To find and simplify the difference quotient for the function f(x) = x² + 5, we will first calculate f(x + h) and then use it to find the difference quotient f(x + h) – f(x) / h.
1. **Calculate f(x + h):**
We start with the function:
f(x) = x² + 5
Now, substituting x + h into the function:
f(x + h) = (x + h)² + 5
This can be expanded:
f(x + h) = x² + 2xh + h² + 5
2. **Find f(x + h) – f(x):**
Next, we compute:
f(x + h) – f(x) = (x² + 2xh + h² + 5) – (x² + 5)
By simplifying this expression, we get:
f(x + h) – f(x) = 2xh + h²
3. **Calculate the difference quotient:**
Now we divide by h:
(f(x + h) – f(x)) / h = (2xh + h²) / h
This simplifies to:
2x + h
4. **Final Expression:**
Since h is not equal to zero, we have successfully simplified the difference quotient:
Thus, the difference quotient of f(x) = x² + 5 is 2x + h.