mfb said:Rate of change with respect to what?
x and z? I think I would calculate those derivatives and evaluate them at the given point.
Maximal change for any direction? This can be evaluated, too (it has to be in the xz-plane here).
This is possible, right.If i remember, the maximal change can be calculated using gradients, right?
No, this is not "bad written" (or even "badly written":tongue:). Since you are asked about its rate on change on y= 2, replace y by 2: f(x, z)= 3x+ 4+ 4z. The rate of change of that, at (1, 2, 3) is the length of the gradient vector evaluated at (1, 2, 3). (If the function really is what you wrote without any powers, its gradient vector is a constant and the point (1, 2, 3) is irrelevant.)supermiedos said:Homework Statement
Given f(x, y, z) = 3x + 2y + 4z, find its rate of change at (1, 2, 3) on the plane y = 2.
Homework Equations
The Attempt at a Solution
Since y is constant, I don't know if I have to find ∂f/∂x or ∂f/∂z. It's a homework problem, but I don't know if it was bad written by the teacher. Could you help me please?
HallsofIvy said:No, this is not "bad written" (or even "badly written":tongue:). Since you are asked about its rate on change on y= 2, replace y by 2: f(x, z)= 3x+ 4+ 4z. The rate of change of that, at (1, 2, 3) is the length of the gradient vector evaluated at (1, 2, 3). (If the function really is what you wrote without any powers, its gradient vector is a constant and the point (1, 2, 3) is irrelevant.)
This is always zero with 2 or more variables, if the function is differentiable.supermiedos said:(since I could also find the lowest rate of change)
RightBut the length of the gradient represents the highest rate of change, right?
The rate of change is calculated by dividing the change in the output variable by the change in the input variable. This is also known as the slope of a line.
The formula for finding the rate of change is (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on a line. This formula can be used for both linear and nonlinear relationships.
The rate of change is represented by the slope of a line on a graph. The steeper the slope, the greater the rate of change. A horizontal line indicates a rate of change of 0, while a vertical line represents infinite rate of change.
The rate of change can be affected by various factors such as time, temperature, pressure, and other physical or environmental variables. It can also be influenced by the starting and ending points of the measurement.
The rate of change provides information about the speed and direction of change in a relationship. A positive rate of change indicates an increase, while a negative rate of change indicates a decrease. The magnitude of the rate of change also indicates the degree of change.