Calculus is one of, if not the most, important branches of mathematics. There are many important calculus terms that all calculus students need to know. This guide will introduce and explain 8 important terms that are the cornerstones of calculus. Some of these are formulas, while others are concepts and definitions.
Introduction to Calculus Terms
Calculus is rich with terminology that can initially seem daunting. However, understanding these calculus terms is crucial for mastering the subject. From limits and derivatives to integrals, each term plays a pivotal role in the broader calculus framework. This article will go through 8 different important calculus terms and will provide an example and either a definition or statement for each.
Limits
Limits are one of the most fundamental concepts in calculus. A limit is an approximation of the value of a function at a certain point using the values of the function near the point. The limit of a function does not necessarily have to equal the actual value of the function, because a limit is purely an approximation. Limits are essential for defining derivatives and integrals.
Definition: The limit of a function f(x) as x approaches a is $lim_{x→a} f(x)=L$ if for every number $ε > 0$ there exists a number $δ > 0$ such that $|f(x) – L| < ε$ when $|x-a| < δ$. To learn more about delta-epsilon notation, check out this article.
Example: For our example, we are given that $f(x) = x+1$. The limit of $f(x)$ as $x$ approaches 3 is: $\displaystyle \lim_{x \to 3} (x + 1) = 4$.
Derivatives
The derivative measures how a function changes as its input changes. It is a function that is equal to the slope of the tangent line at a given point. Not every function has a derivative, as there is a specific set of criteria that a function must meet to be differentiable. Check out Differentiability Rules to learn more about differentiability rules. Calculus students can use many different differentiation methods, three of which are explained later in this article.
Definition: The derivative of $f(x)$ at $x=a$ is given by: $f′(a)=lim_{h→0}\frac{f(a+h)−f(a)}{h}$
Example: The derivative of $f(x)=x^2$ is: $f′(x)=\frac{d}{dx}(x^2)=2x$.
Integrals
Integrals are another core concept in calculus, often described as the inverse operation of differentiation. Integrals are used to calculate areas under curves. There are two types of integrals, those being definite and indefinite integrals. Definite integrals return a value as output, while indefinite integrals return a new function as output. Not every function has an integral, and to learn more about
Definition: The definite integral of $f(x)$ from $a$ to $b$ is: $\displaystyle\int_{a}^{b} f(x) \,dx$. The indefinite integral of $f(x)$ from $a$ to $b$ is: $\displaystyle\int f(x) \,dx$.
Example: The integral of $f(x)=x$ from 0 to 2 is: $\int_0^2 x dx=[\frac{x^2}{2}]_0^2=2$
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus links the concepts of differentiation and integration and shows that they are inverse processes. To learn more about the first fundamental theorem of calculus, check out First Fundamental Theorem of Calculus . To learn more about the second fundamental theorem of calculus, check out xxxx. Also, make sure to read about the inconsistency in the naming of the parts of this important theorem here.
Statement: The theorem has two parts, often called the first and second fundamental theorems of calculus.
The first part of the theorem states that if $f(x)$ is continuous on $[a,b]$, and $F(x) = \int_{a}^{x} f(t) \, dt$, then $F’(x) = f(x)$.
The second part of the theorem states that if $F(x)$ is an antiderivative of $f(x)$ on an interval $[a,b]$, then $\int_{a}^{b} f(x) \, dx = F(b) – F(a)$.
Chain Rule
The Chain Rule is a formula for computing the derivative of the composition of two functions.
Statement: If $g(x)$ and $f(x)$ are functions, then the derivative of $(f \circ g)(x))$, which is their composition, is: $(f \circ g)'(x) = f'(g(x)) \cdot g'(x)$.
Example: If $f(x) = 3x^2 + 2x$ and $g(x) = x^3$, then $f’(x) = 6x + 2$ and $g’(x) = 3x^2$ because of the power rule. Substituting this in, we find $(f \circ g)’(x) = (6(x^3) + 2) \cdot 3x^2 = 18x^5 + 6x^2$.
Product Rule
The Product Rule is used to find the derivative of the product of two functions.
Statement: If $f(x)$ and $g(x)$ are differentiable functions, then: $(fg)′=f′g+fg′$.
Example: If $f(x) = x^2$ and $g(x)=e^x$, then: (f \cdot g)'(x)= (x^2)’e^x + x^2(e^x)’ = 2xe^x + x^2e^x.
Quotient Rule
The Quotient Rule is used to find the derivative of two functions that are in the form $\dfrac{f(x)}{g(x)}$.
Statement: If $f(x)$ and $g(x)$ are differentiable functions, then $\left( \frac{f}{g} \right)’ = \frac{f’g – fg’}{g^2}$
Example: To find the derivative of the function $\frac{x^2}{\sin(x)}$, we will use the quotient rule, with $f(x) = x^2$ and $g(x)=\sin (x)$. We find that $\left( \frac{x^2}{\sin(x)} \right)’ = \frac{2x\sin(x) – x^2\cos(x)}{\sin^2(x)}$.
Implicit Differentiation
Implicit Differentiation is a differentiation technique that uses the chain rule to differentiate equations that cannot be written in the form $y = f(x)$.
Definition: To differentiate an implicit function, take the derivative of both sides of the equation with respect to x, treating y as a function of x.
Example: For the equation $x^2+y^2=1$, differentiate both sides: $2x+2y \cdot \frac{dy}{dx}=0$
Solving for $\frac{dy}{dx}$, we find that $\frac{dy}{dx} = -\frac{x}{y}$.
Conclusion
To conclude, calculus has many important terms and concepts that should be known by all students. Proper knowledge and understanding of these terms are necessary to excel in calculus. While this article only gives a brief overview of each topic, there are many more articles linked in the text that can further improve your understanding.
