### Suppose you are climbing a hill whose shape is given by the equation z=100 – 0.05x^2 – 0.1y^2, where x,y and z are measured in meters, and you are standing at a point with coordinates (60, 50, 1100). The positive x-axis points east and the positive y-axis points north. If you walk due south, will you start to ascend or descend? At what rate?

The question aims to find the direction if the person starts to walk to the south, whether the person will ascend or descend, and at what rate. This question is based on the concept of directional derivatives. The directional derivative is the dot product of the gradient of the function with its unit vector. Expert Answer […]

### V_1 and V_2 are different vectors with lengths V_1 and V_2 respectively. Find the following:

$overrightarrow{V_1} . overrightarrow{V_1}$ Express in terms of $V_1$. $overrightarrow{V_1} . overrightarrow{V_2}$ When they are perpendicular. $overrightarrow{V_1} . overrightarrow{V_2}$ When they are parallel. This question aims to find the dot product of two vectors when they are parallel and also when they are perpendicular. The question can be solved by revising the concept of vector multiplication, […]

### 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 […]