Question

A CAS can be used to calculate and graph partial derivatives. Draw the graphs of each of the following: (a) \(\sin \left(x+y^{2}\right)\) (b) \(D_{x} \sin \left(x+y^{2}\right)\) (c) \(D_{y} \sin \left(x+y^{2}\right)\) (d) \(D_{x}\left(D_{y} \sin \left(x+y^{2}\right)\right)\)

Short answer

Use a CAS to graph the given function and its calculated partial derivatives.

Step-by-step solution

Identify the Function

The function given is \( f(x, y) = \sin(x+y^2) \). We need to graph this function first before finding and graphing its partial derivatives.

Graph the Function

Using a CAS (Computer Algebra System), graph \( f(x, y) = \sin(x+y^2) \). This will display how the function behaves over a range of \( x \) and \( y \) values. Observe the shape and oscillations characteristic of the sine function.

Calculate and Graph \( D_x \sin(x+y^2) \)

The partial derivative with respect to \( x \) is calculated by treating \( y^2 \) as constant: \( D_x = \cos(x+y^2) \cdot 1 = \cos(x+y^2) \). Use the CAS to graph this new function \( g(x, y) = \cos(x+y^2) \).

Calculate and Graph \( D_y \sin(x+y^2) \)

For the partial derivative with respect to \( y \), apply the chain rule: \( D_y = \cos(x+y^2) \cdot 2y \). Thus, the derivative is \( h(x, y) = 2y \cdot \cos(x+y^2) \). Use the CAS to create and view the graph of this derivative.

Calculate and Graph \( D_x(D_y \sin(x+y^2)) \)

First, recall \( D_y \sin(x+y^2) = 2y \cdot \cos(x+y^2) \). Differentiating this with respect to \( x \), we apply the product rule: \( D_x[2y \cdot \cos(x+y^2)] = 0 + 2y \cdot (-\sin(x+y^2)) \cdot 1 = -2y \cdot \sin(x+y^2) \). Use the CAS to graph the function \( j(x, y) = -2y \cdot \sin(x+y^2) \).

Key concepts

Computer Algebra System (CAS)

A Computer Algebra System, or CAS, is a powerful tool that assists with performing complex mathematical computations and graphing functions efficiently. It can handle symbolic manipulation of algebraic expressions, making it incredibly useful for teaching and research. When dealing with functions involving multiple variables, like partial derivatives, CAS provides a reliable way to explore these functions interactively.

• Automation of Calculations: CAS aids in automating tedious and elaborate calculations, allowing you to focus on understanding the concepts rather than getting bogged down with arithmetic.
• Graphing Capabilities: By inputting functions, CAS can generate accurate graphs that help visualize mathematical concepts clearly, showing how variables interact in a function.
• Support for Learning: Through visual representation and step-by-step guidance, CAS can enhance the learning experience, offering hands-on opportunities to dissect complex topics like partial derivatives.

Whether you're studying sine functions or examining the effects of variables on derivatives, CAS is a helpful ally in grasping these intricate ideas.

Sine Function

The sine function, denoted as \( \sin(x) \), is a fundamental trigonometric function widely used in mathematics, especially in calculus and physics. It describes the y-coordinate of a point on the unit circle, providing a periodic oscillating shape that varies between -1 and 1. Its properties are crucial when exploring calculus concepts, including derivatives and integrals.

• Wave-like Behavior: The sine function is known for its repetitive wave pattern. This oscillating nature is key in modeling various phenomena such as sound waves, light waves, and alternating current in electrical applications.
• Periodicity: The function has a period of \( 2\pi \), meaning its values repeat every \( 2\pi \) units. Understanding this concept is vital when working with equations that involve periodic behavior.
• Connection with Angles: In trigonometry, \( \sin(x) \) is particularly valuable as it relates directly to angles, making it essential in solving problems involving triangles and circular motion.

In calculus, sine functions become more intriguing when combined with other variables, as shown in the function \( \sin(x+y^2) \), where new forms of oscillation and interaction with planes can be analyzed.

Graphing Functions

Graphing functions is an indispensable skill in mathematics, helping to visualize and better understand the behavior of mathematical expressions. With the help of a graph, it becomes easier to interpret the function's growth, decline, oscillation, and points of interest like maxima and minima.

• Visual Exploration: By plotting the graph of a function like \( \sin(x+y^2) \), you can see how the function behaves across different values of \( x \) and \( y \). This helps in identifying patterns and critical points.
• Understanding Derivatives: When graphing derivatives such as \( D_x \sin(x+y^2) \) and \( D_y \sin(x+y^2) \), the slopes reveal the rate at which the function changes in different directions.
• Interactive Learning: Utilizing graphing software, students can dynamically adjust variables and immediately observe the effects, fostering a deeper comprehension of multi-variable calculus.

Graphing is not just about creating visual representations; it's an essential tool in conceptualizing mathematical theories and applying them to real-world situations.

Chain Rule

The Chain Rule is a fundamental theorem in calculus that is used to differentiate composite functions. It's particularly useful when finding derivatives of a function in relation to a variable within another function. The integration of inner and outer functions requires the application of this rule to navigate through layers of differentiation.

• Understanding Composite Functions: In our given function \( \sin(x + y^2) \), recognizing it is composite (a function within a function) is key. Here, \( x + y^2 \) is inside the sine function.
• Derivative Application: While calculating \( D_y \sin(x+y^2) \), the chain rule helps take the derivative of the inner function \( y^2 \) and multiply it by the derivative of the sine function.
• Simplifying Complex Calculus: This approach simplifies computing partial derivatives, turning a potentially complex process into manageable steps, especially when dealing with multiple variables.

By mastering the chain rule, you unlock the ability to handle intricate functions that occur in advanced calculus, paving the way for successful problem-solving and analysis.