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Chapter 7: Problem 5

Draw a computational graph to compute the function \(f(x, y)=x^{3}(x-y)\). Use the graph to compute \(f(2,3)\).

Short Answer

Expert verified
The computed value of \(f(2,3)\) using the graph is \(-8\).

Step by step solution

01

Define Sub-expressions

Before drawing a computational graph, breakdown the function into sub-expressions. Let's represent them symbolically: Let \(a = x^3\) and \(b = x - y\). Therefore, the function can be expressed as \(f(x, y) = a \cdot b\).
02

Draw the Computational Graph

Draw nodes for each sub-expression and operation: \(x\) and \(y\) are input nodes. Then, a node for \(a = x^3\) and a node for \(b = x - y\). Finally, an output node for \(f = a \cdot b\). Draw directed edges from inputs through operations to the output.
03

Compute Node Values

Now we'll compute the values for each node: Start with input values \(x = 2\) and \(y = 3\). First, compute \(a = 2^3 = 8\) and then compute \(b = 2 - 3 = -1\).
04

Compute the Output

Using the results from the nodes, compute the final output: Multiply the values from Step 3, \(f = a \cdot b = 8 \cdot (-1) = -8\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Computation
Function computation is the process of evaluating a given function by performing a series of specific arithmetic operations. In your problem, the function is given as \(f(x, y) = x^3(x-y)\). Before diving into calculations, we first need to understand how to approach this function.
  • Determine the key operations involved, like exponentiation and multiplication.
  • Recognize how these operations are combined to form the function.
Breaking down the function into a sequence of simpler operations forms the basis of computational approaches. This simplification helps in systematically handling and computing the complex function, leading to accurate results.
Sub-expressions
Sub-expressions are smaller parts of a larger expression that can be evaluated separately. They are crucial in simplifying complex computations. For the function \(f(x, y) = x^3(x-y)\), splitting it into sub-expressions can make the computation process much easier.
  • For the given function, identify sub-expressions: let \(a = x^3\) and \(b = x - y\).
  • By isolating parts of a function, focus is shifted to understanding each sub-expression's role and evaluation.
This approach allows the function to be handled one part at a time, simplifying not only the computation of the entire function but also the visualization and error diagnosis.
Node Values
Node values refer to the specific numerical results obtained from evaluating sub-expressions at different nodes in a computational graph. These are obtained by substituting variable values into each sub-expression.
For \(x = 2\) and \(y = 3\):
  • First, calculate \(a = x^3 = 2^3 = 8\).
  • Next, calculate \(b = x - y = 2 - 3 = -1\).
Each sub-expression evaluation results in a node value, serving as a building block in the graph. These values are crucial as they feed directly into the next calculation steps, ensuring correctness and completeness of the final output.
Output Computation
Output computation is the final step where the function's result is determined by combining all evaluated sub-expressions. Once node values are computed, they are used to calculate the final output of the function.
  • In the given problem, use the node values to find the final result of the function: \(f(x, y) = a \cdot b = 8 \cdot (-1) = -8\).
  • This step highlights the importance of accurate sub-expression evaluation, as each step builds upon previous calculations.
This process helps in understanding the flow of computation from initial sub-expressions to the final result, providing insights into both the function's nature and the computational method used.

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