Find the gradient of a function at a given point, sketch level curve

In summary, the conversation discusses finding the gradient of a function at a given point and sketching the corresponding level curve. The answer in the back of the book uses the equation \(4=2x+3y\) to represent the line passing through the given point, and this can also be written as \(2=\sqrt{2x+3y}\), which is where the 2 comes from.
  • #1
skate_nerd
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So I have a function \(f(x,y)=\sqrt{2x+3y}\) and need to find the gradient at the point (-1,2). I got this part already, its \(\frac{1}{2}\hat{i}+\frac{3}{4}\hat{j}\). The part I'm having trouble with is when it asks me to sketch the gradient with the level curve that passes through (-1,2).
The back of the book has the answer and it calls the line passing through the point given point \(4=2x+3y\). I know that in relation to the given function this would also equal \(2=\sqrt{2x+3y}\). So where does that 2 come from?
 
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  • #2
Re: find the gradient of a function at a given point, sketch level curve

The 2 comes from:

$f(-1,2)=\sqrt{2(-1)+3(2)}=\sqrt{4}=2$
 

1. What is the definition of a gradient?

The gradient of a function at a given point is a vector that points in the direction of the steepest ascent of the function at that point. It represents the rate of change of the function in the direction of the vector.

2. How do I find the gradient of a function at a given point?

To find the gradient of a function at a given point, you will need to take the partial derivative of the function with respect to each variable, and then evaluate the derivatives at the given point. The resulting values will make up the components of the gradient vector.

3. What is the purpose of finding the gradient of a function at a given point?

Finding the gradient of a function at a given point is useful in many areas of science, including physics, engineering, and economics. It can help determine the direction of maximum increase or decrease of a function, which can be used to optimize processes or make predictions.

4. What is a level curve?

A level curve, also known as a contour curve, is a curve on a graph where all points on the curve have the same function value. In other words, it represents all the points where the function is equal to a constant value. On a two-dimensional graph, a level curve is a line, while on a three-dimensional graph, it is a curve on the surface.

5. How do I sketch a level curve?

To sketch a level curve, you will need to plot several points on the graph where the function has the same value. These points can be found by setting the function equal to a constant and solving for one of the variables. Once you have enough points, you can connect them to create the level curve. It is also helpful to use a graphing calculator or software to visualize the curve accurately.

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