Problem 9 - Active Calculus Multivariable

Question

If \(z_{x y}=5 y\) and all of the second order partial derivatives of \(z\) are continuous, then (a) \(z_{y x}=\) __________. (b) \(z_{x y x}=\) __________. (c) \(z_{x y y}=\) __________.

Step-by-Step Solution

Answer

(a) \(z_{yx} = 0\) (b) \(z_{xyx} = 0\) (c) \(z_{xyy} = 5\)

Step 1: (a) Compute \(z_{yx}\)

Since \(z_{xy} = 5y \), we need to take the partial derivative of this with respect to x to obtain \(z_{yx}\): \(z_{yx} = \frac{\partial}{\partial x}(5y) = 0\) The answer for (a) is \(z_{yx} = 0 \).

Step 2: (b) Compute \(z_{xyx}\)

Now we need to compute the second order mixed partial derivative, \(z_{xyx}\), by applying the partial derivative with respect to x on our previously obtained \(z_{xy}\): \(z_{xyx} = \frac{\partial}{\partial x}(z_{xy}) = \frac{\partial}{\partial x}(5y) = 0\) The answer for (b) is \(z_{xyx} = 0 \).

Step 3: (c) Compute \(z_{xyy}\)

Lastly, let's compute the second order mixed partial derivative, \(z_{xyy}\), by applying the partial derivative with respect to y on our initial \(z_{xy}\): \(z_{xyy} = \frac{\partial}{\partial y}(z_{xy}) = \frac{\partial}{\partial y}(5y) = 5\) The answer for (c) is \(z_{xyy} = 5 \).

Key Concepts

Calculus

Calculus is a branch of mathematics that focuses on change and motion, which make it immensely beneficial in modeling real-world situations.
There are two fundamental operations in calculus: **differentiation** and **integration**. Differentiation involves finding the rate of change of a function, while integration is about finding the total value, often related to area under curves.
Both of these operations are deeply connected and stem from the concept of limits, which define the foundational ideas of calculus.

• Differentiation is often utilized to find derivatives, which can show how a quantity changes in relation to another.
• Understanding the rate of change is crucial in many fields like physics, engineering, and economics.

Calculus doesn't stop at first derivatives; rather it extends to higher order derivatives which provide deeper insights into the behavior of functions.

Second Order Derivatives

Second order derivatives are a deeper exploration of the concept of derivatives. They involve taking the derivative of a derivative, often written as \((f')'\) or \((f'')\).
In the context of partial derivatives, second order derivatives examine how the first order changes further in multiple dimensions.

• These derivatives are labeled such as \(z_{xyx}\) or \(z_{xyy}\), indicating the sequence of differentiation.
• The notation \(z_{xy}\) represents the derivative of \(z\) first with respect to \(y\) and then \(x\), reflecting a more complex interaction of change in multivariable calculus.

Second order derivatives are important for analyzing concavity and inflection points in functions, and in multivariate cases, they help understand the function's curvature.

Continuity of Derivatives

The continuity of derivatives is an essential idea in calculus, particularly for higher order derivatives.
If the derivatives are **continuous**, it means that there are no abrupt changes or breaks in their values across the domain.
This is crucial for ensuring the predictability and smoothness of a function's behavior.

• For functions of multiple variables, continuous partial derivatives imply harmonic regulation in change rates.
• Functions that possess continuous second order partial derivatives can often be approximated accurately using Taylor expansions. Such functions behave predictably and are more manageable analytically.

In practice, ensuring the continuity of derivatives aids in stability and in creating reliable models for real-world phenomena. This continuity often guarantees that we can apply certain mathematical theorems, like Clairaut's theorem, which allows us to equate mixed partial derivatives if they are continuous.