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1.1 Four Ways to Represent a Function Introduction to Functions What is the domain of a function?, Domain and range of a function, Domain and range of a function given a formula, Domain of a function, How to find the domain and the range of a function given its graph (example), What is the range of a function? Finding the Domain of a Function - Made Easy!, Finding Domain and Range of a Function using a Graph, Finding the Domain of a Function Algebraically (No graph!), Domain and Range - Basic Idea - Two Graph Examples, Reading Domain and Range of a Relation From a Graph The Difference Quotient - Example 1, The Difference Quotient - Example 2, Graphing Piecewise Defined Functions, Finding the Domain and Range of a Piecewise Function, Graphing a Piece-Wise Defined Function - Another Example, Find the Formula for a Piecewise Function from Graph Section 1.1 Determining Domain of Functions, Identifying Domain and Range from a Graph, Difference Quotient, Piecewise Functions Domain of a function, the 3 common cases to be careful!, Domain of a function, 3 harder examples, Functions and Graphs, Domain and Range Functions & Graphs - Linear, Quadratic, Rational, Logarithmic & Square Root, Difference Quotient, Evaluating Piecewise Functions, Graphing Piecewise Functions
1.2 Essential Functions Section 1.2 How To Graph Equations - Linear, Quadratic, Cubic, Radical, & Rational Functions
1.3 New Functions from Old Functions Combining and Composition of Functions, Introduction to Graph Transformations, How to Graph with Transformations Shifting functions introduction, Reflecting functions introduction, Scaling functions introduction , Introduction to function composition, Evaluating composite functions example, Creating new function from composition, Evaluating composite functions: using graphs, Evaluating composite functions Horizontal and Vertical Graph Transformations, Graphing Exponential Functions w/ Graph Transformations, Graphing Using Graph Transformations - Example 1, Graphing Using Graph Transformations - Example 2, Graph Transformations about the X-axis and Y-axis, Graphing f(x) = (1/x) + 5 ; Rational Functions and Graph Transformation, Composition of Functions, Domain of a Composition of Functions, Example 1, Domain of a Composition of Functions, Example 2, Domain of a Composition of Functions, Example 3 - Common Mistake Section 1.3 Transformation of Functions, Composition of Functions, Combining Functions & Function Operations, Transformations of Functions, Composite Functions
1.4 Exponential Functions Graphing and Solving Exponential Functions, Solving Exponential and Logarithmic Equations. Exponential growth functions, Solving exponential equation, Graphing exponential functions Graphing Exponential Functions, Solving Exponential Equations - Some Basic Examples, Finding the Equation of an Exponential Function Section 1.4 Graphing Exponential Functions w/ t-table or Transformations, Exponential Growth Decay Functions & their Graphical Attributes Graphing Exponential Functions, Solving Exponential Equations, Graphing Exponential Functions With e, Transformations, Domain and Range, Asymptotes, Exponential Growth and Decay Word Problems & Functions
1.5 Inverse Functions An Introduction to Inverse Functions Introduction to function inverses, Understanding inverse functions Inverse Functions - The Basics!, Finding the Inverse of a Function or Showing One Does Not Exist, Ex 1, Finding the Inverse of a Function or Showing One Does not Exist, Ex 2, Finding the Inverse of a Function or Showing One Does not Exist, Ex 3, Finding the Inverse of a Function or Showing One Does not Exist, Ex 4 Section 1.5 Inverse Functions, Evaluating Inverse Trigonometric Functions, Inverse of Power Functions Inverse Functions, How To Find The Inverse of a Function, Inverse Functions - Domain & range- With Fractions, Square Roots, & Graphs
2.1 The Tangent and Velocity Problem Tangent and Velocity Problems Section 2.1
2.2 The Limit of a Function An Introduction to Limits Introduction to Limits(HD), Proof: lim (sin x)/x, One-sided limits Basic Idea of Limits, One Side Limits, Example 1, OSL, Example 2, OSL, Example 3 Finding Limits From a Graph, Calculating a Limit Involving sin(x)/x as x approaches zero, Limits Involving Absolute Value Limits, Continuity, Trigonometric Limits Section 2.2 Why Limits are Important in Calculus, Finding Real limits Graphical & Numerical Approach, Limits of Piecewise Function Limits of Trigonometric Functions with Correction, One-sided Limits Introduction to Limits
2.3 Calculating Limits Using Limit Laws Calculating Limits Using the Limit Laws Properties of Limits, Techniques of Limit Computation Limit Ex (Pt 1), Limit Ex (Pt. 2), Limit Ex (Pt. 3), Squeeze Theorem, More Limits Finding Limits From a Graph, Finding Limit By Factoring, Finding Limit by Expanding, Finding Limits by Getting a Common Denominator, Calculating a Limit Involving sin(x)/x as x approaches zero, Limits Involving Absolute Value, The Squeeze Theorem for Limits Limits, Continuity, Trigonometric Limits Section 2.3 Properties of Limits, Finding Limits- Examples include Quotients, Limits of Trigonometric Functions with Correction, One-sided Limits Properties of Limits, Limits, Evaluating Limits by Factoring, Evaluating Limits With Fractions and Square Roots Limits of Rational Functions - Fractions and Square Roots, Squeeze Theorem
2.5 Continuity Continuity Continuity of Functions Limits to definite continuity, Limit at a point of discontinuity Continuity - Part 1 of 2, Continuity - Part 2 of 2, Intermediate Value Theorem Limits, Continuity Section 2.5 Continuity Open & Closed Intervals, Intermediate Value Theorem Investigating discontinuities from a Piecewise Function Intermediate Value Theorem Example Continuity Intro, 3 Step Continuity Test, Discontinuity, Piecewise Functions & Limits, Limits and Continuity, Intermediate Value Theorem, Intermediate Value Theorem Explained - To Find Zeros, Roots or C value - Calculus, Piecewise Functions - Limits and Continuity
2.6 Limits at Infinity; Horizontal Asymptotes Limits at Infinity and Horizontal Asymptotes (part 1), Limits at Infinity and Horizontal Asymptotes (part 2) Limits and infinity, Limits at positive and negative infinity, More limits at infinity Finding horizonal and vertical asymptotes Infinite Limits, Limits at Infinity - Basic Idea and Shortcuts, Shortcut to Find Horizontal Asymptotes of Rational Functions, Calculating a Limit at Infinity with a Radical Section 2.6 Infinite Limits & Vertical Asymptotes Limit at Infinity Example 1, Limit at Infinity Example 2, Limit at Infinity Example 3 How To Find The Limit At Infinity, Limits at Infinity & Horizontal Asymptotes, Infinite Limits and Vertical Asymptotes
2.7 Derivatives and Rates of Change Derivatives and Rates of Change Slope of a Curve, Velocity, and Rates of Change Equation of a Tangent Line, Velocity and position from acceleration Finding the Equation of a Tangent Line, Position, Velocity, Acceleration using Derivatives Instantaneous Rate of Change, Slope, Velocity, Rate of Change Section 2.7 Slope of Tangent Line at a Point, Instantaenous Velocity and Speed of Linear Motion Slope and Equation of Normal & Tangent Line of Curve at Given Point , Definition of the Derivative
2.8 The Derivative as a Function The Derivative as A Function Introduction to the Derivative of a Function Derivatives 1, Derivatives 2, Derivatives 2.5 Introduction to Derivatives, Cusp Points and Derivatives Section 2.8 Finding Derivative with Defintion of Derivative, Differentiation Continuity and Differentiability,
3.1 Derivatives of Polynomials and Exponential Functions Derivatives of Polynomials and Exponential Functions Techniques of Differentiation (Finding Derivatives of Functions Easily), Derivatives and Integrals of Exponential Functions Proofs of Derivatives of Ln(x) and e^x Derivative of Exponential Functions Derivative of Exponential and Logarithms Section 3.1 Derivative of Exponential Functions with Base a, what is e, and the derivative of exponential functions Derivatives of Polynomial Functions, Derivatives of Exponential Functions
3.2 The Product and Quotient Rules Product and Quotient Rules The Product and Quotient Rules for Derivatives of Functions Product Rule, Quotient Rule Quotient Rule, Product Rule Derivatives of products, quotients, sine, cosine Section 3.2 Basic Differentiation, Product Rule, Quotient Rule Product rule for derivative example 1, Product rule for derivative example 2, Quotient Rule Proof, Quotient Rule Example 1, Quotient Rule Example 2, Quotient Rule Example 3, Quotient Rule Example 4 Product Rule for Derivatives, Quotient Rule For Derivatives
3.3 Derivatives of Trigonometric Functions Derivatives of Trigonometric Functions Finding Derivatives of Trigonometric Functions Derivatives of sin x, cos x, tan x, e^x and ln x Derivative of Trig Functions, Derivative of Trig Functions 2 Derivatives of products, quotients, sine, cosine Section 3.3 Derivative of Sine and Cosine Derivative of sin(x) and cos(x), PROOF, trigonometric derivative with quotient rule, cos(x)/(1-sin(x)), derivatives of tan(x) and cot(x), quotient rule Derivatives of Trigonometric Functions
3.4 The Chain Rule The Chain Rule Discussion of the Chain Rule for Derivatives of Functions Chain Rule, Chain Rule Examples, More Chain Rule Chain Rule for Finding Derivatives, More Chain Rules Example #1, Harder Example #1, Harder Example #2 Lesson on Chain Rule Section 3.4 Chain Rule for Derivatives & General Power Rule, Chain Rule Harder Algebraic Examples, Chain Rule with Trigonometric Functions, Chain Rule with Trig Functions Harder Examples Chain Rule Explained, My way to introduce the chain rule, Chain Rule Example 1, Chain Rule Example 2, Chain Rule Example 3, Chain Rule Example 4 Chain Rule For Finding Derivatives, Derivatives - Power, Product, Quotient and Chain Rule - Functions & Radicals, Derivatives of Trigonometric Functions - Product Rule Quotient & Chain Rule
3.5 Implicit Differentiation Implicit Differentiation (Derivatives of Inverse Trig Functions) Implicit Differentiation Implicit Differentiation, Implicit Differentiation Part 2, More implicit differentiation, Chain rule and implicit, Trig Implicit, Derivative of inverse sine, Derivative of inverse cosine, Derivative of inverse tangent Implicit Differentiation, Extra Examples, Even More Examples, Derivative of Inverse Tangent, Inverse Trigonometric Functions - Derivatives, Derivatives involving Inverse Trigonometric Functions, Example 2, Example 3 Implicit Differentiation Section 3.5 Introduction to Implicit Differentiation, Implicit Differentiation Examples, Derivative of Inverse Trigonometric Derivative of x^y=y^x, by implicit differentiation, derivative of ln(x), by definition & implicit differentiation, Second derivative with implicit differentiation 9x^2+y^2=9, implicit differentiation y*cos(x)=x^2+y^2, Implicit Differentiation Hack, derivative of x^y=y^x, implicit differentiation 4cos(x)sin(y)=1, implicit differentiation arctan(x^2*y)=x+x*y^2, Equation of the tangent line with implicit differentiation Implicit Differentiation, Implicit Differentiation Explained - Product Rule, Quotient & Chain Rule, Implicit Differentiation Second Derivative Trig Functions & Examples, Derivatives of Inverse Trigonometric Functions, Derivatives of inverse trigonometric functions sin-1(2x), cos-1 (x^2), tan-1 (x/2) sec-1 (1+x^2)
3.6 Derivatives of Logarithmic Functions Derivatives of Log Functions The Natural Log Function, Logarithmic Differentiation Derivative of a Fraction Proofs of Derivatives of Ln(x) and e^x Derivative of Logarithms, Logarithmic Differentiation, Logarithmic Differentiation - Example 2, Logarithmic Differentiation - Example 3 Derivative of Exponential and Logarithms Section 3.6 Derivative of Logarithmic Functions with Base a, Derivative of Natural Logarithmic Functions Derivative of ln(1/x), derivative of log base 10 of (x^3+1), with the box, derivative of ln(x*sqrt(x^2-1)) vs. derivative of ln(x+sqrt(x^2-1)), derivative of ln((2y+1)^5/sqrt(y^2+1)), derivative of ln(ln(s)), Derivative of ln(e^-x + x*e^-x), derivative of 2x*log_10(sqrt(x)), econd derivative of x^2*ln(2x), second derivative of ln(x+sqrt(1+x^2)), equation of the line tangent to ln(x^2-3x+1) at x=3, derivative of sqrt((x-1)/(x^4+1)), logarithmic differentiation, derivative of x^sin(x) derivative of x^x regular derivative vs. logarithmic derivative Derivative of Logarithmic Functions, Logarithmic Differentiation - Rules, Examples, Exponential Functions, Logarithmic Differentiation
3.9 Related Rates Related Rates . Related Rates Implicit Diffrentiation Related Rates Using Cones Related Rates A Point on the Graph Related Rates Involving Trigonometry Related Rates Involving Baseball! Related Rates Changing Circumference of a Circle Related Rates Ladder Sliding Down Wall Related Rates Cars Traveling from an Intersection Lecture 12 - Related Rates Section 3.9 Related Rates Part 1, Related Rates Part 2, Related Rates Example: Volume of a Cone related rates problems! (7 examples), related rates, heating up a circular metal plate, related rates, water rising in an inverted circular cone, related rates, ladder sliding down against the wall, related rates, two cars approaching to the same intersection, related rates, one ship sails south and another sails north, related rates, angle of sight of a rising ballon, related rates, angle increases & area increases, related rates, particle moving along a hyperbola, related rates: rate of change of the radius of a spherical ballon, Related Rates: AREA OF A TRIANGLE, Related Rates: TWO CARS, Related Rates: SURFACE AREA OF A SPHERE (BALL), Related rates, streetlight and shadow, Related rates, area of an equilateral triangle Introduction to Related Rates, Related Rates - Conical Tank, Ladder Angle & Shadow Problem, Circle & Sphere , Related Rates - The Shadow Problem, Relates Rates - Gravel Being Dumped & Conical Tank Problem, Related Rates - The Ladder Problem, Related Rates - Inflated Balloon & Melting Snowball Problem - Surface Area & Volume Related Rates - Airplane Problems, Related Rates - Distance Problems, Related Rate Problems - The Cube - Volume, Surface Area & Diagonal Length, Related Rates - Angle of Elevation Problem
3.10 . Using Differentials to Approximate Change Section 3.10 delta y vs. dy (differential), local linear approximation, e^-0.015, Approximating (1.998)^4 by using differential, tangent line approximation for cbrt(1001) Differentials and Derivatives - Local Linearization, Linear Approximation, Differentials, Tangent Line, Linearization, Finding The Linearization of a Function Using Tangent Line Approximations, Estimating Function Values Using Differentials and Local Linearization
3.11 Hyperbolic Functions A Discussion of Hyperbolic Functions Hyperbolic function inspiration Hyperbolic Functions - The Basics, Hyperbolic Functions - Derivatives, Inverse Hyperbolic Functions - Derivatives Section 3.11 Hyperbolic Functions Introduction, Hyperbolic Functions Derivatives, Inverse Hyperbolic Functions Derivative Introduction to Hyperbolic Trig Functions, hyperbolic function derivatives, inverse sinh(x)
4.1. Maximum and Minimum Values Maximum and Minimum Extreme Value Theorem, Minimum and maximum values on an interval, Applying the EVT, Testing critical points for local extrema Finding Critical Numbers - Example 1, Finding critical numbers - example 2 Lecture 10 - Approximations (cont) Curve Sketching, Lecture 11 - Max-Min Problems Section 4.1 Extrema Intro- Extrema on an Interval Critical Numbers Example 1, Critical Numbers Example 2, Critical Numbers Example 3, Critical Numbers Example 4, Critical Numbers Example 5 Finding Critical Numbers, Extreme Vale Theorem
4.2 The Mean Value Theorem The Mean Value Theorem A BRIEF Discussion of Rolle's Theorem and Mean-Value Theorem Mean value theorem, Getting a ticket because of the mean value theorem Finding increase/decrease local max/mins, Mean Value Theorem Lecture 14 - Mean Value Theorem, Inequalities Section 4.2 Mean Value Theorem for Derivatives Rolle's Thoerem, Rolle's Theorem Explained and Mean Value Theorem For Derivatives, Mean Value Theorem
4.3 How Derivatives Affect the Shape of a Graph How Derivatives Affect the Shape of a Graph Increasing/Decreasing and Concavity of Functions, The First Derivative Test for Increasing and Decreasing, The Second Derivative Test for Concavity of Functions Increasing/Decreasing + Local Max and Mins using First Derivative Test, Finding Local Maximums/Minimums - Second Derivative Test , Concavity, Inflection Points and Second Derivatives Section 4.3 First Derivative Test, Concavity and Second Derivative Test Concavity, inflection points and second derivatives, Second derivative test First Derivative Test, Relative Extrema, Local Maximum and Minimum, First Derivative Test, Critical Points, Concavity, Inflection Points, Increasing Decreasing, First & Second Derivative, Second Derivative Test
4.4 l'Hospital's Rule Indeterminate Forms and L'Hospital's Rule Evaluating Limits of Indeterminate Forms Introduction to l'Hôpital's rule, L'Hôpital's rule example 1, L'Hôpital's rule example 2, L'Hôpital's rule example 3 , L'Hopital's Rule to solve for variable, Tricky L'Hopital's Rule problem, Proof of special case of l'Hôpital's rule L'Hospital's Rule - Indeterminate Products, L'Hospital's Rule and Indeterminate Quotients, L'Hospital's Rule - Indeterminate Powers, L'Hospital's Rule - Indeterminate Differences Section 4.4 L'Hospital's Rule L'Hospital Rule for the 0/0 indeterminate form, limit of cos(x)/(1-sin(x)) as x goes to (pi/2)+, limit of (e^(2t)-1)/sin(t) as t goes to 0, Limit of x^(1/(1-x) as x goes to 1+, Limit of 1/x-1/(e^x-1) as x goes to 0+, limit of (ln(x))^2/x as x goes to infinity, Limit of x/(x-1)-1/ln(x) as x goes to 1, Limit of ln(x^7-1)-ln(x^5-1) as x goes to 1+, limit of ln(x)/x as x goes to 0+, Be Careful with L'Hospital's Rule, Limit of ln(x)/x^p as x goes to infinity L'hopital's rule, L'hospital's Rule Indeterminate Forms, Limits at Infinity, Ln, Trig, Limits of Exponential Functions
4.5 Summary of Curve Sketching Summary of Curve Sketching How to Sketch Graphs of Functions Infection points and concavity intuition, Graphing using derivatives, Graphing with derivatives example Graphing with calculus Section 4.5 Graphing f(x)=x^2-x-ln(x), graph of y=x*e^x, Graph of y=x^x Curve Sketching, Curve Sketching - First & Second Derivatives - Graphing Rational Functions & Asymptotes
4.7 Optimization Problems Optimization Problems Optimization; Max/Min Application Problems Optimization with calculus 1, Optimization 2, Optimization 3, Optimization 4 Optimization problem #1, Optimization #2, Optimization #3 Optimization: Minimizing Length, Optimization: Volume Section 4.7 Example 1, Example 2, Example 3, Example 4, Example 5 Optimization Problems, Optimization Calculus - Fence Problems, Cylinder, Volume of Box, Minimum Distance & Norman Window
4.8 Newton's Method Newton’s Method, Newton’s Method - More Examples part 1 of 3, Newton’s Method - More Examples part 2 of 3, Newton’s Method - How it Can FAIL - More Examples part 3 of 3 Lecture 13 - Newton’s Method and Other Applications Section 4.8 Newton's Method Linear Approximation Estimating a Zero of a Function Newton's method and Omega Constant, Solve x^x^x=2 Newton's Method
4.9 Antiderivatives Antiderivatives An Introduction to the Indefinite Integral The Indefinite Integral or Anti-derivative, Indefinite integrals 2, Indefinite integrals 3, Basic Integration Problems Antiderivatives Section 4.9 Antiderivatives & Indefinite Integration, Antiderivatives Part 2, Indefinite Integration Word Problem Antiderivatives
5.1 Areas Areas and Distances Area Under a Curve, Limit Approach, Riemann Sums Section 5.1 Estimating Area with Rectangles Part 1 of 2 Definition of Area of Riemann Sum Limit of Sums Part 2 of 2, Estimating Area with Riemann Sums Finite Rectangles Midpoint, Left & Right Endpoints, Area, Definite Integral, Sigma Notation, Calculus
5.2 The Definite Integral The Definite Integral Area Under a Curve, Limit Approach, Riemann Sums Riemann sums and integrals, Introduction to definite integrals, Definite integrals (part II), Definite integrals (part III), Definite Integrals (part IV), Definite Integrals (part V) The Definite Integral: Understanding the Definition, Calculating a Definite Integral Using Riemann Sums - Part 1, Calculating a Definite Integral Using Riemann Sums - Part 2 Definite Integrals Section 5.2 Estimating Area with Rectangles & Riemann Limit of Sums Definition of Area, Definite Integrals Defined w. Riemann Limit of Sums Example, Definite Integrals Common Geometric Area Definite Integral, Properties of Definite Integrals Examples - Basic Overview, Calculus
5.3 The Fundamental Theorem of Calculus and Indefinite Integrals The Fundamental Theorem of Calculus The Fundamental Theorem of Calculus Fundamental Theorem of Calculus The Fundamental Theorem of Calculus Part 1, The Fundamental Theorem of Calculus Part 2 Lecture 19: First fundamental theorem of calculus, Lecture 20: Second fundamental theorem, Lecture 21: Applications to logarithms and geometry Section 5.3 First Fundamental Theorem of Calculus, Definite Integral as a Function of x, Second Fundamental Theorem of Calculus FTC Example 1, FTC Example 2 Fundamental Theorem of Calculus Part 1, Fundamental Thereom of Calculus Explained - Part 1 & 2 Examples, Fundamental Theorem of Calculus Part 2
5.4 The Fundamental Theorem of Calculus and Indefinite Integrals The Indefinite Integral and Net Change Theorem The Evaluation of Definite Integrals Section 5.4
5.5 The Substitution Rule Substitution Rule Integration by Substitution Definite Integral with Substitution, U-Substitution, U-substitution example 2, U-substitution Example 3, Another u-substitution example, U-substitution and back substitution, Indefinite Integration (part IV) Integration using U-Substitution, Integration by U-Substitution, Definite Integral, U-substitution - Indefinite Integral, Another 2 examples, U-Substitution - More Complicated Examples Definite Integral by Substitution | MIT 18.01SC Single Variable Calculus, Fall 2010 , Lec 28 | MIT 18.01 Single Variable Calculus, Fall 2007 Section 5.5 Indefinite Integration by U Substitution, Definite Integration with U Substitution how to integrate using u substitution How To Integrate Using U-Substitution, U-substitution With Definite Integrals
6.1 Area Between Curves Introduction to Areas Between Curves, Basic example on finding the area between two curves, Another example on finding the area between two curves, Finding Area Between Curves by Integrating With Respect To y Finding Area Between Two Curves Area between curves Finding Areas Between Curves, Area Between Curves - Integrating with Respect to y Area between curves Section 6.1 Area Between 2 Curves using Vertical and Horizontal Representative Rectangles Area Between Two Curves
6.2 Volume Volume of Solids By Disks and Washers Method Solid of Revolution (part 1), Solid of Revolution (part 2), Solid of Revolution (part 3), Solid of Revolution (part 4), Solid of Revolution (part 5), Solid of Revolution (part 6), Solid of Revolution (part 7), Solid of Revolution (part 8) Volumes of Revolution - Disk/Washers Example 1 Lecture 22 Volume of Solid of Revolution Disk Method and Washer Method Volume of Solid of Revolution Disk & Washer Method
6.3 Calculus 1 Lecture 5.3: Volume of Solids By Cylindrical Shells Method Shell method for rotating around horizontal line Longer Version - Volumes using Cylindrical Shells, Volumes of Revolution - Cylindrical Shells Volume of Revolution via Shells | MIT 18.01SC Single Variable Calculus, Fall 2010 Volume of Solid of Revolution Shell Method 3 Examples Shell Example 1, Shell Example 2, Shell Example 3, Shell Example 4 Shell Method - Volume of Revolution
6.4 Calculating the Work Required to Drain a Tank, Finding Work using Calculus - The Cable/Rope Problem, Finding Work using Calculus - The Cable/Rope Problem - Part b, Lecture 23 Work Done by a Variable Force Work Example 1, Work Example 2, Work Example 3, Work Example 4, Work Example 5 Work Problems, Work Done By a Variable Force Physics Problems, Force Displacement Graphs