Exam-style application: In a derivative-definition setting, use average rate of change, limit definition of derivative, tangent slope, differentiability, and basic derivative rules. Identify the correct limit expression that represents the derivative of $f(x) = x^2$ at any point $x$.

Exam-style application: In a derivative-definition setting, use average rate of change, limit definition of derivative, tangent slope, differentiability, and basic derivative rules. A function $f(x)$ is continuous at x = a but has a sharp corner (cusp) at that point. Which of the following is true about $f'(a)$? Provide a concise written explanation.

Exam-style application: In a derivative-definition setting, use average rate of change, limit definition of derivative, tangent slope, differentiability, and basic derivative rules. If $f(x) = x^2 - 4x$, find $f'(x)$ using basic derivative rules.

Exam-style application: In a derivative-definition setting, use average rate of change, limit definition of derivative, tangent slope, differentiability, and basic derivative rules. For $f(x) = x^2 - 4x$, calculate the instantaneous rate of change at x = 3. Provide a concise written explanation.

Exam-style application: In a derivative-definition setting, use average rate of change, limit definition of derivative, tangent slope, differentiability, and basic derivative rules. Find the equation of the tangent line to $f(x) = x^2 - 4x$ at the point where x = 3.

Exam-style application: In a derivative-definition setting, use average rate of change, limit definition of derivative, tangent slope, differentiability, and basic derivative rules. Calculate the average rate of change of $f(x) = x^2$ on the closed interval [1, 3]. Provide a concise written explanation.