Calculus

Featured Answers

Introduction to Calculus

• What is Calculus?
• Prologue and Historical Context
• Understanding the Gradient function

Limits

• 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

Derivatives

• 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

Basic Differentiation Rules

• 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

Differentiating Trigonometric Functions

• 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

Differentiating Exponential Functions

• From First Principles
• Differentiating Exponential Functions with Calculators
• Differentiating Exponential Functions with Base e
• Differentiating Exponential Functions with Other Bases

Differentiating Logarithmic Functions

• Differentiating Logarithmic Functions with Base e
• Differentiating Logarithmic Functions without Base e
• Overview of Different Functions

Graphing with the First Derivative

• 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

Graphing with the Second Derivative

• 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

Applications of Derivatives

• 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

Introduction to Integration

• 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

Techniques of Integration

• Evaluating the Constant of Integration
• Integration by Substitution
• Integration by Parts
• Integration by Trigonometric Substitution
• Integral by Partial Fractions

Using Integrals to Find Areas and Volumes

• Calculating Areas using Integrals
• Calculating Volume using Integrals
• Deriving Formulae Related to Circles using Integration
• Symmetrical Areas
• Definite Integrals with Substitution

Methods of Approximating Integrals

• Integration Using Euler's Method
• RAM (Rectangle Approximation Method/Riemann Sum)
• Integration Using Simpson's Rule
• Analyzing Approximation Error
• Integration Using the Trapezoidal Rule

Applications of Definite Integrals

• 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

Parametric Functions

• 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

Polar Curves

• 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

Power Series

• 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

Tests of Convergence / Divergence

• 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

Uncategorized Questions

Next
• What is Leibniz Notation ?
• Question #1a2d5
• Question #70fcd
• Question #f387d
• Question #5eee2
• Question #3ee60
• Question #b4d3e
• Question #5b573
• Question #4071a
• Question #babaa
• Question #cb9af
• Question #44d51
• Question #727ae
• Question #db99f
• Question #dc253
• Question #175de
• Question #4b7de
• Question #a180b
• Question #6dcf7
• Question #c1a8e
• Question #59074
• Question #6b550
• Question #4daad
• 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?
• Question #15d7c
• Question #2dca7
• 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?
• Question #5eeee
• 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#?
• Question #64b3c
• Question #35a2f
• Question #4b937
• Question #36b8c
• Question #2577a
• Question #7e44e
• 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?
Next