Calculus
Featured Answers
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
- How do you find the antiderivative of a power series?
- What is a Taylor series?
- What are the most important power series to memorise?
- How do I use a power series to calculate a limit?
- How do I find #lim_(x->oo)(3sin(x))/e^x# using power series?
- What uses do products of power series have?
- What is the link between binomial expansions and Pascal's Triangle?
- If I want to test the series #sum_(n=1)^oo(n^2+2^n)/(1-e^(n+1))# for convergence, what would be the best test to use and why?
- When using integration to find an area, exactly which "area" is found?
- Question #88d66
- Question #8b077
- Question #74cf4
- Question #45b1d
- Question #753ae
- Question #f9d47
- Question #d4260
- Question #b9121
- Question #df498
- Question #17d65
- Question #b8c9a
- Question #782a3
- Question #0ffbf
- Question #102a4
- Question #f9fc8
- Question #b00c7
- Question #1973d
- Question #17c77
- Question #f57de
- Question #ee2f5
- Question #d1deb
- Question #db33c
- Question #bb479
- Question #5f293
- Question #c3696
- Question #6027a
- Question #830d3
- Question #61176
- Question #5c268
- Question #801ba
- Question #c6955
- Question #91801
- Question #58e63
- Question #375c6
- Question #38661
- Question #d1261
- Question #f0469
- Question #86163
- Question #84eab
- Question #df048
- Question #b08a5
- Question #a9115
- Question #27740
- Question #5003d
- Question #21968
- Water is leaking out of an inverted conical tank at the rate of 10,000cm^3/min cm/min at the same time that water is being pumped into the tank at a constant rate. The tank has a height of 6m and the diameter at the top is 4m. If the water level is rising at a rate of 20 cm/min when the height of the water is 2m, how can I find the rate at which water is being pumped into the tank.?
- Question #11039
- Question #492b7
- What is the area bounded by #y_1=x,y_2=2-x,y_3=0#?
- Question #bfb05
- Question #a297a
- Question #a4188
- Question #d2043
- Question #2de8a
- If #h(x)=p(x) * q(x) * r(x)# what is #h'#?
- Question #1092f
- Question #360e8
- What is computing derivative definition and how do you use #f(x+h)-f(x)/h#?
- For the following function: g(x) = 2x + 2 / x^2 - 6x - 7 (a) Find the domain. Answer .................. (b) horizontal asymptotes ? (c) vertical asymptotes ? (d) is there discontinuity?
- For the following function: g(x) = x^2 + 4x - 5 / x^2 + 7x + 10 (a) Find the domain. Answer .................. (b) horizontal asymptotes? (c) vertical asymptotes ? (d) is there discontinuity ?
- Question #a7325
- Question #b676d
- How to find f '(a). f(x) = √ 2-6x ?
- Question #7e240
- Question #4ddba
- Question #fdf1d
- Question #a27d0
- How do I change #int_0^1int_0^sqrt(1-x^2)int_sqrt(x^2+y^2)^sqrt(2-x^2-y^2)xydzdydx# to cylindrical or spherical coordinates?
- Question #25a9e
- Question #9ef51
- Question #7dc4b
- Question #87ae7
- Question #35f3f
- How can you evaluate: #int_0^2(1/(1+x^4))dx# ?
- How would you evaluate #int_0^pi/2 sqrt(sinx)dx# . ?
- For the following function: g(x) = (3x+9) / (x^2-x-12) (a) Find the domain. Answer .................. (b) horizontal asymptotes? (c) vertical asymptotes ? (d) is there discontinuity ?
- Question #7ece0
- How do you integrate #x^3 * sqrt(x^2 + 4) dx#?
- Question #c82fe
- Question #e296f
- Question #e2c4d
- Question #5ab6e
- Question #5ac74
- What is the integral of #(1/sqrt(4-x^2))#?
- Question #1f501
- Question #6d109
- Question #6d140
- What is #lim_(x->0) xlnx - x^2#?
- Question #7515d
- I'm only in algebra II but we're learning the rational root theorem and the factor theorem, apparently those topics fall under calculus. Can someone please explain them to me and provide examples?
- Question #56b43