# Calculus

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### Introduction to Calculus

- What is Calculus?
- Prologue and Historical Context
- Understanding the Gradient function
- Introduction to Limits
- Determining One Sided Limits
- Determining When a Limit does not Exist
- Determining Limits Algebraically
- Infinite Limits and Vertical Asymptotes
- Limits at Infinity and Horizontal Asymptotes
- Definition of Continuity at a Point
- Classifying Topics of Discontinuity (removable vs. non-removable)
- Determining Limits Graphically
- Formal Definition of a Limit at a Point
- Continuous Functions
- Intemediate Value Theorem
- Limits for The Squeeze Theorem
- Tangent Line to a Curve
- Normal Line to a Tangent
- Slope of a Curve at a Point
- Average Velocity
- Instantaneous Velocity
- Limit Definition of Derivative
- First Principles Example 1: x²
- First Principles Example 2: x³
- First Principles Example 3: square root of x
- Standard Notation and Terminology
- Differentiable vs. Non-differentiable Functions
- Rate of Change of a Function
- Average Rate of Change Over an Interval
- Instantaneous Rate of Change at a Point
- Power Rule
- Chain Rule
- Sum Rule
- Product Rule
- Proof of the Product Rule
- Quotient Rule
- Implicit Differentiation
- Summary of Differentiation Rules
- Proof of Quotient Rule
- Limits Involving Trigonometric Functions
- Intuitive Approach to the derivative of y=sin(x)
- Derivative Rules for y=cos(x) and y=tan(x)
- Differentiating sin(x) from First Principles
- Special Limits Involving sin(x), x, and tan(x)
- Graphical Relationship Between sin(x), x, and tan(x), using Radian Measure
- Derivatives of y=sec(x), y=cot(x), y= csc(x)
- Differentiating Inverse Trigonometric Functions
- From First Principles
- Differentiating Exponential Functions with Calculators
- Differentiating Exponential Functions with Base e
- Differentiating Exponential Functions with Other Bases
- Differentiating Logarithmic Functions with Base e
- Differentiating Logarithmic Functions without Base e
- Overview of Different Functions
- Interpreting the Sign of the First Derivative (Increasing and Decreasing Functions)
- Identifying Stationary Points (Critical Points) for a Function
- Identifying Turning Points (Local Extrema) for a Function
- Classifying Critical Points and Extreme Values for a Function
- Mean Value Theorem for Continuous Functions
- Relationship between First and Second Derivatives of a Function
- Analyzing Concavity of a Function
- Notation for the Second Derivative
- Determining Points of Inflection for a Function
- First Derivative Test vs Second Derivative Test for Local Extrema
- The special case of x⁴
- Critical Points of Inflection
- Application of the Second Derivative (Acceleration)
- Examples of Curve Sketching
- Introduction
- Solving Optimization Problems
- Using the Tangent Line to Approximate Function Values
- Using Newton's Method to Approximate Solutions to Equations
- Using Implicit Differentiation to Solve Related Rates Problems
- Sigma Notation
- Integration: the Area Problem
- Formal Definition of the Definite Integral
- Definite and indefinite integrals
- Integrals of Polynomial functions
- Determining Basic Rates of Change Using Integrals
- Integrals of Trigonometric Functions
- Integrals of Exponential Functions
- Integrals of Rational Functions
- The Fundamental Theorem of Calculus
- Basic Properties of Definite Integrals
- Evaluating the Constant of Integration
- Integration by Substitution
- Integration by Parts
- Integration by Trigonometric Substitution
- Integral by Partial Fractions
- Calculating Areas using Integrals
- Calculating Volume using Integrals
- Deriving Formulae Related to Circles using Integration
- Symmetrical Areas
- Definite Integrals with Substitution
- Integration Using Euler's Method
- RAM (Rectangle Approximation Method/Riemann Sum)
- Integration Using Simpson's Rule
- Analyzing Approximation Error
- Integration Using the Trapezoidal Rule
- Solving Separable Differential Equations
- Slope Fields
- Exponential Growth and Decay Models
- Logistic Growth Models
- Net Change: Motion on a Line
- Determining the Surface Area of a Solid of Revolution
- Determining the Length of a Curve
- Determining the Volume of a Solid of Revolution
- Determining Work and Fluid Force
- The Average Value of a Function
- Introduction to Parametric Equations
- Derivative of Parametric Functions
- Determining the Length of a Parametric Curve (Parametric Form)
- Determining the Surface Area of a Solid of Revolution
- Determining the Volume of a Solid of Revolution
- Introduction to Polar Coordinates
- Determining the Slope and Tangent Lines for a Polar Curve
- Determining the Length of a Polar Curve
- Determining the Surface Area of a Solid of Revolution
- Determining the Volume of a Solid of Revolution
- Calculating Polar Areas
- Introduction to Power Series
- Differentiating and Integrating Power Series
- Constructing a Taylor Series
- Constructing a Maclaurin Series
- Lagrange Form of the Remainder Term in a Taylor Series
- Determining the Radius and Interval of Convergence for a Power Series
- Applications of Power Series
- Power Series Representations of Functions
- Power Series and Exact Values of Numerical Series
- Power Series and Estimation of Integrals
- Power Series and Limits
- Product of Power Series
- Binomial Series
- Power Series Solutions of Differential Equations
- Geometric Series
- Nth Term Test for Divergence of an Infinite Series
- Direct Comparison Test for Convergence of an Infinite Series
- Ratio Test for Convergence of an Infinite Series
- Integral Test for Convergence of an Infinite Series
- Limit Comparison Test for Convergence of an Infinite Series
- Alternating Series Test (Leibniz's Theorem) for Convergence of an Infinite Series
- Infinite Sequences
- Root Test for for Convergence of an Infinite Series
- Infinite Series
- Strategies to Test an Infinite Series for Convergence
- Harmonic Series
- Indeterminate Forms and de L'hospital's Rule
- Partial Sums of Infinite Series
- What is Leibniz Notation ?
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- Why is calculus important?
- What is the gradient function used for?
- Why do we need the gradient function?
- Question #3a6f3
- Question #12541
- What is an infinite limit?
- How do you find limits as x approaches infinity?
- Is the function #(x^2-6x+9)/(x-3)# continuous?
- How can I prove that a function is continuous?
- What is the "rate of change" of a function?
- Why is it important to know rates of change?
- Are there different kinds of rate of change?
- What is the slope of a curve?
- How do I find the equation for a tangent line without derivatives?
- How do you find the equation of a normal line if you know the equation of the tangent line?
- If my tangent line at point (4,8) has the equation #y=5x/6 - 9#, what is the equation of the normal line at the same point?
- How do I find the derivative of #f(x)=x^3# from first principles?
- How do I find the derivative of #x^2 + 7x -4# using first principles?
- How do I find the derivative of #x^3 - 2x^2 + x/4 +6# using first principles?
- How do I find the derivative of #f(x)=sqrt(x)# using first principles?
- How do I find the derivative of #f(x) = sqrt(x+3)# using first principles?
- What is the derivative of #x^n#?
- How do I find derivatives of radicals like #sqrt(x)#?
- What is the quotient rule?
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- Is there a way to find the derivative of sin(x) without limits?
- How are sin(x), tan(x), and x related graphically?
- How can I find the derivative of #y=c^x# using first principles, where c is an integer?
- What is the derivative of #log_e(x)#?
- What are the derivatives of the inverse trigonometric functions?
- What are the derivatives of exponential functions?
- What are the derivatives of logarithmic functions?
- What is a stationary point, or critical point, of a function?
- What is special about a turning point?
- How do I find local maxima and minima of a function?
- How can I use derivatives to find acceleration, given a position-time function?
- Do all functions have points of inflection?
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- What is Newton's Method?
- How do I evaluate definite integrals?
- How do I evaluate indefinite integrals?
- What is the antiderivative of a polynomial?
- How do you find the antiderivative of #x^2+5x#, if the point (0,5) exists on the graph of the antiderivative?
- How do you evaluate the integral #int_0^4x^3+2x^2-8x-1#?
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- How can you find a function, if you already know the rate of change of the function?
- What are the antiderivatives of #sin(x)# and #cos(x)#?
- What is the antiderivative of #tan(x)#?
- What is the antiderivative of #sec^2(x)#?
- What are the antiderivatives of #sec(x)#, #csc(x)# and #cot(x)#?
- What is the antiderivative of #e^x#?
- What is the antiderivative of #n^x#?
- What is a rational function?
- How do I find the integral of a rational function?
- How do I divide one polynomial by another?
- Question #3b716
- What is the constant of integration and why is it so important?
- How do I evaluate constants of integration?
- When integrating by trigonometric substitution, what are some useful identities to know?
- Why do we need to approximate integrals when we can work them out by hand?
- Why is the error of approximation of an integral important?
- How do I integrate with Euler's method by hand?
- How do I find the surface area of a solid of revolution using parametric equations?
- How do you find areas bounded by polar curves using calculus?
- How do I find the surface area of a solid of revolution using polar coordinates?
- How do I find the surface area of the solid defined by revolving #r = 3sin(theta)# about the polar axis?
- How do I determine the volume of the solid obtained by revolving the curve #r=3sin(theta)# around the polar axis?
- How do I determine if the alternating series #sum_(n=1)^oo(-1)^n/sqrt(3n+1)# is convergent?
- How do you know when to use the Root Test for convergence of a series?
- When testing for convergence, how do you determine which test to use?
- What is the radius of convergence?