Sage-Tips

How do you find the center of a curve?

In geometry, the center of curvature of a curve is found at a point that is at a distance from the curve equal to the radius of curvature lying on the normal vector. It is the point at infinity if the curvature is zero. The osculating circle to the curve is centered at the centre of curvature.

What does the partial derivative tell us?

The partial derivative f y ( a , b ) tells us the instantaneous rate of change of with respect to at ( x , y ) = ( a , b ) when is fixed at .

Is center and Centre the same?

Center and centre have the same meaning. Center is the correct spelling in American English, while in British English centre is correct. Notice that center (and centre) can be a noun, adjective, or a verb.

How do you find the slope of a partial derivative?

Partial derivatives are the slopes of traces. The partial derivative fx(a,b) f x ( a , b ) is the slope of the trace of f(x,y) f ( x , y ) for the plane y=b at the point (a,b) . Likewise the partial derivative fy(a,b) f y ( a , b ) is the slope of the trace of f(x,y) f ( x , y ) for the plane x=a at the point (a,b) .

What are the interpretations of partial derivatives?

Interpretations of Partial Derivatives – In the section we will take a look at a couple of important interpretations of partial derivatives. First, the always important, rate of change of the function. Although we now have multiple ‘directions’ in which the function can change (unlike in Calculus I).

How do you find marginal rate of substitution with partial derivatives?

For such functions, partial derivatives can be used to measure the rate of change of the function with respect to x x divided by the rate of change of the function with respect to y y, which is fx fy f x f y. If there exists level curves for the function f (x,y) f (x, y), the ratio fx fy f x f y is called the marginal rate of substitution.

What is the difference between partial derivatives and tangent lines?

The difference here is the functions that they represent tangent lines to. Partial derivatives are the slopes of traces. The partial derivative f x(a,b) f x (a, b) is the slope of the trace of f (x,y) f (x, y) for the plane y = b y = b at the point (a,b) (a, b).

What are directional derivatives?

Directional Derivatives – In the section we introduce the concept of directional derivatives. With directional derivatives we can now ask how a function is changing if we allow all the independent variables to change rather than holding all but one constant as we had to do with partial derivatives.