \(6x + 3h - 1\).

Step 1: Evaluate \(f(x + h)\)

Substitute \(x + h\) into the function \(f(x) = 3x^2 - x + 5\). This gives us \(f(x+h) = 3(x + h)^2 - (x + h) + 5\). Expanding the equation yields: \(f(x + h) = 3x^2 + 6hx + 3h^2 - x - h + 5\).

Step 2: Substitute \(f(x + h)\) and \(f(x)\) into the formula.

Substitute \(f(x+h)\) and \(f(x)\) into the difference quotient formula \(\frac{f(x+h)-f(x)}{h}\). This yields: \(\frac{3x^2 + 6hx + 3h^2 - x - h + 5 - f(x)}{h}\). As \(f(x) = 3x^2 - x + 5\), substituting \(f(x)\) makes the equation: \(\frac{3x^2 + 6hx + 3h^2 - x - h + 5 - 3x^2 + x - 5}{h}\).

Step 3: Simplify the result

Simplify the result as much as possible. Terms \(3x^2\), \(-x\), and \(5\) in the numerator on both sides cancel out, yielding: \(\frac{6hx + 3h^2 - h}{h}\). Factoring out \(h\) gives: \(h(6x + 3h - 1)\). Since \(h \neq 0\), the factor \(h\) can be removed, leaving us with the solution: \(6x + 3h - 1\).