What is Calculus?

We define calculus as the study of rates of continuous change, especially instantaneous change or change over short time intervals. Calculus enables individuals to graph and create models of change. It also helps in making predictions of future change. Mathematical operations are used to study and model rates of change. It also helps determine the consequences of change.

Calculus developed over many centuries. Eudoxus and Archimedes in ancient Greece, as well as mathematicians in ancient China, helped lay the foundations of calculus. By the 15th century, Indian mathematicians were using some calculus-type processes. In more recent times, Johannes Kepler contributed to the understanding of advanced math. Isaac Newton and Gottfried Leibniz then developed modern calculus. Newton focused on the geometric aspects of calculus, while Leibniz concentrated on analysis and created a notational system.

Key terms to understand in calculus math are function, derivative, and integral. Functions show relationships or connections between variables. Each input will result in one output. In calculus, they describe motion. Derivatives demonstrate the rate of change with respect to a variable. Integrals deal with areas. The two primary types of calculus are differential calculus and integral calculus. Basic calculus deals with single variables, while more advanced calculus studies the impact of multiple variables.

Calculus is commonly used in physics, engineering, medicine, economics, and computer science. It is also used in navigation and space travel. It is a common math requirement in many college programs and utilizes all the math skills students have previously developed, including those in algebra, geometry, and trigonometry.

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Calculus Topics

Convergence Tests

Learn what a convergence test is. See how they differ from divergence tests and examine the different types, including p-series tests and geometric series tests.

Curve sketching

Learn about curve sketching in calculus and understand how it is used. Explore the process of curve sketching with steps and see curve sketching examples.

Differential calculus

This lesson explores differential calculus. It defines a differential and delves into the many uses of differential equations.

Differentiation rules

Learn about differentiation rules, or derivative rules, and understand how they are used. Explore basic derivative rules in calculus and study some examples.

History of calculus

Learn to define calculus. Explore the history of the discovery of calculus and examine the two mathematicians who chiefly contributed to the development of calculus.

Integral calculus

Understand that an integral measures the area under a curve, and learn how to evaluate linear and polynomial integrals. Explore different applications of integrals with examples.

Integration techniques

Learn about integration techniques in calculus and understand how they are used. Explore the different types of integration methods and study some examples.

Limit (mathematics)

How To Find Limit of a Function | Limit Types & Examples

Linear approximations

Read about the concept of linear approximation. See a derivation of the linearization formula and some of its applications to learn how to use the linear approximation formula.

Mathematical series

Explore the difference between a sequence and a series in mathematics. Understand how to evaluate the sum of finite and infinite series with different examples.

Calculus Resources

Many resources are available to help students study and understand precalculus as well as problems in differential calculus and topics in integral calculus. Whether students are in high school or college or pursuing independent learning goals, Study.com has the tools to make these advanced math concepts accessible. There are full classes and skill practice provided. Students who need additional help can access our 24/7 tutoring service. Experts in math are eager to help students with their higher math problems. Students are sure to find what they need to help them succeed in their study of higher math.

Calculus Courses

Study.com's collection of precalculus and calculus courses is designed for high school or college students as well as independent learners who want to learn more about higher math. Teachers may also find these courses helpful for reviewing or developing new skills. They could also be used to create supplementary lessons for existing precalculus, differential calculus, or integral calculus classes. Students who pursue these classes can rest assured that they will have the skills they need to meet high school or college requirements and a foundation to study higher-level math courses.

Calculus Test Prep for Teachers and Students

Calculus Test Prep

Students and prospective teachers can take advantage of Study.com's calculus test prep courses. Students hoping to earn college credit can study using the CLEP, AP Calculus AB, AP Calculus BC, or AP Calculus AB & BC courses. Prospective teachers planning to teach in Georgia can prepare for the GACE Math Test 1 while those hoping to teach in California can take advantage of the CSET Math Subtest Prep.

  • AP Calculus AB: Exam Prep

    Prepare to earn college credit via tests with these reviews of differential and integral calculus including derivatives, integrals, and applications.

  • AP Calculus BC: Exam Prep

    Study for the AP Calculus BC exam by reviewing key topics in differential and integral calculus, such as derivatives and basic integration.

  • AP Calculus AB & BC: Exam Prep

    Review important concepts in calculus, including functions, derivatives, and integration, to get ready for the AP Calculus AB & BC exam.

For Teachers

Calculus Skills and Practice

Study.com's precalculus and calculus courses, as well as additional skills practice, provides students with the tools they need to succeed in higher math. Some of these courses and practices are designed to meet common high school academic standards, while others are tailored to expectations for college-level work.

Those who complete the precalculus courses will be able to:

  • Graph various types of functions and their transformations
  • Prove polynomial identities
  • Know and apply the Binomial Theorem
  • Understand and rewrite rational expressions
  • Solve logarithmic and exponential equations
  • Understand continuity and discontinuity
  • Analyze characteristics of functions and asymptotes
  • Make connections between algebraic and geometric concepts

Students will have a firm grasp of these concepts and be able to apply them to their current coursework or be ready to continue their mathematical studies with calculus.

Calculus students will have a firm foundation in both differential and integral calculus. By working through various calculus examples and problems, students will:

  • Understand and define limits as well as estimate limits and find them by substitution
  • Work with continuity and discontinuity
  • Understand maximum and minimum values
  • Build functions to model relationships
  • Compose and combine functions
  • Find and work with inverse functions
  • Produce invertible functions
  • Understand the unit circle
  • Tackle trigonometric functions and equations
  • Prove and apply trigonometric identities
  • Express geometric properties with an equation
  • Work with and understand complex numbers and vectors

The links in this section provide many practice problems geared to meeting specific math standards in both Florida and Texas. Many other states have similar standards, so the practice skills are easily transferable regardless of the student's location. These practice equations and problems allow students to practice key skills or to work on areas in which they may need extra help. Step-by-step guides on how to solve the equations will help students develop a better understanding of how to answer problems. Teachers may also find these practice problems helpful in building their own skills and understanding as well as developing material for their courses.

With the skills and practice that Study.com offers, students who wish to pursue higher math studies beyond calculus will be well-prepared to do so.

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Calculus Homeschool Curriculum

Advanced homeschooled high school students can take advantage of Study.com's excellent courses in precalculus and calculus. High School Precalculus: Homeschool Curriculum provides a solid foundation for the study of higher math. Students study graphing with functions, working with inequalities and complex numbers, polynomial functions, trigonometry, vectors, matrices, analytic geometry, limits, and more. They can then prepare for the AP Calculus AB or BC exams with our AP Calculus AB & BC: Homeschool Curriculum(info), which covers both differential and integral calculus. Topics included are graphing and functions, limits, continuity, exponents and polynomials, derivatives, integrals, and rate of change. After completing these courses, students will be well-prepared for the AP exams and to tackle college-level math.

Homeschool solutions
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Frequently Asked Questions

  • What is calculus really used for?

    Calculus has many applications. It is used in the construction of buildings and bridges, and meteorologists use it to help forecast the weather. Biologists employ calculus to track rates of change in bacteria and cellular growth, and it is also widely used in the computer field.

  • What math is used in calculus?

    Students studying calculus will want a solid understanding and knowledge of all math studies that usually come before it in an academic program. These include geometry, algebra, and trigonometry. Understanding quantitative reasoning and functions are also important.