How to Find the Linear Approximation of the Function f(x, y, z) = x² + y² + z² at (3, 2, 6)?
To find the linear approximation of the function f(x, y, z) = x² + y² + z² at the point (3, 2, 6), we use the formula for the linear approximation, which is: L(x, y, z) = f(a, b, c) + fx(a, b, c)(x – a) + fy(a, b, c)(y – b) + fz(a, b, […]
Solve for x: 4x² + 4a²x + a⁴ + b⁴ = 0
To solve the equation 4x² + 4a²x + a⁴ + b⁴ = 0, we first need to rearrange it into a form suitable for using the quadratic formula. The equation is a quadratic in terms of x. We can identify the coefficients in the standard quadratic form Ax² + Bx + C = 0: A […]
How do you find the value of y when y varies directly with x?
To determine the value of y when it varies directly with x, we can use the formula: y = kx where k is a constant. First, we need to find the value of k using the provided information. We know that when y = 52, x = 84. Substituting these values into the equation: 52 […]
How to Find the Total Differential of z = x * cos(y) * cos(x)?
To find the total differential of the function z = x * cos(y) * cos(x), we need to compute the partial derivatives of z with respect to each variable: x and y. 1. **Partial Derivative with respect to x**: Using the product rule, we calculate: ∂z/∂x = cos(y) * cos(x) + x * cos(y) * […]
If the ratio of the length to the breadth of a rectangle be 53, what does that mean?
The ratio of the length to the breadth of a rectangle being 53 means that for every 53 units of length, there is 1 unit of breadth. This ratio indicates that the rectangle is quite elongated, making it much longer than it is wide. To put this into perspective, if we denote the length of […]
A 5 card hand is dealt from a deck of 52 cards. What is the probability that all are hearts?
To find the probability that all 5 cards dealt from a standard deck of 52 cards are hearts, we can break it down step-by-step. Firstly, there are 13 hearts in a deck of 52 cards. The probability of the first card being a heart is: Probability of 1st card: 13/52 Once the first card is […]
Name the Angles in the Given Figure
To name the angles in a given figure, you first need to identify the points that form the angles. Typically, angles are represented by a vertex and two rays. For example, if you have a triangle with vertices labeled A, B, and C, the angles are named as follows: Angle A: The angle formed at […]
When Does the Function gx = 6x² + 23x + 4 Equal Zero?
To determine when the function gx = 6x² + 23x + 4 equals zero, we need to solve the equation: 6x² + 23x + 4 = 0 This is a quadratic equation in the standard form of ax² + bx + c = 0, where a = 6, b = 23, and c = 4. […]
The equation of a graph is 4x + 3y + 5. What is the x-intercept?
To find the x-intercept of the equation 4x + 3y + 5 = 0, we need to determine the value of x when y is equal to 0. We start by substituting y = 0 into the equation: 4x + 3(0) + 5 = 0 4x + 5 = 0 Next, we solve for x: […]
Solve the system of equations: 3r + 4s = 0 and 2r + 5s = 23
To solve the system of equations, we have: Equation 1: 3r + 4s = 0 Equation 2: 2r + 5s = 23 We can use the substitution or elimination method to find the values of r and s. Here, we’ll use the elimination method. First, let’s solve for one variable in terms of the other […]