Problem 24
Question
Find the range of each function \(f(x, y)\), when defined on the specified
domain \(D\).
\(\left.f(x, y)=x^{2} y ; D=(x, y):-2 \leq x \leq 1,0 \leq y \leq 1,
y
Step-by-Step Solution
Verified Answer
The range of \( f(x, y) \) is \( [0, 16) \).
1Step 1: Understand the Function and Domain
The function given is \( f(x, y) = x^2 y \). The domain \( D \) is such that \( -2 \leq x \leq 1 \), \( 0 \leq y \leq 1 \), and additionally \( y < x^2 \). Our task is to find the range of \( f(x, y) \) over this domain.
2Step 2: Consider the Restrictions on \( y \)
The condition \( y < x^2 \) implies that \( y \) is bounded by \( 0 \leq y < x^2 \). Since \( x^2 \) is always non-negative, this further refines the range for \( y \).
3Step 3: Determine the Range of \( x^2 \)
Within the bounds \( -2 \leq x \leq 1 \), the square of \( x \) gives us \( 0 \leq x^2 \leq 4 \). This helps determine the upper limit of \( y \), which is \( x^2 \).
4Step 4: Identify the Range of \( x^2 y \)
As the function is \( f(x, y) = x^2 y \) where \( 0 \leq y < x^2 \), let's substitute this into the function. The expression \( x^2 y \) becomes \( 0 \leq x^2 y < x^2 \cdot x^2 = x^4 \).
5Step 5: Establish the Range of \( f(x, y) \)
Since \( x^2 y \) can vary from 0 up to just under \( x^4 \), consider the maximum possible value of \( f(x, y) \). The maximum value occurs near the boundary \( y = x^2 \), reaching close to \( x^4 \). Varying the \( x \) from -2 to 1 gives \( 0 \leq x^4 < 16 \). Hence, \( f(x, y) \) ranges from 0 to slightly below 16. Therefore, the range is \( [0, 16) \).
Key Concepts
Function RangeDomain and RangeCalculus for Biology
Function Range
The range of a function refers to the set of all possible output values it can produce. To determine the range of a multivariable function like \( f(x, y) = x^2 y \), we must understand how the function behaves over its domain. This is crucial, especially in multivariable calculus as the output varies with more than one input variable.
In our exercise, we are dealing with a function defined over a specific domain \( D \) where \( x \) and \( y \) are bounded by certain inequalities. The range is influenced by these conditions, since \( f(x, y) = x^2 y \) depends on both variables. As \( x \) varies from -2 to 1, its square \( x^2 \) varies from 0 to 4. Meanwhile, \( y \) is restricted by \( y < x^2 \), meaning it ranges from 0 to just under \( x^2 \).
Thus, the output of \( x^2 y \), and consequently the range of the function, is from 0 up to just less than \( x^4 \). In this problem, this means from 0 to slightly below 16. Understanding this helps us map out the function's behavior within the given constraints.
In our exercise, we are dealing with a function defined over a specific domain \( D \) where \( x \) and \( y \) are bounded by certain inequalities. The range is influenced by these conditions, since \( f(x, y) = x^2 y \) depends on both variables. As \( x \) varies from -2 to 1, its square \( x^2 \) varies from 0 to 4. Meanwhile, \( y \) is restricted by \( y < x^2 \), meaning it ranges from 0 to just under \( x^2 \).
Thus, the output of \( x^2 y \), and consequently the range of the function, is from 0 up to just less than \( x^4 \). In this problem, this means from 0 to slightly below 16. Understanding this helps us map out the function's behavior within the given constraints.
Domain and Range
In calculus, when analyzing a function, two important concepts are its domain and range. The domain represents all the input values \( (x, y) \) that the function can accept without running into undefined operations. For \( f(x, y) = x^2 y \), the domain \( D \) is specified by \( -2 \leq x \leq 1 \), \( 0 \leq y \leq 1 \), and \( y < x^2 \).
Because of these conditions, the domain dictates how the function behaves and thus affects the range. By restricting \( y < x^2 \), we ensure \( y \) always stays less than \( x^2 \), adding another layer of complexity in bounding the function's output. This restriction ensures that \( f(x, y) \) never equals or exceeds \( x^4 \).
The interaction between the domain and the range forms the framework within which the function operates. Understanding the limitations or boundaries set by the domain is key to predicting and determining the range in a multivariable context.
Because of these conditions, the domain dictates how the function behaves and thus affects the range. By restricting \( y < x^2 \), we ensure \( y \) always stays less than \( x^2 \), adding another layer of complexity in bounding the function's output. This restriction ensures that \( f(x, y) \) never equals or exceeds \( x^4 \).
The interaction between the domain and the range forms the framework within which the function operates. Understanding the limitations or boundaries set by the domain is key to predicting and determining the range in a multivariable context.
Calculus for Biology
Calculus, and specifically multivariable calculus, plays a vital role in biological studies, where systems and models often involve more than one variable. Understanding the range and domain of functions is essential when applying calculus to biological problems, where relationships between different biological factors are explored.
For example, let's consider a simple model of population growth that depends on various nutrients and environmental factors. The outputs of the model (population sizes) can be analyzed in terms of their range, given the domain of input conditions, such as nutrient levels (\( x \)) and temperature (\( y \)).
This exercise's principles can help students model and understand how different biological variables interact—predicting conditions under which certain biological thresholds might be reached or exceeded. Hence, these exercises are not just mathematical practices; they form foundational skills for predicting and analyzing complex biological systems.
For example, let's consider a simple model of population growth that depends on various nutrients and environmental factors. The outputs of the model (population sizes) can be analyzed in terms of their range, given the domain of input conditions, such as nutrient levels (\( x \)) and temperature (\( y \)).
This exercise's principles can help students model and understand how different biological variables interact—predicting conditions under which certain biological thresholds might be reached or exceeded. Hence, these exercises are not just mathematical practices; they form foundational skills for predicting and analyzing complex biological systems.
Other exercises in this chapter
Problem 24
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