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• Number Theory
• Probability
• Everyday Math
• Classical Mechanics
• Electricity and Magnetism
• Computer Science
• Quantitative Finance

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
• Introduction to Calculus

Multivariable Calculus

• Vector Bootcamp
• Multivariable Functions
• Limits with Many Variables
• Optimization
• Multiple Integrals

Differential Equations I

• First-Order Separable Equations
• Advanced First-Order Equations
• Basics of Linear Systems
• Higher-Order Equations

Community Wiki

Browse through thousands of Calculus wikis written by our community of experts.

Sequences and Limits

• 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?
• Continuous Functions
• Epsilon-Delta Definition of a Limit
• Squeeze Theorem
• Extreme Value Theorem
• Intermediate Value Theorem
• Is infinity at the end of the real number line?
• If F(x) is the antiderivative of f(x), is it true that \(\int_a^b\)f(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
• Product Rule
• Quotient 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
• 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 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

Applications of Integration

• Disc Method
• Shell Method
• Pappus's Centroid Theorems

Parametric Equations Calculus

• Parametric Equations
• Polar Coordinates
• Converting Polar Coordinates to Cartesian
• Polar Curves
• Parametric Derivative
• Parametric Equations - Velocity and Acceleration
• Polar Equations - Area
• Differential Equations
• Separable Differential Equations
• Homogeneous Linear Differential Equations
• Euler's Method
• Differential Equations - Euler's Method - Small Step Size
• Logistic Differential Equations
• Bernoulli Equation
• Systems of Linear Differential Equations
• Chaos Theory

Numerical Methods

• Root Approximation - Bisection
• Newton Raphson Method
• Integral Approximation - Trapezium Rule
• Integral Approximation - Simpson's Rule
• Chebyshev's Formula

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Unit 1: Limits and continuity

Unit 2: derivatives: definition and basic rules, unit 3: derivatives: chain rule and other advanced topics, unit 4: applications of derivatives, unit 5: analyzing functions, unit 6: integrals, unit 7: differential equations, unit 8: applications of integrals.

Calculus Questions, Answers and Solutions

Calculus questions with detailed solutions are presented. The questions are about important concepts in calculus.

Calculus Concepts Questions

• Questions and Answers on Functions . A set of questions on the concepts of a function, in calculus, are presented along with their answers and solutions.
• Properties of the Graphs of Functions . Questions designed to help you gain deep understanding of the properties of the graphs of functions which are of major importance in calculus.
• Optimization Problems for Calculus 1 with detailed solutions.
• Calculus 1 Practice Question with detailed solutions.
• Antiderivatives in Calculus . Questions on the concepts and properties of antiderivatives in calculus are presented.
• Fundamental Theorems of Calculus . Questions on the two fundamental theorems of calculus are presented.
• Questions and Answers on Derivatives in Calculus . A set of questions on the concepts of the derivative of a function in calculus are presented with their answers and solutions.
• Computation and Properties of the Derivative in Calculus . Questions on the computation and properties of the derivative of a function in calculus are presented. These questions have been designed to help you gain deep understanding of the properties of the first derivative. Answers to the questions are also presented.
• Applications of Derivatives . Questions and answers on the applications of the first derivative are presented. These questions have been designed to help you understand the applications of derivatives in calculus.
• Critical Numbers of Functions . Questions on the critical numbers of functions are presented. The present questions have been designed to help you better understand the concept of a critical number of a function as defined in calculus. Answers to these questions are also presented.
• Questions and Answers on Limits in Calculus . A set of questions on the concepts of the limit of a function in calculus are presented along with their answers.
• Questions and Answers on Continuity of Functions . Questions on the concepts of continuity and continuous functions in calculus are presented along with their answers.

Calculus Analytical Questions

• Questions on Inverse Functions with Solutions . Questions on inverse functions are presented along with detailed solutions and explanations.
• Express a Function as the Sum of an Even and an Odd Functions . Show that any function f may be expressed as the sum of an even and an odd functions.
• Derivative of Even and Odd Functions . Questions, with answers, explanations and proofs, on derivatives of even and odd functions are presented.
• Calculus Questions with Answers (1) . The uses of the first and second derivative to determine the intervals of increase and decrease of a function, the maximum and minimum points, the interval(s) of concavity and points of inflections are discussed.
• Calculus Questions with Answers (2) . The behaviors and properties of functions, first derivatives and second derivatives are studied graphically .discussed.
• Calculus Questions with Answers (3) . Approximate graphically the first derivative of a function from its graph. Questions are presented along with solutions.
• Calculus Questions with Answers (4) . Calculus questions, on differentiable functions, with detailed solutions are presented. We first present two important theorems on differentiable functions that are used to discuss the solutions to the questions.
• Calculus Questions with Answers (5) . Calculus questions, on tangent lines, are presented along with detailed solutions.
• Questions with Answers on the Second Fundamental Theorem of Calculus . Questions with detailed solutions on the second theorem of calculus are presented.
• Questions on Functions (with Solutions) . Several questions on functions are presented and their detailed solutions discussed.
• Questions on Composite Functions with Solutions . Questions on composite functions are presented along with their detailed solutions.
• Questions on Concavity and Inflection Points . Questions with detailed solutions on concavity and inflection point of graphs of functions.
• Derivatives in Calculus: Questions with Solutions . Questions on derivatives of functions are presented and their detailed solutions discussed.

More References and links on Calculus

• Calculus Tutorials and Problems .

• Solve equations and inequalities
• Simplify expressions
• Factor polynomials
• Graph equations and inequalities
• Advanced solvers
• All solvers
• Arithmetics
• Determinant
• Percentages
• Scientific Notation
• Inequalities

The calculus section of QuickMath allows you to differentiate and integrate almost any mathematical expression.

What is calculus?

Calculus is a vast topic, and it forms the basis for much of modern mathematics. The two branches of calculus are differential calculus and integral calculus.

Differential calculus is the study of rates of change of functions. At school, you are introduced to differential calculus by learning how to find the derivative of a function in order to determine the slope of the graph of that function at any point.

Integral calculus is often introduced in school in terms of finding primitive functions (indefinite integrals) and finding the area under a curve (definite integrals).

Differentiate

The differentiate command allows you to find the derivative of an expression with respect to any variable. In the advanced section, you also have the option of specifying arbitrary functional dependencies within your expression and finding higher order derivatives. The differentiate command knows all the rules of differential calculus, including the product rule, the quotient rule and the chain rule.

Go to the Differentiate page

The integrate command can be used to find either indefinite or definite integrals. If an indefinite integral (primitive function) is sought but cannot be found for a particular function, QuickMath will let you know. Definite integrals will always be given in their exact form when possible, but failing this QuickMath will use a numerical method to give you an approximate value.

Go to the Integrate page

The Fundamental Theorem of Calculus

Integrals were evaluated in the previous tutorial by identifying the integral with an appropriate area and then using methods from geometry to find the area. This procedure will succeed only for very simple integrals. The main result of this section, the fundamental theorem of calculus, includes a very important formula for evaluating integrals. This theorem shows us how to evaluate integrals by first evaluating antiderivatives. The theorem establishes an amazing relationship between the integral, which may be interpreted as an area, and the antiderivative, which is inversely related to the derivative; that is, it relates area and the derivative.

Let f be a continuous function on [a, b ], and define a function F by

The following theorem is called the fundamental theorem and is a consequence of Theorem 1 . The Fundamental Theorem of Calculus

0 = F (a) = G (a) + c

Math Topics

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Practice Test

For the following exercises, determine whether each of the following relations is a function.

y = 2 x + 8 y = 2 x + 8

{ ( 2 , 1 ) , ( 3 , 2 ) , ( − 1 , 1 ) , ( 0 , − 2 ) } { ( 2 , 1 ) , ( 3 , 2 ) , ( − 1 , 1 ) , ( 0 , − 2 ) }

For the following exercises, evaluate the function f ( x ) = − 3 x 2 + 2 x f ( x ) = − 3 x 2 + 2 x at the given input.

f ( −2 ) f ( −2 )

f ( a ) f ( a )

Show that the function f ( x ) = − 2 ( x − 1 ) 2 + 3 f ( x ) = − 2 ( x − 1 ) 2 + 3 is not one-to-one.

Write the domain of the function f ( x ) = 3 − x f ( x ) = 3 − x in interval notation.

Given f ( x ) = 2 x 2 − 5 x , f ( x ) = 2 x 2 − 5 x , find f ( a + 1 ) − f ( 1 ) . f ( a + 1 ) − f ( 1 ) .

Graph the function f ( x ) = { x + 1    if − 2 < x < 3     − x     if   x ≥ 3 f ( x ) = { x + 1    if − 2 < x < 3     − x     if   x ≥ 3

Find the average rate of change of the function f ( x ) = 3 − 2 x 2 + x f ( x ) = 3 − 2 x 2 + x by finding f ( b ) − f ( a ) b − a . f ( b ) − f ( a ) b − a .

For the following exercises, use the functions f ( x ) = 3 − 2 x 2 + x  and  g ( x ) = x f ( x ) = 3 − 2 x 2 + x  and  g ( x ) = x to find the composite functions.

( g ∘ f ) ( x ) ( g ∘ f ) ( x )

( g ∘ f ) ( 1 ) ( g ∘ f ) ( 1 )

Express H ( x ) = 5 x 2 − 3 x 3 H ( x ) = 5 x 2 − 3 x 3 as a composition of two functions, f f and g , g , where ( f ∘ g ) ( x ) = H ( x ) . ( f ∘ g ) ( x ) = H ( x ) .

For the following exercises, graph the functions by translating, stretching, and/or compressing a toolkit function.

f ( x ) = x + 6 − 1 f ( x ) = x + 6 − 1

f ( x ) = 1 x + 2 − 1 f ( x ) = 1 x + 2 − 1

For the following exercises, determine whether the functions are even, odd, or neither.

f ( x ) = − 5 x 2 + 9 x 6 f ( x ) = − 5 x 2 + 9 x 6

f ( x ) = − 5 x 3 + 9 x 5 f ( x ) = − 5 x 3 + 9 x 5

f ( x ) = 1 x f ( x ) = 1 x

Graph the absolute value function f ( x ) = − 2 | x − 1 | + 3. f ( x ) = − 2 | x − 1 | + 3.

Solve | 2 x − 3 | = 17. | 2 x − 3 | = 17.

Solve − | 1 3 x − 3 | ≥ 17. − | 1 3 x − 3 | ≥ 17. Express the solution in interval notation.

For the following exercises, find the inverse of the function.

f ( x ) = 3 x − 5 f ( x ) = 3 x − 5

f ( x ) = 4 x + 7 f ( x ) = 4 x + 7

For the following exercises, use the graph of g g shown in Figure 1 .

On what intervals is the function increasing?

On what intervals is the function decreasing?

Approximate the local minimum of the function. Express the answer as an ordered pair.

Approximate the local maximum of the function. Express the answer as an ordered pair.

For the following exercises, use the graph of the piecewise function shown in Figure 2 .

Find f ( 2 ) . f ( 2 ) .

Find f ( −2 ) . f ( −2 ) .

Write an equation for the piecewise function.

For the following exercises, use the values listed in Table 1 .

Find F ( 6 ) . F ( 6 ) .

Solve the equation F ( x ) = 5. F ( x ) = 5.

Is the graph increasing or decreasing on its domain?

Is the function represented by the graph one-to-one?

Find F − 1 ( 15 ) . F − 1 ( 15 ) .

Given f ( x ) = − 2 x + 11 , f ( x ) = − 2 x + 11 , find f − 1 ( x ) . f − 1 ( x ) .

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Access for free at https://openstax.org/books/precalculus/pages/1-introduction-to-functions

• Authors: Jay Abramson
• Publisher/website: OpenStax
• Book title: Precalculus
• Publication date: Oct 23, 2014
• Location: Houston, Texas
• Book URL: https://openstax.org/books/precalculus/pages/1-introduction-to-functions
• Section URL: https://openstax.org/books/precalculus/pages/1-practice-test

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Differential Calculus Questions

Differential calculus questions with solutions are provided for students to practise differentiation questions. Differential calculus is a branch of Calculus in mathematics that studies the instantaneous rate of change in a function corresponding to a given input value. Geometrically, it represents the slope of the tangent line to the graph of the function at a given particular point, provided that the function exists and is differentiable at that point. The derivative of a real-valued function at a given point in its domain represents the closest linear approximation of that function at that point.

Learn more about differential calculus in maths .

In the above figure, y = f(x) is a continuous and differentiable function between x and x + h, then derivative of y, dy/dx is the slope of the tangent to the graph of y at x.

Then, the derivative of a function is defined as:

Derivatives of Some Functions in Differential Calculus

Some rules of differential calculus.

Let us study some essential rules that we are going to require while differentiating any function.

• Differential of addition or subtraction of functions
• Differential of product of functions
• Quotient rule of differentiation
• Chain rule of differentiation

Let y = f(u) and u = g(x), then

Watch the Video on Theorems of Differentiation

Differential Calculus Questions with Solutions

Solve the following differential calculus question and check your solution with the one given here. Practising these questions will improve your understanding of differentiation and help you score better in examinations. These questions are provided keeping in view the syllabus of Classes XI and XII.

Question 1:

Differentiate the following functions with respect to x:

(ii) cos x 3

(iii) x 3 + tan x

(i) Let f(x) = sin 4x, put 4x = t, then

(ii) Let f(x) = cos x 3 , put x 3 = u, then

(iii) Let h(x) = x 3 + tan x, then

Question 2:

If f(x) = [cos x –sin x]/[cos x + sin x], then prove that f’(x) + [f(x)] 2 = –1.

Divide both the denominator and numerator by cos x, and we get

f(x) = [1 – tan x]/[1 + tan x] = tan ( 𝜋/4 – x)

And [f(x)] 2 = tan 2 ( 𝜋/4 – x)

f’(x) + [f(x)] 2 = –sec 2 ( 𝜋/4 – x) + tan 2 ( 𝜋/4 – x) = –sec 2 ( 𝜋/4 – x) + sec 2 ( 𝜋/4 – x) –1 = –1

∴ f’(x) + [f(x)] 2 = –1

Question 3:

Differentiate with respect to x: (2x + 1)/(2x + 3)

Let y = (2x + 1)/(2x + 3) = 1 – 2/(2x + 3)

Question 4:

If y = tan –1 [(4x)/(1 + 5x 2 )] + tan –1 [(2 + 3x)/(3 – 2x)], find dy/dx.

Differentiating both sides with respect to x, we get

Question 5:

Differentiate the following function:

Differentiating with respect to x, we get

Also check:

• Maxima and Minima
• Application of Derivatives
• Increasing and Decreasing Function
• Linear Approximations

Question 6:

Find the differential of the following exponential and logarithmic functions:

(i) 5 x – 3 cos x + log x

(ii) e x sec x

(iii) 3 x /(2 + sin x)

(i) Let y = 5 x – 3 cos x + log x

dy/dx = 5 x log 5 + 3 sin x + 1/x.

(ii) Let y = e x sec x

dy/dx = d/dx [e x sec x] = sin x {d/dx (e x } + e x {d/dx (sec x)}

= e x sec x + e x. sec x tan x

= e x sec x (tan x + 1)

(iii) Let y = 3 x /(2 + sin x)

Question 7:

Differentiate x 3 /(1 – x 3 ) with respect to x 3 .

Let y = x 3 /(1 – x 3 ) and put t = x 3 , thus we have to find the value of dy/dt

And dt/dx = 3x 2

Question 8:

Differentiate with respect to x:

Taking logarithm on both sides, we get

Check out: Derivative calculator

Question 9:

Given function

Prove that differential of y with respect to x is

Taking logarithms on both sides, we get

log y = y x log x, again taking logarithm on both sides, we have

log (log y) = x log y + log (log x)

Let us differentiate both sides with respect x; we get

Question 10:

If x = sin u and y = sin bu, where b is any real constant. Prove that

First, we shall determine dx/du and dy/du. Thus, differentiating both x and y with respect to u, we get

On squaring both sides, we get,

Differentiating both sides of (i) with respect to x, we get

Dividing both sides by 2(dy/dx), we get,

Recommended Video

Practice Questions on Differential Calculus Questions

1. Find the differential of the following functions with respect to x.

2. Find the derivative of sin (sin x 3 ) at x = 𝜋/2.

3. Find the second order derivative with respect to x of the function x 3 + 24xy + y 3 = 8.

4. If y = 𝛼e ax + 𝛽e –ax , prove that y” – a 2 y = 0.

5. Differentiate with respect to x: tan (x x ).

Download BYJU’S – The Learning App to solve more practice questions on various concepts of higher mathematics with proper explanations, solved examples and video lessons. Register yourself today!

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Computer Science > Computation and Language

Title: large language models are unconscious of unreasonability in math problems.

Abstract: Large language models (LLMs) demonstrate substantial capabilities in solving math problems. However, they tend to produce hallucinations when given questions containing unreasonable errors. In this paper, we study the behavior of LLMs when faced with unreasonable math problems and further explore their potential to address these problems. First, we construct the Unreasonable Math Problem (UMP) benchmark to examine the error detection ability of LLMs. Experiments show that LLMs are able to detect unreasonable errors, but still fail in generating non-hallucinatory content. In order to improve their ability of error detection and correction, we further design a strategic prompt template called Critical Calculation and Conclusion(CCC). With CCC, LLMs can better self-evaluate and detect unreasonable errors in math questions, making them more reliable and safe in practical application scenarios.

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Netflix's hit sci-fi series '3 Body Problem' is based on a real math problem that is so complex it's impossible to solve

• The three-body problem is a centuries-old physics question that puzzled Isaac Newton .
• It describes the orbits of three bodies, like planets or stars, trapped in each other's gravity.
• The problem is unsolvable and led to the development of chaos theory.

While Netflix's "3 Body Problem" is a science-fiction show, its name comes from a real math problem that's puzzled scientists since the late 1600s.

In physics, the three-body problem refers to the motion of three bodies trapped in each other's gravitational grip — like a three-star system.

It might sound simple enough, but once you dig into the mathematics, the orbital paths of each object get complicated very quickly.

Two-body vs. three- and multi-body systems

A simpler version is a two-body system like binary stars. Two-body systems have periodic orbits, meaning they are mathematically predictable because they follow the same trajectory over and over. So, if you have the stars' initial positions and velocities, you can calculate where they've been or will be in space far into the past and future.

However, "throwing in a third body that's close enough to interact leads to chaos," Shane Ross, an aerospace and ocean engineering professor at Virginia Tech, told Business Insider. In fact, it's nearly impossible to precisely predict the orbital paths of any system with three bodies or more.

While two orbiting planets might look like a ven diagram with ovular paths overlapping, the paths of three bodies interacting often resemble tangled spaghetti. Their trajectories usually aren't as stable as systems with only two bodies.

All that uncertainty makes what's known as the three-body problem largely unsolvable, Ross said. But there are certain exceptions.

The three-body problem is over 300 years old

The three-body problem dates back to Isaac Newton , who published his "Principia" in 1687.

In the book, the mathematician noted that the planets move in elliptical orbits around the sun. Yet the gravitational pull from Jupiter seemed to affect Saturn's orbital path.

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The three-body problem didn't just affect distant planets. Trying to understand the variations in the moon's movements caused Newton literal headaches, he complained.

But Newton never fully figured out the three-body problem. And it remained a mathematical mystery for nearly 200 years.

In 1889, a Swedish journal awarded mathematician Henri Poincaré a gold medal and 2,500 Swedish crowns, roughly half a year's salary for a professor at the time, for his essay about the three-body problem that outlined the basis for an entirely new mathematical theory called chaos theory .

According to chaos theory, when there is uncertainty about a system's initial conditions, like an object's mass or velocity, that uncertainty ripples out, making the future more and more unpredictable.

Think of it like taking a wrong turn on a trip. If you make a left instead of a right at the end of your journey, you're probably closer to your destination than if you made the mistake at the very beginning.

Can you solve the three-body problem?

Cracking the three-body problem would help scientists chart the movements of meteors and planets, including Earth, into the extremely far future. Even comparatively small movements of our planet could have large impacts on our climate, Ross said.

Though the three-body problem is considered mathematically unsolvable, there are solutions to specific scenarios. In fact, there are a few that mathematicians have found.

For example, three bodies could stably orbit in a figure eight or equally spaced around a ring. Both are possible depending on the initial positions and velocities of the bodies.

One way researchers look for solutions is with " restricted " three-body problems, where two main bodies (like the sun and Earth) interact and a third object with much smaller mass (like the moon) offers less gravitational interference. In this case, the three-body problem looks a lot like a two-body problem since the sun and Earth comprise the majority of mass in the system.

However, if you're looking at a three-star system, like the one in Netflix's show "3 Body Problem," that's a lot more complicated.

Computers can also run simulations far more efficiently than humans, though due to the inherent uncertainties, the results are typically approximate orbits instead of exact.

Finding solutions to three-body problems is also essential to space travel, Ross said. For his work, he inputs data about the Earth, moon, and spacecraft into a computer. "We can build up a whole library of possible trajectories," he said, "and that gives us an idea of the types of motion that are possible."

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