In this article, we will review the final exam of Calculus 1 in detail and chapterwise. The syllabus of Calculus 1 includes Limit, Continuity, Derivative, Integration, and their applications. Limit Final Exam Review Question 1: Find $\lim\limits_{x \to 2} \dfrac{x^2-5x+6}{x^2-4}$. Solution: At first, we will factorize both the numerator and the denominator. Note that x2-5x+6 … Read more
As an application (or an example) of the intermediate value theorem, we can prove the fixed point theorem (FPT) for continuous function which is given below. Let us first recall the intermediate value theorem. Intermediate Value Theorem: If f(x) is a real-valued continuous function on the closed interval [a, b] with f(a) ≠ f(b), then … Read more
A function is discontinuous if we cannot sketch its graph without lifting the pen. In this article, we will study discontinuous functions with their types, examples, and a few solved problems. At first, we recall the definition of the continuity of a function. A function f(x) is called continuous at x=a if we have limx→af(x) … Read more
By continuity of a function, we mean that we can sketch the graph of the function without lifting the pencil. In this article, we will learn the definition of the continuity of a function along with its properties, examples, and solved problems. Definition of Continuity of a Function Let f(x) be a real-valued function where … Read more
The nth derivative of a function is obtained by the successive differentiation of the same function till n times. The n-th differentiation is referred to the higher order derivatives. In this article, we will learn the definition of the nth derivative along with its formulas, properties, and examples. nth Derivative Definition Let f(x) be a … Read more
The derivative of xlog x is equal to 1+log x. Note that x log x is the product of x with its logarithm. In this post, we will learn how to find the derivative of xlog x. Derivative of xlogx Formula The formula for the derivative of xlogx is given as follows: d(xlogx)/dx = 1+logx. … Read more
The derivative of xex is equal to (x+1)ex. Note that xex is the product of two functions x and ex. In this post, we will find the derivative of xex by the first principle and the product rule of derivatives. Derivative of xex Formula The formula for the derivative of xex is given by d(xex)/dx … Read more
The derivative of xcos x is equal to cosx – xsinx. The function xcos x is the product of x with its cosine. In this article, we will learn how to find the differentiation of xcos x using the following methods: Derivative of xcosx Formula The derivative of xcosx formula is given by d(xcosx)/dx = … Read more
The derivative of 1/x3 is equal to -3/x4. In this article, we will learn how to find the derivative of 1 divided by x3 using the power rule, product rule, and the definition of derivatives. Derivative of 1/x3 Formula The derivative of 1/x3 can be expressed mathematically as d/dx(1/x3) or (1/x3)$’$. The derivative formula of … Read more
The derivative of 1/x2 is equal to -2/x3. In this article, we will learn how to find the derivative of 1 divided by x2 using the power rule, product rule, and the definition of derivatives. Derivative of 1/x2 Formula Note that 1/x2 is an algebraic function. The derivative of 1/x2 can be expressed as d/dx(1/x2) … Read more