Question

Calculate the instantaneous rate of change of the function f(x)=x2−x+3 at x=2 using the limit definition of the derivative:

f ′(x)=limₕ→₀ ((f(a+h)−f(a))÷ h)

Answer 1

The instantaneous rate of change of the function f(x) at x=2 is calculated using the limit definition of the derivative. After simplifying the expression that represents the derivative, the result is found to be 3.

Step-by-step explanation:

To calculate the instantaneous rate of change of the function f(x) = x^2 - x + 3 at x = 2 using the limit definition of the derivative, we use the formula:

f'(x) = limh → 0 ((f(a+h) - f(a)) / h)

Substituting the given function and a = 2:

f'(2) = limh → 0 (( (2+h)^2 - (2+h) + 3 - (2^2 - 2 + 3) ) / h)

This simplifies to:

f'(2) = limh → 0 ((4 + 4h + h^2 - 2 - h + 3 - 3 + 2) / h)

f'(2) = limh → 0 ((h^2 + 3h) / h)

Canceling out the h, we have:

f'(2) = limh → 0 (h + 3)

As h approaches 0, we are left with:

f'(2) = 3

Therefore, the instantaneous rate of change of f(x) at x = 2 is 3.