Directional Derivative of Lake Depth at Point (-1, 2) in Direction (4, 1)

In summary, the problem involves determining if the water is getting deeper or shallower at a specific point on the surface of a lake. The depth of the lake is given by a function, and the solution involves using the gradient and a unit vector. After some mistakes, the correct answer is 28/sqrt(17).
  • #1
RaulTheUCSCSlug
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Having a melt down as I have done this problem twice now and my exam is tomorrow and I can't seem to figure it out anymore... ugh. 1. Homework Statement
The depth of a lake at the point on the surface with coordinates (x, y ) is given by D(x, y ) = 100−4x 2 −y 2 . a) If a boat at the point (−1, 2) is sailing in the direction 4~i +~j, is the water getting deeper or shallower ?

Homework Equations


I know that I need to use the gradient and I am going to need to have the direction scaled to a unit vector.

The Attempt at a Solution


So I took the gradient of D(x,y), plugged in (-1,2) then dotted it with the unit vector of (4,1) the answer is supposed to be 28/sqrt(17) and I can't seem to get that answer.
 
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  • #2
never mind, I just kept doing the unit vector wrong. Simple arithmetic mistake.
 

Related to Directional Derivative of Lake Depth at Point (-1, 2) in Direction (4, 1)

1. What is a directional derivative?

A directional derivative is a measure of how a function changes in a particular direction. It is used to calculate the rate of change of a function along a specific direction.

2. How is a directional derivative different from a partial derivative?

A directional derivative is a generalization of a partial derivative, which only considers changes in one variable while holding all others constant. A directional derivative takes into account changes in all variables, including the direction in which the function is changing.

3. How is a directional derivative calculated?

A directional derivative is calculated using a formula that involves the gradient of the function and the unit vector representing the direction in which the derivative is being calculated. The formula is: Df(v) = ∇f · v, where ∇f is the gradient of the function and v is the unit vector.

4. What is the significance of the directional derivative in real-world applications?

The directional derivative is used in various fields of science and engineering, such as physics, economics, and computer graphics. It is used to calculate the slope of a function at a given point and to determine the direction in which the function is changing the fastest.

5. Can a directional derivative be negative?

Yes, a directional derivative can be negative. This indicates that the function is decreasing in the direction of the derivative. A positive directional derivative indicates that the function is increasing in the direction of the derivative.

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