Glossary
Concavity
Describes the direction in which the graph of a function opens; a function is concave up if its graph opens upwards and concave down if it opens downwards.
Example:
The graph of always exhibits concavity up because its second derivative is always positive.
Critical Points
Points in the domain of a function where the first derivative is either zero or undefined, which are candidates for local extrema.
Example:
For , setting yields and , which are the function's critical points.
First Derivative Test
A calculus method used to find local extrema (maximum or minimum values) of a function by analyzing the sign changes of its first derivative.
Example:
If the velocity of a particle changes from positive to negative, the First Derivative Test indicates a local maximum displacement.
Optimization Problems
Problems that involve finding the best possible solution, typically by maximizing or minimizing a specific quantity, often using calculus techniques.
Example:
Designing a cylindrical can to hold a specific volume with the least amount of material is an example of an optimization problem.
Second Derivative Test
A calculus method used to determine if a critical point corresponds to a local maximum or minimum by evaluating the sign of the second derivative at that point.
Example:
If at a critical point , the Second Derivative Test confirms a local maximum at that point.
Substitution
An algebraic technique used in optimization problems to reduce the number of variables in an equation, allowing differentiation with respect to a single variable.
Example:
When maximizing the area of a rectangular garden with a fixed perimeter, you can use substitution to express the width in terms of the length.