Glossary
Constant of Integration
An arbitrary constant, denoted as '+C,' introduced when finding the indefinite integral of a function, accounting for the loss of information about the original constant during differentiation.
Example:
When integrating f'(x) = 2x, the result is f(x) = x^2 + C, where C can be any real number, representing the vertical shift of the antiderivative.
Critical Points
Points on a function where its derivative (slope) is either zero or undefined, often corresponding to local maxima, minima, or points of inflection.
Example:
In a slope field, if you see horizontal line segments along y=5, it indicates that critical points of the original function occur where y=5, as the slope is zero there.
Differential Equations
Equations that relate a function with its derivatives, describing how a quantity changes with respect to one or more independent variables.
Example:
The equation dy/dx = ky models exponential growth or decay, where the rate of change of a quantity is proportional to the quantity itself, a classic differential equation in calculus.
Family of Functions
A set of functions that are all solutions to the same differential equation, differing only by the value of the constant of integration.
Example:
The solutions to dy/dx = cos(x) form a family of functions given by y = sin(x) + C, where each value of C produces a different curve that satisfies the differential equation.
Initial Conditions
Specific values of a function at a particular point, used to determine the unique constant of integration and thus identify a particular solution from a family of functions.
Example:
To find the specific path of a projectile, you need its starting height and initial velocity, which serve as initial conditions to solve the differential equation describing its motion.
Particular Solution
A unique function from a family of functions that satisfies a given differential equation and also meets specific initial conditions.
Example:
If the general solution is y = x^2 + C, and we're given the initial condition y(1)=3, then substituting these values (3 = 1^2 + C) allows us to find C=2, yielding the particular solution y = x^2 + 2.
Slope Fields
Graphical representations that show the slope of a function at various points in the coordinate plane, providing a visual understanding of differential equation solutions.
Example:
A slope field for dy/dx = x/y would show horizontal segments along the x-axis (where y=0) and vertical segments along the y-axis (where x=0), helping visualize the flow of solutions.