Find the planes tangent to the following surfaces at the indicated points.

– $x^2 + 2y^2 + 3xz = 1-$, at the point $(1, 2, dfrac{1}{3})$ – $y^2 – x^2 = 3$, at the point (1,2,8) This problem aims to find the 2D planes that are tangent to the given surfaces. To better understand the problem, you must be familiar with tangents, normal lines, and linear approximation techniques. […]

A piece of wire 10 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be cut so that the total area enclosed is a maximum?

This question aims to find the total area enclosed by a wire when it is cut down into two pieces. This question uses the concept of the area of a rectangle and an equilateral triangle. The area of a triangle is mathematically equal to: [Area space of space triangle space = space frac{Base space times space […]

Which of these functions from R to R are bijections?

$f(x)=-3x+4$ $f(x)=-3x^2+7$ $f(x)=dfrac{x+1}{x+2}$ $f(x)=x^5+1$ This question aims to identify the bijective functions from the given list of functions. In mathematics, functions are the foundation of calculus representing various kinds of relationships.  A function is a rule, expression, or law that specifies an association between a variable known as an independent variable and a dependent variable. […]

Solve the system of equations below.

(begin{align}& 2x+3y=7\& y=-x+3end{align}) In this question, a system of two equations is given. We are required to find the solution to the given system. A set or collection of simultaneous linear or non-linear equations is called a system of equations. This set or collection is finite and usually has common solutions. A system of equations […]