How do I calculate the upper bound of a rectangle?

Answer 1

I will assume that you mean "How do I calculate the upper bound of a function over a rectangle?".

I am not sure what you mean by the question, but I will answer the question: Given a continuous differentiable function #f:RR^2 -> RR#, how do you find the upper bound of the value of #f# over a rectangle #[x_1, x_2] xx [y_1, y_2]#.

The upper bound will be the maximum value of the function, occurring at one of the following locations:

(1) At a corner of the rectangle, i.e. #(x_1, y_1)#, #(x_1, y_2)#, #(x_2, y_1)# or #(x_2, y_2)#.

(2) Along a horizontal edge, where the partial derivative is zero:

#del/(del x) f(x, y_1) = 0# or #del/(del x) f(x, y_2) = 0#
and #x in (x_1, x_2)#

(3) Along a vertical edge, where the partial derivative is zero:

#del/(del y) f(x_1, y) = 0# or #del/(del y) f(x_2, y) = 0#
and #y in (y_1, y_2)#

(4) Inside the body of the rectangle at a point where both partial derivatives are zero:

#del/(del x) f(x, y) = del/(del y) f(x, y) = 0#
and #(x, y) in (x_1, x_2) xx (y_1, y_2)#
Evaluate #f(x, y)# at each of these possible locations and pick the maximum value found.
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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