Calculus
Study of continuous change through derivatives and integrals.
Applications of Derivatives
Using Derivatives in Real Life
Derivatives are used everywhere! They help us find where things reach their highest or lowest points, how fast things change, and where curves are steepest or flattest.
Optimization Problems
Want to build the biggest box from a fixed piece of material? Or minimize fuel costs? Use derivatives to find where the slope is zero—these are your maximum or minimum points.
Motion and Physics
In physics, velocity is the derivative of position, and acceleration is the derivative of velocity. Derivatives help us model how things move, fall, or speed up.
Curve Sketching
By analyzing where the derivative is positive, negative, or zero, you can sketch the shape of a function and predict its behavior.
Key Formula
\[f'(a) = 0 \Rightarrow \text{possible max/min at } x = a\]
Examples
Finding the minimum cost of materials for a can with a fixed volume.
Determining when a car reaches its top speed by finding where acceleration is zero.
In a Nutshell
Derivatives help us optimize, analyze motion, and understand the shape of graphs.