Final answer:
The instantaneous rate of change of the function f(x) at x=3 is calculated using the limit definition of the derivative, which yields a derivative function f'(x) = 10x + 3. Substituting x=3 gives an instantaneous rate of change of 33.
Step-by-step explanation:
The question asks us to use the limit definition of the derivative to find the instantaneous rate of change of the function f(x) = 5x²+3x+5 at x=3. The limit definition of the derivative of a function f at a point x is:
f'(x) = lim₀ (Δx)→ [f(x + Δx) - f(x)] / Δx
Applying this definition to our function, we set up the following limit:
f'(x) = lim₀ (Δx)→ [(5(x + Δx)² + 3(x + Δx) + 5) - (5x² + 3x + 5)] / Δx
After expanding and simplifying the expression within the limit, we can cancel out any terms that are not multiplied by Δx and divide by Δx. This will leave us with a simplified expression that we can evaluate the limit of as Δx approaches 0:
f'(x) = lim₀ (Δx)→ [10xΔx + 5Δx² + 3Δx] / Δx = 10x + 3
To find the rate of change at x=3, we substitute 3 into our derived derivative function:
f'(3) = 10(3) + 3 = 33
The instantaneous rate of change of the function at x=3 is 33.