Chapter 4
Differential Equations, Parametric Equations, and Sequences and Series
By Boundless
Differential equations are solved by finding the function for which the equation holds true.
![Thumbnail](../../../../../figures.boundless-cdn.com/17695/square/px-elmer-pump-heatequation.jpg)
Differential equations can be used to model a variety of physical systems.
![Thumbnail](../../../../../figures.boundless-cdn.com/17711/square/250px-slope-field.jpg)
Direction fields and Euler's method are ways of visualizing and approximating the solutions to differential equations.
![Thumbnail](../../../../../figures.boundless-cdn.com/17760/square/ave-packet-28dispersion-29.gif)
Separable differential equations are equations wherein the variables can be separated.
![Thumbnail](../../../../../figures.boundless-cdn.com/17705/raw/logistic-curve.jpg)
A logistic equation is a differential equation which can be used to model population growth.
![Thumbnail](../../../../../figures.boundless-cdn.com/17706/raw/linear-function-graph.jpg)
Linear equations are equations of a single variable.
![Thumbnail](../../../../../figures.boundless-cdn.com/17707/square/volterra-lotka-dynamics.jpg)
The relationship between predators and their prey can be modeled by a set of differential equations.
![Thumbnail](../../../../../figures.boundless-cdn.com/17688/raw/butterfly-trans01.jpg)
Parametric equations are a set of equations in which the coordinates (e.g.,
![Thumbnail](../../../../../figures.boundless-cdn.com/17708/square/inclinedthrow.gif)
Calculus can be applied to parametric equations as well.
![Thumbnail](../../../../../figures.boundless-cdn.com/17652/raw/polar-to-cartesian.jpg)
Polar coordinates define the location of an object in a plane by using a distance and an angle from a reference point and axis.
![Thumbnail](../../../../../figures.boundless-cdn.com/17653/raw/dinates-integration-region.jpg)
Area and arc length are calculated in polar coordinates by means of integration.
![Thumbnail](../../../../../figures.boundless-cdn.com/17687/raw/conic-sections-with-plane.jpg)
Conic sections are defined by intersections of cones with planes.
![Thumbnail](../../../../../figures.boundless-cdn.com/17684/square/arclength-2.jpg)
Arc length and speed in parametric equations can be calculated using integration and the Pythagorean theorem.
Conic sections are sections of cones and can be represented by polar coordinates.
![Thumbnail](../../../../../figures.boundless-cdn.com/17803/raw/onverging-sequence-example.jpg)
A sequence is an ordered list of objects and can be considered as a function whose domain is the natural numbers.
![Thumbnail](../../../../../figures.boundless-cdn.com/17804/square/3119ebc8f7c539b9686bf51869.jpg)
A series is the sum of the terms of a sequence.
![Thumbnail](../../../../../figures.boundless-cdn.com/17839/raw/integral-test.jpg)
The integral test is a method of testing infinite series of nonnegative terms for convergence by comparing them to an improper integral.
![Thumbnail](../../../../../figures.boundless-cdn.com/17840/raw/onverging-sequence-example.jpg)
Comparison test may mean either limit comparison test or direct comparison test, both of which can be used to test convergence of a series.
![Thumbnail](../../../../../figures.boundless-cdn.com/17846/square/lternating-harmonic-series.jpg)
An alternating series is an infinite series of the form
![Thumbnail](../../../../../figures.boundless-cdn.com/18223/raw/ratio-test-proof.jpg)
An infinite series of numbers is said to converge absolutely if the sum of the absolute value of the summand is finite.
![Thumbnail](../../../../../figures.boundless-cdn.com/17850/raw/integral-test.jpg)
Convergence tests are methods of testing for the convergence or divergence of an infinite series.
![Thumbnail](../../../../../figures.boundless-cdn.com/17790/square/exp-series.gif)
A power series (in one variable) is an infinite series of the form
![Thumbnail](../../../../../figures.boundless-cdn.com/17798/raw/sintay.jpg)
A power function is a function of the form
![Thumbnail](../../../../../figures.boundless-cdn.com/17781/square/exp-series.gif)
Taylor series represents a function as an infinite sum of terms calculated from the values of the function's derivatives at a single point.
![Thumbnail](../../../../../figures.boundless-cdn.com/17783/raw/sintay.jpg)
Taylor series expansion can help approximating values of functions and evaluating definite integrals.
![Thumbnail](../../../../../figures.boundless-cdn.com/17805/raw/chy-sequence-illustration2.jpg)
Infinite sequences and series can either converge or diverge.
![Thumbnail](../../../../../figures.boundless-cdn.com/17838/square/3119ebc8f7c539b9686bf51869.jpg)
For a sequence