Final answer:
To evaluate the difference quotient for the function h(x), the expression for h(x + Δx) is first expanded, then subtracted by h(x), and divided by Δx. The simplified result is 2x + 3, which is also the derivative of the original function.
Step-by-step explanation:
To evaluate the difference quotient and simplify the result for the function h(x) = x^2 + 3x + 5, we need to follow several steps. First, we calculate h(x + Δx) by substituting x + Δx into our function, giving us (x + Δx)^2 + 3(x + Δx) + 5. Expanding this, we get x^2 + 2xΔx + (Δx)^2 + 3x + 3Δx + 5. Next, we find h(x + Δx) - h(x) which simplifies to 2xΔx + (Δx)^2 + 3Δx. Finally, we divide by Δx to get the difference quotient: 2x + Δx + 3. As Δx approaches 0, the difference quotient becomes 2x + 3, which is the derivative of h(x).