graph

Examples of graph in the following topics:

• Graphing on Computers and Calculators

  • They can be created with graphing calculators.
  • Graphs are often created using computer software.
  • GraphCalc includes many of the standard features of graphing calculators, but also includes some higher-end features.
  • c) Three-dimensional graphing: While high-end graphing calculators can graph in 3-D, GraphCalc benefits from modern computers' memory, speed, and graphics acceleration.
  • It also includes tools for visualizing and analyzing graphs.

• Real Numbers, Functions, and Graphs

  • Graphs can be used to represent these relationships pictorially.
  • The graph of a function $f$ is the collection of all ordered pairs $(x, f(x))$.
  • Graphing on a Cartesian plane is sometimes referred to as curve sketching.
  • If the function input $x$ is an ordered pair $(x_1, x_2)$ of real numbers, the graph is the collection of all ordered triples $(x_1, x_2, f(x_1, x_2))$ and its graphical representation is a surface (see the three-dimensional graph below).
  • This is a graph of the function $f(x,y) = \sin{x^2} \cos{y^2}$

• Using Calculators and Computers

  • For numerical calculations and graphing, scientific calculators and personal computers are commonly used in classes and laboratories.
  • In certain contexts such as higher education, scientific calculators have been superseded by graphing calculators , which offer a superset of scientific calculator functionality along with the ability to graph input data and write and store programs for the device.
  • These days, scientific and graphing calculators are often replaced by personal computers or even by supercomputers.
  • A typical graphing calculator built by Texas Instruments, displaying a graph of a function $f(x)=2x^2-3$.
  • Describe how calculators and computers can assist with arithmetic, graphing, and other complicated operations.

• Derivatives and the Shape of the Graph

  • The shape of a graph may be found by taking derivatives to tell you the slope and concavity.
  • If $x$ and y are real numbers, and if the graph of $y$ is plotted against $x$, the derivative measures the slope of this graph at each point.
  • The simplest case is when $y$ is a linear function of $x$, meaning that the graph of $y$ divided by $x$ is a straight line.
  • If the function $f$ is not linear (i.e. its graph is not a straight line), however, then the change in $y$ divided by the change in $x$ varies: differentiation is a method to find an exact value for this rate of change at any given value of $x$.
  • Sketch the shape of a graph by using differentiation to find the slope and concavity

• Linear and Quadratic Functions

  • Linear and quadratic functions make lines and a parabola, respectively, when graphed and are some of the simplest functional forms.
  • Linear and quadratic functions make lines and parabola, respectively, when graphed.
  • Although affine functions make lines when graphed, they do not satisfy the properties of linearity.
  • The graph of a quadratic function is a parabola whose axis of symmetry is parallel to the y-axis .

• Vector-Valued Functions

  • The graph shows a visual representation of
  • If you were to take a cross section, with the cut perpendicular to any of the three axes, you would see the graph of that function.
  • For example, if you were to slice the three-dimensional shape perpendicular to the $z$-axis, the graph you would see would be of the function $z(t)=t$.The domain of a vector valued function is a domain that satisfies all of the component functions.
  • This a graph of a parametric curve (a simple vector-valued function with a single parameter of dimension $1$).
  • The graph is of the curve: $\langle 2 \cos(t), 4 \sin(t),t \rangle$ where $t$ goes from $0$ to $8 \pi$.

• Derivatives and Rates of Change

  • The simplest case is when $y$ is a linear function of x, meaning that the graph of $y$ divided by $x$ is a straight line.
  • If the function $f$ is not linear (i.e., its graph is not a straight line), however, then the change in $y$ divided by the change in $x$ varies: differentiation is a method to find an exact value for this rate of change at any given value of $x$.
  • If $x$ and $y$ are real numbers, and if the graph of $y$ is plotted against $x$, the derivative measures the slope of this graph at each point.
  • Describe the derivative as the change in $y$ over the change in $x$ at each point on a graph

• Area Between Curves

  • The area between the graphs of two functions is equal to the integral of a function, $f(x)$, minus the integral of the other function, $g(x)$: $A = \int_a^{b} ( f(x) - g(x) ) \, dx$.
  • The area between the graphs of two functions is equal to the integral of one function, $f(x)$, minus the integral of the other function, $g(x)$: A=∫ba(f(x)−g(x))dxA = \int_a^{b} ( f(x) - g(x) ) \, dx where $f(x)$ is the curve with the greater y-value .
  • The area between two graphs can be evaluated by calculating the difference between the integrals of the two functions.

• Derivatives of Exponential Functions

  • The slope of the graph at any point is the height of the function at that point.
  • Graph of the exponential function illustrating that its derivative is equal to the value of the function.

• Inverse Trigonometric Functions: Differentiation and Integration

  • The usual principal values of the $\text{arctan}(x)$ and $\text{arccot}(x)$ functions graphed on the Cartesian plane.
  • Principal values of the $\text{arcsec}(x)$ and $\text{arccsc}(x)$ functions graphed on the Cartesian plane.
  • The usual principal values of the $\arcsin(x)$ and $\arccos(x)$ functions graphed on the Cartesian plane.