### Find the points on the surface y^2 = 9 + xz that are closest to the origin.

This question aims to learn the basic methodology for optimizing a mathematical function (maximizing or minimizing). Critical points are the points where the value of a function is either maximum or minimum. To calculate the critical point(s), we equate the first derivative’s value to 0 and solve for the independent variable. We can use the […]

### If f and g are both even functions, is f + g even? If f and g are both odd functions, is f+g odd? What if f is even and g is odd? Justify your answers.

The main objective of this question is to check whether the addition of the given two functions when both the functions are odd, even or one is odd and the other is even results in even or odd function. This question shows the concept of even and odd functions. An even function is mathematically represented as: [f(-x) […]

### Determine the longest interval in which the given initial value problem is certain to have a unique twice differentiable solution. Do not attempt to find the solution.

( x + 3 ) y”  +  x y’  +  ( ln|x| ) y  =  0,  y(1) =  0,  y'(1)  =  1  The aim of this question is to qualitatively find the possible interval of the differential equation’s solution. For this we need to convert any given differential […]

### Explain why the function is differentiable at the given point. Then find the linearization L(x, y) of the function at that point.

f(x,y) = 1 + x ln (xy – 5), (2,3) This problem explains why the given function is differentiable at a point, and to find the linearization at that point. The concept required to solve […]

### 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 […]

### For the equation, write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. Keeping the restrictions in mind, solve the equation.

(dfrac{4}{x+5}+dfrac{2}{x-5}=dfrac{32}{x^2-25}) This question aims to find the solution to the given equation by taking into consideration the restrictions on the given function. The fraction of two polynomials is said to be a rational expression. Such expression can be expressed as $dfrac{a}{b}$ in which $a$ and $b$ both are polynomials. The product, sum, division, and subtraction […]

### A man 6 feet tall walks at a rate of 5 feet per second away from a light that is 15 feet above the ground.

When he is $10$ feet from the base of the light, at what rate is the tip of his shadow moving? When he is $10$ feet from the base of the light, at what rate is the length of his shadow changing? The purpose of this question is to find the rate of change of […]

### Find the parametric equation of the line through a parallel to b.

(a=begin{bmatrix}3\-4end{bmatrix}, b=begin{bmatrix}-7\8end{bmatrix}) This question aims to find the parametric equation of the line through two given vectors. A parametric equation is an equation that incorporates a parameter that is an independent variable. In this equation, the dependent variables are the continuous functions of the parameter.  Two or more parameters can also be used if needed. […]