Calculus AB covers the fundamental concepts of differential and integral calculus, preparing students for advanced mathematical applications.
Derivatives measure how a function changes—like the speed of a car at a specific instant. Differentiation is the process of finding that rate.
The derivative of \( f(x) \) at a point tells us the function’s instantaneous rate of change there. It’s like asking, "How fast am I going right now?"
For \( f(x) = x^n \), the derivative is \( f'(x) = n x^{n-1} \).
Derivatives help us:
From tracking speed on your fitness app to figuring out how fast a cup of coffee cools, derivatives are everywhere!
The derivative of \( f(x) = x^2 \) is \( 2x \), so at \( x = 3 \), the slope is 6.
The slope of the tangent line to \( y = \sin(x) \) at \( x = 0 \) is 1.
Derivatives tell us how fast things change at any moment.