Estimating partial derivatives/directional derivatives

In summary, To estimate the derivatives at the point (1,2), use the given values to estimate the directional derivatives in the directions parallel to (0.2,0.3) and (-0.1,0.1), which will give you estimates for the partial derivatives dz/dx and dz/dy, respectively. Then, use these estimates to estimate the directional derivative in the directions (1,0) and (0,1), which are the two partial derivatives.
  • #1
playa007
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Homework Statement


Let z = f(x,y) be a differentiable function on R^2 such that f(1, 2) = 3,
f(1.2, 2.3) = 3.4 and f(0.9, 2.1) = 3.2.

a) Estimate dz/dx and dz/dy at (1,2) (dz/dx and dz/dy are partial derivatives)
b) Estimate the value of the directional derivative of z = f(x,y) at the point (1,2) as you move towards (2,3)


Homework Equations





The Attempt at a Solution


I've tried using the formal definitions of directional and partial derivatives to do these but I just can't get how the estimates come in. Help would be very much appreciated
 
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  • #2
A finite difference quotient estimates the derivative, since the derivative is the limit of the difference quotient as the difference goes to zero. The given values will let you estimate the directional derivative in the direction parallel to (0.2,0.3) and (-0.1,0.1). Use those to estimate the directional derivatives in the directions (1,0) and (0,1). Which are the two partial derivatives.
 

What is the purpose of estimating partial derivatives/directional derivatives?

The purpose of estimating partial derivatives/directional derivatives is to calculate the rate of change of a multivariable function with respect to one or more of its variables. This allows us to understand how the function behaves in different directions and make predictions about its behavior at a specific point.

How do you estimate a partial derivative?

To estimate a partial derivative, you hold all other variables constant and calculate the derivative of the function with respect to the variable of interest. This can be done using the limit definition of a derivative or by using various differentiation rules.

What is the difference between a partial derivative and a directional derivative?

A partial derivative measures the rate of change of a function with respect to one of its variables, while a directional derivative measures the rate of change of a function in a specific direction. The directional derivative takes into account the slope of the function in addition to the slope of the chosen direction.

When is it necessary to use the chain rule when estimating partial derivatives?

The chain rule is used when estimating partial derivatives of composite functions, where the variable of interest is a function of another variable. In this case, the chain rule allows us to break down the function into smaller parts and calculate the partial derivatives separately before combining them to get the overall partial derivative.

How are partial derivatives/directional derivatives used in real-world applications?

Partial derivatives and directional derivatives are used in a variety of fields such as physics, economics, and engineering. They can be used to optimize processes, analyze the behavior of complex systems, and make predictions about future outcomes. For example, in economics, partial derivatives can be used to calculate marginal costs and marginal revenue, which are important concepts in understanding the behavior of a market.

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