Question

Use the limit definition of the derivative to find the instantaneous rate of change of f(x)=6x² +5x+2 at x=2

Answer 1

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:

1. Expand the function f(x + h) = 6(x + h)² + 5(x + h) + 2.
2. Subtract f(x) from f(x + h), and divide by h.
3. 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.