Glossary
Analytical Method (Optimization)
A technique for solving optimization problems by using calculus, specifically derivatives, to find critical points and classify them as extrema.
Example:
Using the first and second derivative tests to find the dimensions of a box that maximize its volume is an application of the analytical method.
Asymptote
A line that a curve approaches as it heads towards infinity, but never actually touches or crosses.
Example:
The function f(x) = 1/x has a vertical asymptote at x=0 and a horizontal asymptote at y=0.
Average Rate of Change
The slope of the secant line connecting two points on a function, calculated as the change in the function's output divided by the change in its input over an interval.
Example:
The average rate of change of a car's position from t=0 to t=5 seconds is its total displacement divided by 5 seconds.
Candidates Test
A method for finding the absolute extrema of a continuous function on a closed interval by evaluating the function at all critical points within the interval and at the interval's endpoints.
Example:
To find the absolute maximum of f(x) = x^3 - 3x on [0, 2], you would use the Candidates Test by checking f(0), f(2), and f(1) (since x=1 is a critical point).
Concave Down
A portion of a function's graph that opens downwards, resembling a frown, where the second derivative is negative.
Example:
The function f(x) = -x^2 is concave down everywhere because its second derivative f''(x) = -2 is always negative.
Concave Up
A portion of a function's graph that opens upwards, resembling a smile, where the second derivative is positive.
Example:
The parabola f(x) = x^2 is concave up everywhere because its second derivative f''(x) = 2 is always positive.
Concavity
Describes the curvature of a function's graph, indicating whether it is 'bending up' (concave up) or 'bending down' (concave down).
Example:
The graph of f(x) = x^3 changes its concavity at x=0, moving from concave down to concave up.
Continuous Function
A function whose graph can be drawn without lifting the pen, meaning there are no breaks, jumps, or holes in its domain.
Example:
A polynomial function like f(x) = x^3 - 2x + 1 is always a continuous function across all real numbers.
Critical Point
A point 'c' in the domain of a function f(x) where the first derivative f'(c) is either zero or undefined.
Example:
For f(x) = x^(2/3), x=0 is a critical point because f'(0) is undefined, even though f(0) exists.
Decreasing Function
A function is decreasing on an interval if its y-values generally fall as its x-values increase, which corresponds to a negative first derivative.
Example:
The function f(x) = e^(-x) is a decreasing function for all real x because its derivative f'(x) = -e^(-x) is always negative.
Differentiable Function
A function for which the derivative exists at every point in its domain, implying the graph has no sharp corners, cusps, or vertical tangent lines.
Example:
The function f(x) = |x| is continuous at x=0 but not differentiable there because of the sharp corner.
Domain
The set of all possible input values (x-values) for which a function is defined.
Example:
The domain of f(x) = 1/x is all real numbers except x=0, as division by zero is undefined.
Extreme Value Theorem (EVT)
If a function is continuous on a closed interval [a, b], then it must attain both an absolute maximum and an absolute minimum value on that interval.
Example:
For the function f(x) = x^2 on [-1, 2], the Extreme Value Theorem guarantees both a highest and lowest point exist within that interval.
First Derivative Test
A method used to determine the intervals where a function is increasing or decreasing, and to identify local extrema, by analyzing the sign of the first derivative.
Example:
Using the first derivative test, if f'(x) changes from positive to negative at x=c, then f(c) is a local maximum.
Global/Absolute Extrema
The highest (maximum) and lowest (minimum) points of a function over its entire domain or a specified closed interval.
Example:
On the interval [0, 4], the function f(x) = (x-2)^2 has a global minimum of 0 at x=2 and a global maximum of 4 at x=0 or x=4.
Graphical Analysis
The study of a function's behavior, properties, and characteristics by interpreting its graph and the graphs of its derivatives.
Example:
Understanding how the slope of f'(x) relates to the concavity of f(x) is a key part of graphical analysis.
Increasing Function
A function is increasing on an interval if its y-values generally rise as its x-values increase, which corresponds to a positive first derivative.
Example:
The function f(x) = x^2 is an increasing function on the interval (0, ∞) because its derivative f'(x) = 2x is positive there.
Inflection Point
A point on a curve where the concavity changes (from concave up to concave down or vice versa), and the second derivative is typically zero or undefined.
Example:
For f(x) = x^3, the point (0,0) is an inflection point because the concavity changes there.
Local Extrema
The highest (maximum) or lowest (minimum) points of a function within a specific subinterval of its domain.
Example:
A cubic function like f(x) = x^3 - 3x can have a local maximum and a local minimum where its derivative is zero.
Maximization Problem
An optimization problem focused on finding the largest possible value of a function.
Example:
Determining the dimensions of a rectangular field that yield the largest area for a fixed perimeter is a maximization problem.
Mean Value Theorem (MVT)
If a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one point 'c' in (a, b) where the instantaneous rate of change (derivative) equals the average rate of change over the interval.
Example:
If you drove 100 miles in 2 hours, the Mean Value Theorem guarantees that at some point during your trip, your instantaneous speed was exactly 50 mph.
Minimization Problem
An optimization problem focused on finding the smallest possible value of a function.
Example:
Finding the shortest distance from a point to a curve is a minimization problem.
Optimization Problems
Mathematical problems that involve finding the maximum or minimum value of a quantity, often subject to certain constraints.
Example:
Designing a cylindrical can to hold a specific volume of liquid while minimizing the amount of material used is a classic optimization problem.
Range
The set of all possible output values (y-values) that a function can produce.
Example:
The range of f(x) = x^2 is [0, ∞), as the output can never be negative.
Second Derivative Test
A method used to determine the concavity of a function and to classify critical points as local maxima or minima by analyzing the sign of the second derivative.
Example:
If f'(c) = 0 and f''(c) > 0, the second derivative test tells us that f(c) is a local minimum.