Question

The partial differential equation from \(z=(c+x)^{2}+y\) is (i) \(z=\left(\frac{d e}{\partial x}\right)^{2}+y\) \((j) z=\left(\frac{\partial z}{\partial y}\right)^{2}+y\) \((i a) z=\frac{1}{4}\left(\frac{\partial z}{\partial x}\right)^{2}+y\) (iw) \(z=\frac{1}{4}\left(\frac{\partial e}{\partial y}\right)^{2}+y_{.}\)

Short answer

Option (ia) is correct.

Step-by-step solution

Determine Partial Derivative with Respect to x

The function given is \( z = (c + x)^2 + y \). To find the partial derivative of \( z \) with respect to \( x \), we treat \( y \) as a constant and differentiate \( (c + x)^2 \). The result is:\[ \frac{\partial z}{\partial x} = 2(c + x). \]

Substitute Partial Derivative Into Option (i)

Option (i) suggests checking whether \( z = \left(\frac{de}{\partial x}\right)^2 + y \). Since the differentiation was incorrect in the option title, we rectify it with:\( \left(\frac{\partial z}{\partial x}\right)^2 = (2(c + x))^2 = 4(c + x)^2 \) and substitute:\[ z = 4(c + x)^2 + y. \]Comparing this with the original equation \( z = (c + x)^2 + y \), they do not match since \( 4(c + x)^2 eq (c + x)^2 \). So, option (i) is incorrect.

Determine Partial Derivative with Respect to y

Now find \( \frac{\partial z}{\partial y} \) treating \( x \) as a constant, yielding:\[ \frac{\partial z}{\partial y} = 1. \]

Substitute Partial Derivative Into Option (j)

Option (j) suggests \( z = \left(\frac{\partial z}{\partial y}\right)^2 + y \). Substitute the derivative to get:\[ z = 1^2 + y = 1 + y. \]This doesn't match \( z = (c + x)^2 + y \), so option (j) is incorrect.

Check Option (ia)

For option (ia), compute \( \frac{1}{4}\left(\frac{\partial z}{\partial x}\right)^2 + y \).Use \( \frac{\partial z}{\partial x} = 2(c + x) \) from before, leading to:\[ \frac{1}{4}(2(c + x))^2 + y = \frac{1}{4}\cdot4(c + x)^2 + y = (c + x)^2 + y. \]This matches the original equation \( z = (c + x)^2 + y \). So, option (ia) is correct.

Check Option (iw)

For option (iw), compute \( \frac{1}{4}\left(\frac{\partial e}{\partial y}\right)^2 + y \).The correct form is really \( \frac{1}{4}(1)^2 + y = \frac{1}{4} + y \). This doesn't match the original equation \( z = (c + x)^2 + y \) since \( \frac{1}{4} + y eq (c + x)^2 + y \). Thus, option (iw) is incorrect.

Key concepts

Partial Derivatives

Partial derivatives are an essential tool in calculus, particularly when dealing with functions of multiple variables. They represent the rate at which a function changes as one of the variables changes, keeping all other variables constant. For example, in the exercise provided, the function given is \( z = (c + x)^2 + y \). Here, \( z \) depends on both \( x \) and \( y \).
To find the partial derivative of \( z \) with respect to \( x \), one treats \( y \) as a constant and differentiates only concerning \( x \). This process yields:

• \( \frac{\partial z}{\partial x} = 2(c + x) \)

Similarly, we find the partial derivative with respect to \( y \) by treating \( x \) as a constant:

• \( \frac{\partial z}{\partial y} = 1 \)

Understanding how partial derivatives work is crucial in many applications, including optimization problems where we find maxima or minima of functions of several variables.

Mathematical Problem-Solving

Mathematical problem-solving is a critical skill that enables us to approach and solve complex equations methodically. The step-by-step solution provided in the exercise demonstrates this perfectly. By breaking down the problem into smaller, manageable steps, one can systematically verify each option presented in the problem.
The process begins by determining the partial derivatives of the given equation \( z = (c + x)^2 + y \). With the partial derivatives in hand, one then substitutes these into the various options (i), (j), (ia), and (iw) to verify their correctness. This approach ensures that each potential solution is logically evaluated without room for errors.
It's essential to:

• Carefully examine each step
• Use logical reasoning to connect ideas
• Ensure consistency with the original equation

This kind of structured approach is invaluable in engineering, physics, and other scientific fields where precision is key.

Engineering Mathematics

Engineering mathematics involves applying mathematical techniques and principles to solve engineering problems. This field heavily relies on concepts such as partial derivatives, presented in the exercise, as they often appear in several engineering disciplines.
Partial differential equations are fundamental in modeling and solving problems involving changes concerning various variables, such as in the analysis of structural mechanics, heat conduction, fluid dynamics, and more. For instance, equations that describe how temperature varies across a surface or how fluids flow are often partial differential equations.
Within engineering mathematics:

• We see frequent use of deriving relationships between different engineering quantities.
• The methods and principles help engineers design, analyze, and innovate effectively.
• Mastering these techniques opens up possibilities for advanced problem-solving and technological advancements.

Thus, developing a firm grasp of concepts like partial derivatives is not just theoretical but also practical, enabling engineers to enhance their solutions and efficiencies in varied real-world applications.