"Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations."
The mathematical study of change and motion, including derivatives, integrals, and limits.
Limits: An introduction to the idea of limits, including finding limits at infinity and limits from the left and right.
Continuity: An explanation of what it means for a function to be continuous, both at a point and on an interval.
Derivatives: The concept of taking the derivative of a function, which gives information about its rate of change at each point.
Differentiation Rules: Differentiation rules, including the power rule, product rule, quotient rule, and chain rule.
Applications of Derivatives: Looking at applications of derivatives, including curve sketching, optimization, and related rates problems.
Integration: An introduction to the process of integration, including finding antiderivatives and evaluating definite integrals.
Integration Techniques: Techniques for evaluating integrals, including u-substitution, integration by parts, and partial fractions.
Applications of Integrals: Applications of integrals, including finding areas, volumes, and average values.
Differential Equations: An introduction to differential equations, which model many real-world phenomena.
Multivariable Calculus: Working with calculus in two or more dimensions, including partial derivatives, gradients, and optimization in multiple variables.
Vector Calculus: An introduction to vector calculus, which includes topics such as vector fields, line integrals, and flux.
Series and Sequences: An introduction to series and sequences, including the convergence and divergence of infinite series.
Taylor Series: A deeper look at Taylor series, which allow functions to be approximated by polynomials.
Fourier Series: An introduction to Fourier series, which represent periodic functions as a sum of sine and cosine functions.
Laplace Transform: An introduction to the Laplace transform, which is a powerful tool for solving differential equations.
Differential Calculus: This branch of calculus deals with the study of how things change. The derivative is a key concept in differential calculus, and it is used to find the instantaneous rate of change of a function. Many physical phenomena can be modeled using differential calculus, such as the motion of objects under the influence of gravity.
Integral Calculus: This branch of calculus deals with the study of how things accumulate. The integral is a key concept in integral calculus, and it is used to find the total amount of something over a range of values. Integral calculus is used in many areas of physics, such as finding the total amount of work done by a force acting over a distance.
Vector Calculus: This branch of calculus deals with the study of vector fields. Vector calculus is used to study phenomena that have both magnitude and direction, such as fluid flow and electrical fields. Key concepts in vector calculus include the gradient, divergence, and curl of a vector field. Vector calculus is used extensively in fields like electromagnetism and fluid mechanics.
"It has two major branches, differential calculus and integral calculus..."
"The former concerns instantaneous rates of change, and the slopes of curves..."
"The latter concerns accumulation of quantities, and areas under or between curves."
"These two branches are related to each other by the fundamental theorem of calculus..."
"They make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit."
"Infinitesimal calculus was developed independently in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz."
"Later work, including codifying the idea of limits, put these developments on a more solid conceptual footing."
"Today, calculus has widespread uses in science, engineering, and social science."
"...geometry is the study of shape..."
"The former concerns instantaneous rates of change, and the slopes of curves, while the latter concerns accumulation of quantities, and areas under or between curves."
"These two branches are related to each other by the fundamental theorem of calculus..."
"Infinitesimal calculus was developed independently in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz."
"They make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit."
"Later work, including codifying the idea of limits, put these developments on a more solid conceptual footing."
"Today, calculus has widespread uses in science, engineering, and social science."
"It has two major branches, differential calculus and integral calculus..."
"The former concerns instantaneous rates of change, and the slopes of curves..."
"The latter concerns accumulation of quantities, and areas under or between curves."
"These two branches are related to each other by the fundamental theorem of calculus..."