For either the graphic solution or the algebraic solution it is important to realize that cos^{2}x = (cos x)(cos x) = (cos x)^{2}. To enter cos^{2}x into your calculator, calculate cos x and square it. For an algebraic solution write y = cos x and then the equation becomes 2 y^{2} – 3 y – 4 = 0.

Cos 2pi is the value of the cosine function cos x when x = 2π. We know that the period of the cosine function is 2π, that is, the cycle of cos x repeats after every 2π radians. Hence, if we subtract 2π from 2π, the value of cos x will be the same as cos 2pi and we get back to 0 radians.

A function f is periodic of period T if f(x+T) = f(x) for all x in the domain of f. The smallest positive value of T is called the fundamental period. For example, both sin x and cos x have fundamental period 2π, whereas tan x has fundamental period π. A constant function is periodic with arbitrary period T.

Answer in 60sInputResultKey moments in videoFrom 00:00 -- Intuitive IdeaFrom 01:54 -- DefinitionFrom 03:31 -- Graphing by HandFrom 06:10 -- Graphing by ComputerFrom 09:12 -- Vector Fields in 3DKey moments in videoFrom 01:25 -- Changing the number of vectors along the x and y axesFrom 01:46 -- Show the flow line from a particular pointFrom 02:21 -- Change the color of the vectorsRelated queriesWhich function will you use to plot a vector of a field?Use VectorDensityPlot and StreamDensityPlot to visualize the field densities: Copy to clipboard. Use VectorPlot3D to plot a three-dimensional vector field (vectors are colored depending on their…

Answer in 60sInput interpretationSeries expansion at x=0Related queriesWhat is the linear approximation of a function?The linear approximation of a function f(x) is the linear function L(x) that looks the most like f(x) at a particular point on the graph y = f(x).How do you calculate approximate value?If a number has digits to the right of the decimal less than 5. Then just drop the digits to the right of the decimals. The number so obtained will be the approximated value.What is the linearization equation?The Linearization of a function f(x,y) at (a,b) is L(x,y) = f(a,b)+(x−a)fx(a,b)+(y−b)fy(a,b). This is very similar to…

Answer in 60sInput interpretationVenn diagramBoolean formBoolean truth tableMinimal formsOther formsBoolean Logic circuitBoolean truth densityRelated queriesWhat is a ∩ B C?A intersection B intersection C represents the common elements of the sets A, B, and C respectively. This is generally represented as A n B n C. The symbol 'n' represents intersection and gives the common element of the two sets.What is Buc in math?A intersection B union C is represented as A n B U C. A intersection B union C. The set A n B U C can be obtained by taking the intersection of set A and the…

Answer in 60sInput interpretationInput valueProbabilities that people have the same birthdayKey moments in videoFrom 00:04 -- The Birthday ProblemFrom 01:11 -- Taking a look at the sample spaceFrom 03:00 -- StudentsFrom 07:52 -- Finding the Sample SpaceFrom 09:07 -- Summary of findingsFrom 13:13 -- People?From 13:55 -- Probability of having a group of 23 peopleKey moments in videoFrom 01:05 -- WHAT ARE THE ODDS TWO OF THEM SHARE A BIRTHDAY?From 01:11 -- NO TWINS NO LEAP YEAR NO PATTERNSFrom 02:14 -- VERIDICAL PARADOXFrom 06:02 -- THE BIRTHDAY ATTACKFrom 06:46 -- HASH COLLISIONFrom 07:15 -- BRUTE FORCERelated queriesWhat is the probability…

Answer in 60sInput interpretationResultAlternate formsAlternate form assuming ω>0Trigonometric transformRelated queriesWhat is the Fourier transform of sine wave?The Fourier Transform of the Sine and Cosine Functions Equation [2] states that the fourier transform of the cosine function of frequency A is an impulse at f=A and f=-A. That is, all the energy of a sinusoidal function of frequency A is entirely localized at the frequencies given by |f|=A.What is sine Fourier transform?In mathematics, the Fourier sine and cosine transforms are forms of the Fourier transform that do not use complex numbers or require negative frequency. They are the forms originally used…

Answer in 60sInput interpretationRelated queriesWhat is Dirac delta function in quantum mechanics?The Dirac delta function is the name given to a mathematical structure that is intended to represent an idealized point object, such as a point mass or point charge. It has broad applications within quantum mechanics and the rest of quantum physics, as it is usually used within the quantum wavefunction.What is a delta function in calculus?The delta function is a generalized function that can be defined as the limit of a class of delta sequences. The delta function is sometimes called "Dirac's delta function" or the "impulse symbol"…

Answer in 60sInput interpretationInfinite sumSum convergencePartial sum formulaPartial sumsSeries representationsKey moments in videoFrom 06:39 -- Grand FinaleFrom 08:53 -- ConclusionKey moments in videoFrom 00:42 -- Analyzing the SeriesFrom 01:12 -- Comparison TestFrom 04:48 -- Why is this significant?From 06:38 -- Geometric Series ExampleFrom 07:27 -- What does this convergent series converge to?Related queriesHow do you find the sum of n terms?The formula to find the sum of n terms in AP is Sn = n/2 (2a+(n−1)d), in which a = first term, n = number of terms, and d = common difference between consecutive terms.Is the sequence 2 n convergent?So…

Answer in 60sInput interpretationResultKey moments in videoFrom 01:27 -- The Equation for the Tangent at the Point X Equal Minus 1From 03:32 -- Plotting ProgramFrom 04:58 -- Wolfram AlphaFrom 06:01 -- ReviewFrom 06:18 -- The Tangent Equation and the Normal EquationKey moments in videoFrom 00:03 -- Example 1From 04:21 -- Example 2Key moments in videoFrom 00:49 -- What Derivatives AreFrom 03:41 -- Derivative of More than One FunctionFrom 04:31 -- Higher-Order DerivativesFrom 05:17 -- Find the Derivative with Respect to More than One VariableFrom 07:19 -- Visualize the DerivativeFrom 08:02 -- Find the Slope of an Algebraic FunctionFrom 08:47 --…

Answer in 60sInputResultDiagonalizationKey moments in videoFrom 00:45 -- Create a matrixFrom 01:31 -- Convert matrix into transpose matrixFrom 03:19 -- Inverse matrix in mathematicaFrom 04:34 -- Summary of what we have seen so farRelated queriesWhat is a transpose operator?In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix A by producing another matrix, often denoted by AT (among other notations).What is the transpose of a tensor?To transpose a tensor, we need two dimensions to be transposed. If a tensor is…

Answer in 60secInput interpretationMath solution sin^2(x) graph Graph sin^2(x) graph Related questionsWhat is the period of the graph of the function y sin 2x?The equation is y=sin2x , so it is in the form y=sin(b⋅x) The period of this graph is 2πb , which equals 2π2 or π . Therefore, the "important points" (quarter points) when graphing this function will be π4 apart.What is the amplitude of sin 2x?(c) y = sin 2x has amplitude 1, period π and phase shift 0.What is the formula for sin 2x?so that sin2x = 2 sin x cos x.What is sin squared of…

Answer in 60sInputRootProperties as a real functionDerivativeIndefinite integralAlternative representationsKey moments in videoFrom 00:45 -- Log definitionFrom 00:55 -- Definition of LogarithmFrom 03:17 -- Example: LogarithmsFrom 04:05 -- Common Bases for LogarithmsFrom 04:16 -- Common BasesFrom 15:28 -- 6th PropertyFrom 16:00 -- Change of BaseFrom 16:25 -- Quotient Property of LogarithmsFrom 20:50 -- Simplify Logarithms examplesFrom 22:38 -- Using Shift + Enter to Check Your WorkRelated queriesHow do you type log base on a keyboard?Open the document and place the cursor at the point where you want to insert the logarithm. Type "log," followed by the subscript icon given under the…

Answer in 60sInputResultKey moments in videoFrom 00:07 -- Definition of a quadratic residueFrom 00:41 -- Example Checking if v is a Quadratic ResidueFrom 01:57 -- How to check your answer?Related queriesWhat are the quadratic residue of 7?Thus 1,2,4 are quadratic residues modulo 7 while 3,5,6 are quadratic nonresidues modulo 7.How do you calculate residue?We compute the residues at each pole: At z = i: f(z) = 1 2 · 1 z − i + something analytic at i. Therefore the pole is simple and Res(f,i)=1/2. At z = −i: f(z) = 1 2 · 1 z + i + something…

Answer in 60sIndefinite integralRelated queriesWhat is an improper integral examples?For example, ∫ 0 1 1 x d x \displaystyle\int_0^1 \dfrac{1}{\sqrt x}\,dx ∫01x 1dxintegral, start subscript, 0, end subscript, start superscript, 1, end superscript, start fraction, 1, divided by, square root of, x, end square root, end fraction, d, x is an improper integral.What is improper integral of first kind?An improper integral of the first kind is an. integral performed over an infinite domain, e.g. Z 1. a. f(x) dx.How do you find the improper integral?Let f(x) be continuous over (a,b]. Then, ∫baf(x)dx=limt→a+∫btf(x)dx. In each case, if the limit exists, then…

Answer in 60sInputResultAlternate formExpanded formKey moments in videoFrom 01:13 -- Consider the CT signalFrom 05:15 -- Case 2From 12:39 -- Case 6Key moments in videoFrom 03:29 -- Time Shifting OperationsFrom 03:42 -- Convolution ExampleFrom 03:52 -- ExampleFrom 11:54 -- How to find General AnswerRelated queriesWhat does it mean to convolve two functions?In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions (f and g) that produces a third function ( ) that expresses how the shape of one is modified by the other. The term convolution refers to both the result function and to the process…

Answer in 60sInput interpretationResultDimensionsMatrix plotMatrix rankNullityPseudoinverseRelated queriesWhat is row reduction?Row reduction (or Gaussian elimination) is the process of using row operations to reduce a matrix to row reduced echelon form. This procedure is used to solve systems of linear equations, invert matrices, compute determinants, and do many other things.What is the use of reduce command in Mathematica?Reduce[expr,vars,dom] restricts all variables and parameters to belong to the domain dom. If dom is Reals, or a subset such as Integers or Rationals, then all constants and function values are also restricted to be real.How do you type matrix?To enter a matrix, use…

Answer in 60sInput interpretationResultKey moments in videoFrom 00:03 -- Solving Linear Systems with MathematicaFrom 00:11 -- Solve 3 EquationsFrom 01:54 -- Solve for a different variableFrom 02:16 -- (B)From 02:23 -- Variables, 3 equationsFrom 02:39 -- Using a Matrix (Not the Solve Function)From 03:29 -- Using RowReduce to Solve the SystemFrom 04:11 -- (Reduced form)From 04:59 -- Solving a system with only 1 solutionRelated queriesWhy is Gauss-Jordan method used?Gaussian Elimination and the Gauss-Jordan Method can be used to solve systems of complex linear equations. For a complex matrix, its rank, row space, inverse (if it exists) and determinant can all…

Answer in 60sInput interpretationGlobal maximumKey moments in videoFrom 00:53 -- First step: Entering the function in the calculatorFrom 03:43 -- How to draw a sketch of the graphFrom 09:56 -- Answering the question of absolute minimumFrom 11:37 -- Slope Number Line ExampleFrom 14:09 -- Summary & ConclusionRelated queriesWhat is absolute extremum?An absolute extremum (or global extremum) of a function in a given interval is the point at which a maximum or minimum value of the function is obtained. Frequently, the interval given is the function's domain, and the absolute extremum is the point corresponding to the maximum or minimum value…

Answer in 60sInputAlternate formRootsProperties as a real functionSeries expansion at x=0DerivativeIndefinite integralIdentitiesGlobal maximaAlternative representationsKey moments in videoFrom 00:25 -- Recall the unit circle valuesFrom 01:04 -- Identifying and remembering the unit circle valuesFrom 04:37 -- X - axis calibrationFrom 05:30 -- Make your own Trig axisFrom 09:02 -- Complete the table for quadrant 2From 10:46 -- Plot the pointsFrom 11:47 -- Terminology of the GraphFrom 13:09 -- What is an inflection pointFrom 15:42 -- Summary and conclusionFrom 16:10 -- Summary of domain and range of the cos (x) functionRelated queriesWhat is Coshx equal to?cosh x = ex + e−x 2…