Practice
Calculus
Courses
Take a guided, problem-solving based approach to learning Calculus. These compilations provide unique perspectives and applications you won't find anywhere else.
Calculus Fundamentals
What's inside
- Introduction
- Computing Limits
- Derivatives
- Computing Derivatives
- Linear Approximation and Applications
Multivariable Calculus
What's inside
- Introduction
- Vector Bootcamp
- Multivariable Functions
- Limits with Many Variables
- Derivatives
- Optimization
- Multiple Integrals
Differential Equations I
What's inside
- Introduction
- First-Order Separable Equations
- Advanced First-Order Equations
- Basics of Linear Systems
- Higher-Order Equations
Additional Practice
Sharpen your skills with these quizzes designed to check your understanding of the fundamentals.
Sequences and Limits
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Calculus Warmups
The concepts of limits, infinitesimal partitions, and continuously changing quantities paved the way to Calculus, the universal tool for modeling continuous systems from Physics to Economics.
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Sequences and Series
What's the sum of the first 100 positive integers? How about the first 1000?
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Limits of Sequences and Series
Infinitely many mathematicians walk into a bar. The first says "I'll have a beer". The next ones say "I'll have half of the previous guy". The bartender pours out 2 beers and says "Know your limits".
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Limits of Functions
What happens when a function's output isn't calculable directly – e.g., at infinity – but we still need to understand its behavior? That's where limits come in.
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Continuity
If there are no holes in your function, it's continuous! Many powerful theorems in Calculus only apply to these special types of functions.
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Common Misconceptions (Calculus)
How does infinity really work? Is it the biggest number? Is it even a number at all?
Differentiation
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Derivatives
A derivative is simply a rate of change. Whether you're modeling the movement of a particle or a supply/demand model, this is a key instrument of Calculus.
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Differentiation Rules
These are the rules that explain how to take derivatives of any functions: from polynomials to trigonometric functions to logarithms.
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Higher-order Derivatives
The first derivative is the slope of a curve, and the second derivative is the slope of the slope, like acceleration. So what's the third derivative? (Fun fact: it's actually called jerk.)
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Implicit Differentiation
Not all equations can be written like y = f(x), so taking the derivative can be tricky. Save the mess and do it directly with implicit differentiation.
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Differentiability
Those friendly functions that don't contain breaks, bends or cusps are "differentiable". Take their derivative, or just infer some facts about them from the Mean Value Theorem.
Applications of Differentiation
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L'Hôpital's Rule
When you've got a limit that looks like 0/0 or ∞/∞, L'Hôpital's rule can often find its value -- and make it clear that not all infinities are equal!
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Related Rates
How does pressure change in a combustion engine? How fast does the water level rise when filling a pool? Calculus quantifies the impact of change on areas, angles, distances, temperatures, and more.
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Extrema
How can you maximize your happiness under a budget? When does a function reach its minimum value? When does a curve change direction? The calculus of extrema explains these "extreme" situations.
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Curve Sketching
You don't need a calculator or computer to draw your graphs! Derivatives and other Calculus techniques give direct insights into the geometric behavior of curves.
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Displacement, Velocity, Acceleration
How fast are you moving? How fast is how fast you are moving changing? Displacement, velocity, and acceleration form the art of understanding movement, Calculus-style.
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Taylor Series
The Taylor series is a polynomial of infinite degree used to represent functions like sine, cube roots, and the exponential function. They're how some calculators (and Physicists) make approximations.
Integration
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Antiderivatives
This is the opposite of the derivative - and it's an integral part of Calculus. It's uses range from basic integrals to differential equations, with applications in Physics, Chemistry, and Economics.
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Definite Integrals
The definite integral of a function computes the area under the graph of its curve, allowing us to calculate areas and volumes that are not easily done using geometry alone.
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Integration Techniques
Writing an integral down is only the first step. A toolkit of techniques can help find its value, from substitutions to trigonometry to partial fractions to differentiation.
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Area Between Curves
In the language of Calculus, the area between two curves is essentially a difference of integrals. The applications of this calculation range from macroeconomic tax models to signal analysis.
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Advanced Integration
The integrals of these special functions have relationships to probability distributions, factorials, quantum physics, and fluid dynamics.
Applications of Integration
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Volume of Revolution
Integration extends to the third dimension with the means to compute the volumes of shapes obtained by revolving regions around an axis, such as spheres, toroids, and paraboloids.
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Arc Length and Surface Area
Finding the perimeter of arbitrary curves and the area of 3D shapes foils the traditional tools of Geometry, and calls for the help of integrals and derivatives to make these calculations.
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Displacement, Velocity, Acceleration
Derivatives are rates of change, and in the physical world that means things like velocity and acceleration. In fact, studying these quantities played a major role in the invention of Calculus.
Parametric Equations Calculus
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Parametric Equations
As a particle moves, its position can often be written in terms of time. Systems like these can be modeled with parametric equations, which cleverly write coordinates in terms of a parameter.
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Polar Equations
Whether you're charting the seas or just doing some plotting in the complex plane, polar equations are the go-to tool for describing points via a distance and angle with respect to the origin.
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Calculus of Parametric Equations
Parametric equations can be quite handy, and we don't want to unravel them just to do Calculus. Some tricks can bend traditional derivative and integral methods to apply to parametric equations.
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Polar Equations Calculus
Just because your equation is polar doesn't mean you can't do Calculus! Some tricks can extend the toolkit of Calculus to these special equations.
Numerical Methods
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Root Approximation
What are the zeroes of a function? When an exact answer isn't attainable, Calculus provides approximation techniques - the same ones that allow calculators to find your roots.
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Numerical Approximation of Integrals
Whether you want to program a calculator or just approximation an integral on paper, break out some numerical approximation tools like Simpson's Rule and the Trapezoid Rule.
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Numerical Methods for Differential Equations
When differential equations can't be solved explicitly, approximate solutions can be found via clever numerical methods.
Community Wiki
Browse through thousands of Calculus wikis written by our community of experts.
Sequences and Limits
- Sequences
- Series
- Arithmetic Progressions
- Geometric Progressions
- Arithmetic-Geometric Progression
- Telescoping Series - Sum
- Telescoping Series - Product
- Convergence Tests
- Harmonic Number
- Absolutely Convergent
- Sums Of Divergent Series
- Limits of Sequences
- Infimum/Supremum
- Nested Functions
- Dedekind Cuts
- Limits of Functions
- Limits by Substitution
- Limits by Factoring
- Limits by Rationalization
- When Does A Limit Exist?
- Asymptotes
- Continuous Functions
- Epsilon-Delta Definition of a Limit
- Squeeze Theorem
- Extreme Value Theorem
- Intermediate Value Theorem
- Infinity
- Is infinity at the end of the real number line?
- If F(x) is the antiderivative of f(x), is it true that ∫abf(x)dx=F(b)-F(a)?
- Do local extrema occur if and only if f'(x) = 0?
- Is Infinity / Infinity = 1?
- Is infinity times zero = zero?
- What is 1 divided by 0?
- If the limit of a sequence is 0, does the series converge?
Differentiation
- Average and Instantaneous Rate of Change
- Tangent Line to a Curve
- Derivative by First Principle
- Derivatives of Polynomials
- Derivatives of Rational Functions
- Derivatives of Exponential Functions
- Derivatives of Logarithmic Functions
- Partial Derivatives
- Applying Differentiation Rules To Logarithmic Functions
- Power Rule
- Product Rule
- Quotient Rule
- Chain Rule
- Differentiation of Inverse Functions
- Applying Differentiation Rules to Trigonometric Functions
- Calculus With Inverse Trigonometric Functions
- Differentiation Rules
- Higher-order Derivatives
- Increasing / Decreasing Functions
- Inflection Points
- Implicit Differentiation
- Differentiable Function
- Mean Value Theorem
- Rolle's Theorem
Applications of Differentiation
- Indeterminate Forms
- L'Hôpital's Rule
- Related Rates of Change
- Extrema (Local and Absolute)
- Critical Points
- Second Derivative Test
- Optimization
- Lagrange Multipliers
- Vertical Asymptotes
- Average Velocity
- Instantaneous Velocity
- Taylor Series
- Maclaurin Series
- Taylor Series Approximation
- Taylor Series Manipulation
- Interval and Radius of Convergence
- Taylor Series - Error Bounds
- Power Series
- Small-Angle Approximation
- Fourier Series
- Taylor's Theorem (with Lagrange Remainder)
- Analytic Continuation
Integration
- Line Integral
- Integration
- Integration of Algebraic Functions
- Integration of Exponential Functions
- Integration of Trigonometric Functions
- Integration of Rational Functions
- Integration of Logarithmic Functions
- Integration of Radical Functions
- Definite Integrals
- Riemann Sums
- Fundamental Theorem of Calculus
- Improper Integrals
- Multiple Integral
- U-Substitution
- Trigonometric Substitution in Integration
- Integration by Parts
- Differentiation Under the Integral Sign
- Integration Tricks
- Lebesgue Integration
- Stokes' Theorem
- Green’s Theorem
- Area between curves
- Gamma Function
- Beta Function
- Digamma Function
- Riemann Zeta Function
- Cauchy Integral Formula
- Isolated Singularities and Residue Theorem