Multivariable Instantaneous rate of change clarification When you are computing the instantaneous rate of change for f(x,y) what do you take the derivative with respect to? for example, for f(x,y)=(sin(pi^x)cos(pi^y),ye^xy,x^2+y^3) If I was to find the instantaneous rate of change for all 3 of these functions going through (1,2) with the velocity vector (3,-2) would I just take d/dx of all of the functions at (3,-2)?

oopsteekwe 2022-10-13 Answered
Multivariable Instantaneous rate of change clarification
When you are computing the instantaneous rate of change for f(x,y) what do you take the derivative with respect to?
for example, for
f(x,y)=(sin(πx)cos(πy),yexy,x2+y3)
If I was to find the instantaneous rate of change for all 3 of these functions going through (1,2) with the velocity vector (3,-2) would I just take ddx of all of the functions at (3,-2)?
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

HadoHaurrysap3w
Answered 2022-10-14 Author has 10 answers
If you want the change in f as both x and y are changing, then you are going to need something that relates x to y. (a parametric curve perhaps, or a direction vector).
Without that, you can evaluate the sentivity of f to changes in x this is the partial derivative. f x
And there is also a partial derivative with respect to y, i.e. f y
If you have the partials, and a parametric curve then you can find the "total derivative"
d f d t = f x d x d t + f y d y d t and these will correspond to the same derivative you would get if you subsituted x(t),y(t) into your function.
Since f is a vector, you will get a derivatives for each of the i,j,k components.
Did you like this example?
Subscribe for all access

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2022-11-18
For which value of x is the average rate of change equal to the instantaneous rate of change?
The average rate of change for f ( x ) = x 2 + 4 x 6 on the interval [1,3] is 8.
I am not interested in final answer but more how to get there. I am going through calculus right now and already know about derivatives and rate of change. My problem is how to get this word problem into math language and try to solve it.
Inst. rate of change is derivative when lim approaches 0 average f ( x + h ) f ( x ) divided by h.
asked 2022-09-07
Average rate of change?
How would I figure the following problem out?
Find the average rate of change of g ( x ) = x 2 + 3 x + 7 from x=5 to x=9My thought is that I would plug in 5 and 9 for the x values to get the y values. And the use the slope formula y 2 y 1 x 2 x 1
asked 2022-08-06
Find highest rate of change of any function
Is there any way to find where the rate of change of function is maximum/highest?
Suppose we have a sine wave and i want to know where the rate of change of the function is maximum highest?
How can i find that?
asked 2022-11-13
Calculating Rate of Change
At the point (0,1,2) in which direction does the function f ( x , y , z ) = x y 2 z increase most rapidly? What is the rate of change of f in this direction? At the point (1,1,0), what is the derivative of f in the direction of the vector 2 i ^ + 3 j ^ + 6 k ^ ?
I assumed that the rate of change is the same as the gradient of the function, namely f. Calculating this gave me:
f = ( x y 2 z ) x i ^ + ( x y 2 z ) y j ^ + ( x y 2 z ) z k ^
            = y 2 z   i ^ + 2 x z   j ^ + x y 2   k ^
Evaluating at point:
f ( 0 , 1 , 2 ) = 2   i ^
Hence, the function increases most rapidly in the x direction.
I am uncertain of how to approach solving the third part of the question, should I evaluate the rate of change at (1,1,0) and then find the difference between that and the vector 2 i ^ + 3 j ^ + 6 k ^ ?
asked 2022-08-10
What do instantaneous rates of change really represent?The derivative of f ( x ) is the value of the limit of the average rate of change of y with respect to x as the change in x approaches 0. This is the value, in other words, that the average rate of change approaches but NEVER hits.
This means that it is NOT the infintesimal rate of change of y with respect to d y / d x merely approaches the derivative's value. If the rate of change did actually achieve 0 change in x, you'd get 0/0 which is an indeterminant form.
So if the derivative is the literal rate of change at an exact instant -- a rate of change with an interval of 0, what does that actually tell you? Can a specific moment in time really have a rate of change? Is that rate of change ever even maintained, even at a specific instant?
I know that a point by itself can't have a rate of change, you need a continuum of points around it to determine one (hence a limit). What does an instantaneous rate of change tell you?
asked 2022-08-19
Maximum rate of change for a data
I came up with a question when studying calculus. Suppose that we have a data set, say v = v ( t ). Assume that we know about their physical meaning and consequently we know that function V fits the data well. Now the maximum rate of change of the data is better approximated by:
a) maximum rate of change of V
b) average rate of change of V
c) we cannot say
I think (b) gives a more reasonable value. But is there a reasoning behind that? or other options?
asked 2022-07-14
Rate of change of cross-section of cylinder
I got this task on my calculus class and I got stuck at process of figuring it out
A clay cylinder is being compressed so that its height is changing at the rate of 4 millimeters per second, and its diameter is increasing at the rate of 2 millimeters per second. Find the rate of change of the area of the horizontal cross-section of the cylinder when its height is 1 centimeter.
What I know:
Volume is unknown and constant
Rate of change of diameter is
d D d t = 0 , 2 cm / sec
Rate of change of height is
d h d t = 0 , 4 cm / sec
Trying to find d D d t by using formula
V = π r 2 h

New questions