Problem 30 - Applied Calculus

Question

Calculate all four second-order partial derivatives and confirm that the mixed partials are equal. $$ f(x, t)=t^{3}-4 x^{2} t $$

Step-by-Step Solution

Answer

All four second-order derivatives are \( f_{xx} = -8t \), \( f_{tt} = 6t \), and both mixed partials are \( f_{xt} = f_{tx} = -8x \); they are equal.

Step 1: Calculate First-Order Partial Derivative with respect to x

To find the first partial derivative of the function with respect to \( x \), differentiate \( f(x, t) = t^3 - 4x^2t \) with respect to \( x \). The \( t^3 \) term has no \( x \) dependency, so its derivative is zero. Only consider \( -4x^2t \):\[ f_x = \frac{\partial}{\partial x}(t^3 - 4x^2t) = -8xt \].

Step 2: Calculate First-Order Partial Derivative with respect to t

Now, find the first partial derivative of the function with respect to \( t \):\[ f_t = \frac{\partial}{\partial t}(t^3 - 4x^2t) = 3t^2 - 4x^2 \].

Step 3: Calculate Second-Order Partial Derivative with respect to x twice

Differentiate the first derivative with respect to \( x \) again to find \( f_{xx} \):\[ f_{xx} = \frac{\partial}{\partial x}(-8xt) = -8t \].

Step 4: Calculate Second-Order Partial Derivative with respect to t twice

Differentiate the first derivative with respect to \( t \) again to find \( f_{tt} \):\[ f_{tt} = \frac{\partial}{\partial t}(3t^2 - 4x^2) = 6t \].

Step 5: Calculate Mixed Partial Derivative \( f_{xt} \)

Differentiate the first partial derivative with respect to \( x \) with respect to \( t \):\[ f_{xt} = \frac{\partial}{\partial t}(-8xt) = -8x \].

Step 6: Calculate Mixed Partial Derivative \( f_{tx} \)

Differentiate the first partial derivative with respect to \( t \) with respect to \( x \):\[ f_{tx} = \frac{\partial}{\partial x}(3t^2 - 4x^2) = -8x \].

Step 7: Confirm Equality of Mixed Partials

Now, confirm that the mixed partial derivatives \( f_{xt} \) and \( f_{tx} \) are equal. From steps 5 and 6, we found:\[ f_{xt} = -8x \] and \[ f_{tx} = -8x \].Both mixed partial derivatives are equal, confirming Clairaut's theorem for this function.

Key Concepts

First-Order Partial Derivative

A first-order partial derivative is a derivative taken of a function with respect to one variable while holding the other variable constant. This is useful in multivariable calculus where functions depend on more than one variable. For the function \( f(x, t) = t^3 - 4x^2t \), there are two main first-order partial derivatives:

• Partial derivative with respect to \( x \): This involves differentiating the function, treating \( t \) as a constant. In this exercise, this resulted in: \( f_x = -8xt \).
• Partial derivative with respect to \( t \): Here, \( x \) is treated as constant. The derivative is calculated to be \( f_t = 3t^2 - 4x^2 \).

First-order partial derivatives help us understand how the function changes as one variable changes, providing foundational insights into the behavior of multivariable functions.
While the above process may seem similar to taking an ordinary derivative, the key distinction is holding other variables constant. This approach allows us to study each variable's individual effect on the function.

Second-Order Partial Derivative

The second-order partial derivatives involve differentiating the first-order derivatives again, to explore deeper relationships within the function. This process can be conducted twice with respect to the same variable or different variables.

• Differentiating twice with respect to \( x \), known as \( f_{xx} \): For our function, this yields \( f_{xx} = -8t \).
• Differentiating twice with respect to \( t \), known as \( f_{tt} \): The function returns \( f_{tt} = 6t \).

Additionally, mixed derivatives analyze the change with respect to both variables differently, such as:

• First with respect to \( x \), then \( t \), \( f_{xt} \): Resulting in \( -8x \).
• First with respect to \( t \), then \( x \), \( f_{tx} \): Again resulting in \( -8x \).

These second-order derivatives can describe curvature and other higher-level characteristics of the surface described by the function. Understanding these derivatives provides valuable insights into the relationships between variables.

Clairaut's Theorem

Clairaut's theorem states that if the mixed partial derivatives of a function are continuous, then the mixed partial derivatives are equal. For multivariable functions like \( f(x, t) = t^3 - 4x^2t \), this theorem verifies the commutativity of mixed derivatives under certain conditions.In our exercise, the function meets these conditions, allowing us to see that:

• The mixed partial derivative \( f_{xt} \) is \( -8x \).
• The mixed partial derivative \( f_{tx} \) is also \( -8x \).

Since these are equal, Clairaut's theorem holds true for this function. Demonstrating this equality not only confirms the smoothness of the function but also assures us about the predictability in changes of function values across variables. Clairaut's theorem simplifies the analysis of mixed partial derivatives considerably, especially in complex functions.