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Pure Mathematics
Calculus
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Course Content
Category: Pure Mathematics
Topic outline
General
General
General
Calculus Discussion Group
Forum
Precalculus
Precalculus
Precalculus
Functions
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Which functions pass the VERTICAL LINE TEST?!!
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How to find DOMAIN AND RANGE for a function
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Finding values and domain and range from a graph
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Determining if a function is even, odd or neither
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How to classify functions
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Equation Modeling
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Modeling the equation of a piecewise defined function from its graph
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Sketching a graph from a story problem
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Equation of a line in point-slope form
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Equation of the line in slope-intercept form
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Least squares line
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Sketch the graph of a parabola
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How to find the CENTER AND RADIUS of a circle
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Sketch the graph of a circle
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Combinations of functions and their domains
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Composite functions
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Composite functions and their domains
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Describing transformations algebraically
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Using transformations to sketch a graph
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Determine whether a function is 1 to 1
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How to find the INVERSE OF A FUNCTION
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Finding the linear function given two points on its inverse
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Use laws of logarithms to simplify a logarithmic function
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Use the quadratic formula to find roots of the function
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Completing the square of a quadratic function
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Polynomial long division for rational functions
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Hyperbolic identities
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How to remember the unit circle
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Limits & Continuity
Limits & Continuity
Limits & Continuity
Limits and Continuity
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Solving limits with substitution
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How to solve LIMITS BY FACTORING
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CONJUGATE METHOD for solving limits
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Use limit laws to evaluate limits of combination functions
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How to find limits on CRAZY GRAPHS
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Limits at infinity
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Infinite limits
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Limits of trig functions: Everything you need to know!
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How to find the ONE-SIDED LIMITS of a function
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How to prove that the limit does not exist
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The precise definition of the limit EXPLAINED!
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Finding delta from a graph and the epsilon-delta definition of the limit
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How to find the limit using SQUEEZE THEOREM
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Limit of an inequality, with squeeze theorem
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Finding the domain by investigating continuity
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How to make a function continuous (for a piecewise function)
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How to find REMOVABLE DISCONTINUITIES
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How to use the Intermediate Value Theorem
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Intermediate value theorem to prove a root in an interval
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Prove the equation has at least one real root
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Limits of composite functions - How to find them?
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Jump discontinuities - What are they?
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Derivatives
Derivatives
Derivatives
Derivatives
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How to calculate the difference quotient
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DEFINITION OF THE DERIVATIVE to find the derivative
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Power rule for derivatives
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PRODUCT RULE for derivatives
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Product rule - 3+ functions
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Quotient rule
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Reciprocal rule
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Chain rule combined with power rule
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Chain rule for derivatives, with product rule
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Chain rule for derivatives, with quotient rule
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Chain rule for derivatives, with trig functions
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Derivatives of TRIG FUNCTIONS
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Derivative of cot^4(x/2)
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Derivatives of inverse trig functions - arcsin
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Derivatives of inverse trig functions - arccot
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Derivatives of HYPERBOLIC functions
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Derivatives of inverse hyperbolic functions
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Implicit differentiation COMPLETELY EXPLAINED!
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Finding the derivative of a function with e^x
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Derivatives of logarithms
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How to use LOGARITHMIC DIFFERENTIATION to find the derivative
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Derivative of x^x
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Equation of the tangent line
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Value that makes two tangent lines parallel
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DIFFERENTIABILITY and horizontal and vertical tangent lines
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Equation of the normal line at a point
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Average rate of change of a function
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Finding HALF LIFE
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Continuous compounding interest
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SALES DECLINE calculus problems
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Applications of derivatives
Applications of derivatives
Applications of derivatives
Linear approximation
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ESTIMATE THE ROOT with linear approximation
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Linearization of a function at a point
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Optimization
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How to find CRITICAL POINTS
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Where is the function INCREASING AND DECREASING?
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Inflection points and concavity
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FIRST DERIVATIVE TEST to find the max and min
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How to use the SECOND DERIVATIVE TEST
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How to find VERTICAL ASYMPTOTES
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How to find HORIZONTAL ASYMPTOTES
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How to find SLANT ASYMPTOTES
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All about GRAPH SKETCHING
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Maxima and minima on a CLOSED interval
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Dimensions that maximize the area of the rectangle
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OPTIMIZATION: Dimensions that maximize the volume of a cylinder
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OPTIMIZATION: Dimensions of a rectangle that minimize its perimeter
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OPTIMIZATION: Dimensions that maximize the volume of a box
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Dimensions of the rectangle with largest area inscribed in an equilateral triangle
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Largest area of a rectangle inscribed in a semicircle
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Points on the cone closest to a point
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Max volume of a rectangular box inscribed in a sphere
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Maximum volume of a cone shaped cup
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Largest possible volume of a cylinder inscribed in a sphere
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Dimensions that minimize the surface area of a cylinder
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OPTIMIZATION: Dimensions that minimize page size with a given printed area
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OPTIMIZATION: Dimensions that minimize the surface area of an open top box
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OPEN-TOP BOX?!! How to find the smallest possible surface area
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OPTIMIZATION: Point on the line closest to another point
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Minimizing the product of two real numbers
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Finding numbers with the smallest sum of squares
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OPTIMIZATION: Production level, sale price that maximize profit
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OPTIMIZATION: Sales level that maximizes revenue
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OPTIMIZATION: Time when velocity is minimum
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Related Rates
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RELATED RATES: Radius of the balloon
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RELATED RATES: Ladder sliding down the wall
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RELATED RATES: Distance between the person and the plane
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RELATED RATES: Water level in the tank
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RELATED RATES: Price of the product
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MEAN VALUE THEOREM (MVT) for derivatives
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How to use ROLLE'S THEOREM
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How to use NEWTON'S METHOD
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How to use L'HOSPITAL'S RULE
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Position Function
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All about a PARTICLE'S POSITION function
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Instantaneous velocity
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HOW HIGH CAN YOU THROW THE BALL?!! How fast is it going when it hits the ground?
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Dropping a coin from the roof (VERTICAL MOTION!)
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How to find MARGINAL COST, REVENUE AND PROFIT
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Integration
Integration
Integration
Integrals
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Summation Notation - Finding the Sum
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How to find a Riemann sum using LEFT ENDPOINTS
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Riemann sums with right endpoints
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THE MOST ACCURATE WAY TO DO A RIEMANN SUM (Midpoint rule)
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Close only counts in horseshoes, hand grenades, and Riemann sums
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Trapezoidal Rule
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Simpson's Approximation
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Definite Integral
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ARE YOU FINDING THE RIGHT AREA?!! Area under vs. area enclosed by a graph
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Definite integral of an even function
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Definite integral of an odd function
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PART 1 OF THE DREADED FUNDAMENTAL THEOREM OF CALCULUS!
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PART 2 OF THE FUNDAMENTAL THEOREM OF CALCULUS!
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Net change theorem
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Indefinite Integrals
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Properties of integrals
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Find f(x) given f''(x), its second derivative
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Find f(x) given f'''(x), its third derivative
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Initial Value Problem
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Find f given f'' and initial conditions
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Solving Integrals
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U-Substitution
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U-Substitution in Definite Integrals
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Integration by Parts
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Integration by parts two times
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Integration by parts three times
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U-substitution with integration by parts
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Integration by parts to prove the reduction formula
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A TOTALLY DIFFERENT WAY to do integration by parts!
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Partial fractions, distinct linear factors
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Partial fractions, distinct linear factors, example 2, part 1 of 2
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Partial fractions, distinct linear factors, example 2, part 2 of 2
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Partial fractions, distinct linear factors, example 3
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Partial fractions, repeated linear factors
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Using partial fractions with DISTINCT QUADRATIC FACTORS
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Partial fractions, distinct quadratic factors, example 2
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Partial fractions, distinct quadratic factors, example 3
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Partial fractions, repeated quadratic factors
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Partial fractions, rationalizing substitution
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Integrating with partial fractions: how to factor difficult denominators
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Partial fractions, two ways to find the constants
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Integrating a trig function using u-substitution!
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Trigonometric integrals - sin^mcos^n, odd m
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Trigonometric integrals - sin^mcos^n, odd n
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Trigonometric integrals - sin^mcos^n, m and n even
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Integrals of trigonometric functions, tan^msec^n, odd m
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Integrals of trigonometric functions, tan^msec^n, even n
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Integrals of trigonometric functions, sin(mx)cos(nx)
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Integrals of trigonometric functions, sin(mx)sin(nx)
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Integrals of trigonometric functions, cos(mx)cos(nx)
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Hyperbolic integrals... pretty much just like trig integrals
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Integrals of Inverse Hyperbolic Functions
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How to solve EVERY trigonometric substitution problem ever!
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Trigonometric substitution with secant
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Trigonometric substitution with sine
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Trigonometric substitution with tangent
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Integral of a quadratic function
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Improper integrals, case 2
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Improper integrals, case 3
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improper integrals, case 4
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COMPARISON THEOREM WILL FINALLY MAKE SENSE!
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Integrals using reduction formulas
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How to use inverse hyperbolic functions to evaluate integrals
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Partial fractions are SPECIAL! (Repeated linear factors)
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PARTIAL FRACTIONS example with distinct linear factors
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Trig substitution - How to solve?
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Applications of integration
Applications of integration
Applications of integration
Applications of Integration
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Average Value of a Function Example 1
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Area between curves - dx
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Area between curves - dy
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Area between curves - sketching
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Arc Length
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Arc length x=g(y)
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Arc length - Simpson's Rule
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Surface Area of Revolution
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Surface area of revolution - Simpson's Rule
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Surface of Revolution
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Volume of rotation: disk method about the y-axis or x=
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Volume of rotation: disk method about the y-axis or x=
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Volume of rotation: disk method about the x-axis or y=
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Volume of rotation: washer method about the x-axis or y=
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Volume of rotation: washer method about y-axis or x=
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Volume of rotation: washer method about x-axis or y=
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Volume of rotation: cylindrical shells about the y-axis or x=
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Volume of rotation: cylindrical shells about the x-axis or y=
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Mean value theorem for integrals
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Work done using a rope to lift a weight
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Work Done on Elastic Springs
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Work Done by a Variable Force
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Moments of the system
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Moments of the system, x-axis
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Center of mass of the system
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Center of mass of the system, x-axis
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Hydrostatic pressure
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Hydrostatic force
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Vertical Motion (integration)
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Rectilinear Motion
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Centroids of Plane Regions
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Centroid of the plane - Simpson's Rule
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Area of the triangle with the given vertices
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Present and Future Value
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Present and Future Value Example 2
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Present and Future Value Example 3
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Present and Future Value Example 4
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Present and Future Value Example 5
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Consumer and Producer Surplus
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Probability density functions
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Cardiac output
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Cardiac output - Simpson's Rule
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Poiseuille's law
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Theorem of Pappus
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Polar & Parametric
Polar & Parametric
Polar & Parametric
Eliminating the Parameter
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Derivative of a parametric curve
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Second Derivative of a Parametric Curve
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Sketch the parametric curve by plotting points
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Tangent line to the parametric curve
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Area under the parametric curve
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Parametric area under one arc or loop
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Volume of Revolution of a Parametric Curve
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Parametric arc length
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Parametric arc length and the distance traveled by the particle
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Parametric arc length, simpson's rule
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Parametric Curve - Surface Area of Revolution
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Polar coordinates
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Converting Polar Coordinates
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Converting Rectangular Equations to Polar Equations
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Converting Polar Equations to Rectangular Equations
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Distance between two polar points
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Sketching Polar Curves
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Sketching Polar Curves - 2 Examples
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Sketching polar curves from cartesian curves
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Tangent line to the polar curve
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Vertical and horizontal tangent lines to the polar curve
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Polar Area
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Polar Area Bounded by One Loop
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Points of intersection of two polar curves
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Area Between Polar Curves
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Polar area inside both curves
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Arc Length of a Polar Curve
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Polar Parametric Curve - Arc Length
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Polar Parametric Curve - Surface Area of Revolution
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Taking a look at complex numbers in polar form
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Let's multiply some complex numbers!
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Analytic Geometry - Graph of a Single Point or of No Points
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Analytic Geometry - Set of Points Equally Distant from Two Points
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Analytic Geometry- Set of Points Unequally Distant from Two Points
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Eccentricity and directrix of the conic section
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Parabolas -Vertex, Axis, Focus, Directrix
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Equation of a parabola (conic section)
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Polar equation of the parabola (conic section)
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Vertex axis focus directrix of an ellipse
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Equation of an ellipse (conic section)
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Polar equation of the ellipse (conic section)
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Vertex axis focus directrix asymptotes of a hyperbola
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Equation of a hyperbola (conic section)
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Polar equation of the hyperbola (conic section)
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Sequences and series
Sequences and series
Sequences and series
Listing the first terms of the sequence
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Calculating the first terms of the sequence
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Finding a formula for the general term of the sequence (a_n)
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Does the sequence converge or diverge?
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Finding the limit of a convergent sequence
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Increasing, decreasing and not monotonic sequences
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Bounded sequences
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Calculating the first terms in a series of partial sums
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Sum of the series of partial sums
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nth term test, divergence test, zero test
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Convergence of a geometric series
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Convergence and sum of a geometric series, example 1
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Convergence and sum of a geometric series, example 2
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Convergence and sum of a geometric series, example 3
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Values for which the geometric series converges
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Repeating decimal expressed as a ratio of integers
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Convergence of a telescoping series
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Sum of Telescoping Series
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p-Series test for convergence
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How to use the INTEGRAL TEST for series
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Comparison Test
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Limit Comparison Test
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Estimating error/remainder of a series
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Alternating Series Test - Calculus
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Alternating series estimation theorem
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Ratio Test
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Ratio Test with Factorials
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Root Test
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Absolute and conditional convergence
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Difference Between Limit and Sum of the Series
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Radius of convergence
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Interval of Convergence
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Interval of Convergence Example 2
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Power series representation, radius and interval of convergence
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Power series multiplication
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Power series division
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Power series differentiation
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Using power series to estimate a definite integral
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Binomial series
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Taylor Polynomial Example 1 PART 1/2
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Taylor Polynomial Example 1 PART 2/2
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Taylor Polynomial Example 2 PART 1/2
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Taylor Polynomial Example 2 PART 2/2
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Finding radius of convergence of a Taylor series
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Taylor's inequality
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Maclaurin Series
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Sum of the maclaurin series
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Maclaurin series radius of convergence
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Expressing the integral as a power series
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Using maclaurin series to estimate an indefinite integral
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Maclaurin series to estimate a definite integral
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Maclaurin series to evaluate a limit
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