Solving f(x,y) and Sin(x) + cos(y) / 2x - 3y

  • Thread starter Ravic85
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In summary, when evaluating the functions f(x,y) = e^(x+y) + cos(xy) / Ln(xy) and Sin(x) + cos(y) / 2x - 3y, the sets of points (x,y) where the functions are undefined or the limit does not equal the function value are where ln(xy) is undefined or equal to zero, and where 2x - 3y is equal to zero. These restrictions can be lessened by considering all points where the product xy is negative, zero, or equal to one. Additionally, the entire line 3y = 2x can also be considered.
  • #1
Ravic85
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Homework Statement


A: f(x,y) = e^(x+y) + cos(xy) / Ln(xy)

b: Sin(x) + cos(y) / 2x - 3y


Homework Equations



I'm not exactly sure if I'm doing this correctly, my book a little vague.


The Attempt at a Solution



for A I have when x = -1 and y = any real real number

for B I have when x = 3 and y = 2
 
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  • #2


Why restrict the sets so much? The functions will not be continuous anyplace where the functions are undefined (or where the limit is not defined, or where the limit does not equal the function value).

The numerators are always defined, so they do not present a problem. So the ratios are undefined if either the denominator functions are undefined, or where the denominator functions are zero.
 
  • #3


So do you mean on say part A I'd have x = any negative number? I don't quite understand how I would lessen the restrictions on part B.
 
  • #4


Remember that you want the sets of points (x,y), so not everything hinges on x alone.

part A: Where would ln(xy) be undefined? Where is it zero?

part B: 2x - 3y is always defined, but where is this difference equal to zero?
 
  • #5


Where Ln would be undefined is where you get a negative number? when it's zero it's ln(1)? And with B: I thought I answered where the difference is equal to zero with x = 3 and y = 2
 
  • #6


Yes ln is undefined for negative numbers and also for zero.
Furthermore the nominator is zero when it's ln(1).

So any x and y for which xy<0 or xy=0 or xy=1 qualify for the function to be not continuous.

As it is you have given 1 example, but the question asks for all points (x,y) that qualify.

With B: For instance x=6 and y=4 also qualifies.

Edit: Oops. Fixed to y=4.
 
Last edited:
  • #7


For part B, why not the entire line 3y = 2x --> y = (2/3) x ?

I like Serena said:
With B: For instance x=6 and y=2 also qualifies.

Do you mean x = 6 and y = 4?
 

1. What is the purpose of solving f(x,y)?

The purpose of solving f(x,y) is to find the values of x and y that satisfy the given equation. This is important in understanding the relationship between the variables and how they affect the overall function.

2. How do you solve f(x,y) and Sin(x) + cos(y) / 2x - 3y?

To solve this equation, you can use algebraic manipulation and trigonometric identities to simplify the expression. Then, you can substitute different values for x and y to see which combination satisfies the equation.

3. What is the significance of Sin(x) + cos(y) / 2x - 3y in this equation?

The significance of Sin(x) + cos(y) / 2x - 3y is that it represents the relationship between the two variables, x and y. It also demonstrates the use of trigonometric functions in mathematical equations.

4. Can this equation have multiple solutions?

Yes, this equation can have multiple solutions. Depending on the given values and the manipulation of the equation, there can be an infinite number of solutions or a finite set of solutions.

5. What real-world applications can be modeled using this equation?

This equation can be used to model various real-world situations, such as the relationship between two physical quantities or the behavior of a system over time. It can also be applied in fields like physics, engineering, and economics.

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