Calculus Q&A Archives - The Story of Mathematics - A History of Mathematical Thought from Ancient Times to the Modern Day

Find yʹ and yʹʹ. y=xln(x)

In this question, we have to find the first and second derivatives of the given function y=x ln(x) The basic concept behind this question is the knowledge of derivatives and the rules such as the product rule of derivatives and the quotient rule of derivatives. Expert Answer Given function: [y=x ln{ (x)}] For first derivative, […]

Find the curvature of r(t) = 7t, t2, t3 at the point (7, 1, 1).

This question aims to find the curvature of the given equation for the points (7,1,1).This question uses the concept of calculus and curvature. Curvature is used for graphs which tells us how sharply a graph bends. Mathematically it is represented as: [K space= space || space frac{dT}{ds} space ||] Expert Answer We are given the equation: […]

If xy + 3ey = 3e, find the value of y” at the point where x = 0.

This problem aims to familiarize us with higher-order differential equations. The concept required to solve this problem is ordinary differential equations given at a specific point and product rule. Here we are going to find the second-order differential with the help of a reference point. Now, an ordinary differential equation also known as ODE is an […]

Find an equation of the plane tangent to the following surface at the given point:

7xy + yz + 4xz – 48 = 0; ( 2, 2, 2 ) The aim of this question is to understand the partial derivatives of a surface and their significance in terms of finding the tangent planes. Once we have partial derivative equations, we […]

A particle moves along the curve y=2 sin(pi x/2). As the particle passes through the point (1/3, 1), its x-coordinate increases at a rate of sqrt{10} cm/s. How fast is the distance from the particle to the origin changing at this instant?

The question aims to find the rate of change in distance of the particle from the origin as it moves along the given curve and its movement increases. The background concepts needed for this question include basic calculus, which includes derivatives and calculating distance by using distance formula and some trigonometric ratios. Expert Answer The […]