Final answer:
The instantaneous rate of change of the function f(x) = 6x² + 5x + 2 at x = 2, found using the limit definition of the derivative, is 29 units per one unit change in x.
Step-by-step explanation:
To find the instantaneous rate of change of the function f(x) = 6x² + 5x + 2 at x = 2, we use the limit definition of the derivative. The derivative f'(x) can be found by taking the limit as h approaches zero of the difference quotient:
f'(x) = lim (h → 0) [(f(x + h) - f(x)) / h]
Applying the formula:
- Expand the function f(x + h) = 6(x + h)² + 5(x + h) + 2.
- Subtract f(x) from f(x + h), and divide by h.
- Take the limit as h approaches zero.
Here are the calculations:
- f(x + h) = 6(x + h)² + 5(x + h) + 2
- f(x + h) - f(x) = [6(x + h)² + 5(x + h) + 2] - [6x² + 5x + 2]
- Simplify and cancel terms.
- Divide by h and take the limit as h → 0.
After simplification and calculation, the derivative of the function at
x = 2 is 29.
This means the instantaneous rate of change of f(x) at x = 2 is 29 units per one unit change in x.