Problem 28 - Applied Calculus

Question

For Problems 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

The second-order partial derivatives are: \( f_{xx} = -8t \), \( f_{tt} = 6t \), and the mixed partials \( f_{xt} = f_{tx} = -8x \), confirming they are equal.

Step 1: Find the First Partial Derivatives

First, we need to find the first partial derivative of the function with respect to each variable, \( x \) and \( t \). For \( f(x, t) = t^3 - 4x^2t \), the partial derivative with respect to \( x \) is \( f_x = \frac{\partial}{\partial x}(t^3 - 4x^2t) = -8xt \), and the partial derivative with respect to \( t \) is \( f_t = \frac{\partial}{\partial t}(t^3 - 4x^2t) = 3t^2 - 4x^2 \).

Step 2: Find the Second Partial Derivatives with Respect to x

Now we'll find the second derivative of each first derivative with respect to \( x \). The second partial with respect to \( x \) is \( f_{xx} = \frac{\partial}{\partial x}(-8xt) = -8t \). Next, we find the mixed partial \( f_{xt} = \frac{\partial}{\partial t}(-8xt) = -8x \).

Step 3: Find the Second Partial Derivatives with Respect to t

Now find the second derivative of each first derivative with respect to \( t \). The second partial with respect to \( t \) is \( f_{tt} = \frac{\partial}{\partial t}(3t^2 - 4x^2) = 6t \). Then, the mixed partial \( f_{tx} = \frac{\partial}{\partial x}(3t^2 - 4x^2) = -8x \).

Step 4: Confirm that the Mixed Partials are Equal

To confirm that the mixed partial derivatives are equal, we check \( f_{xt} \) against \( f_{tx} \). As calculated, \( f_{xt} = -8x \) and \( f_{tx} = -8x \), confirming that the mixed partial derivatives are indeed equal.

Key Concepts

Second-order Derivatives

Calculating second-order derivatives involves taking the derivative of a first-order derivative. This process is crucial for analyzing how a function behaves and changes, particularly in multiple dimensions. When you have a function like \( f(x, t) = t^3 - 4x^2t \), the first-order partial derivative with respect to a variable, say \( x \), involves differentiating the function while treating the other variable \( t \) as constant.
Once the first-order derivatives are found, like \( f_x = -8xt \) and \( f_t = 3t^2 - 4x^2 \) in this case, the second-order derivatives require differentiating the first-order results again:
- For \( f_{xx} \), differentiate \( f_x = -8xt \) with respect to \( x \), resulting in \( -8t \).
- For \( f_{tt} \), differentiate \( f_t = 3t^2 - 4x^2 \) with respect to \( t \), resulting in \( 6t \).
These second derivatives give you insight into the concavity and the curvature of the function's surface.

Mixed Partials

In calculus, mixed partial derivatives involve differentiating a function first with respect to one variable and then with respect to another. These are critical when dealing with functions of multiple variables, like our example \( f(x, t) = t^3 - 4x^2t \).
The mixed partials are:
- \( f_{xt} \): Differentiate \( f_x = -8xt \) with respect to \( t \), yielding \( -8x \).
- \( f_{tx} \): Differentiate \( f_t = 3t^2 - 4x^2 \) with respect to \( x \), also yielding \( -8x \).
These calculations show a key property: the order of differentiation doesn’t matter—\( f_{xt} \) equals \( f_{tx} \). This property is often used to simplify calculations and confirm correctness in problems.

Calculus Problem Solving

When tackling problems in calculus, breaking down the process can help. Take the problem of finding all second-order derivatives and verifying if the mixed partials are equal. Start by identifying the function and the variables.
Next, calculate the first derivatives with respect to each variable. In our example, \( f_x \) and \( f_t \) were evaluated to be \( -8xt \) and \( 3t^2 - 4x^2 \), respectively.
For the second-order derivatives, differentiate the first-order derivatives again accordingly. This might involve differentiating once more with respect to the same variable, or a different one in the case of mixed partials.

• Compute regular second-order partials: \( f_{xx} = -8t \), \( f_{tt} = 6t \).
• Compute mixed partial derivatives: \( f_{xt} = -8x \) and \( f_{tx} = -8x \).

Finally, verify that the mixed partial derivatives are equal, affirming the symmetry in the problem, which can act as a check for correctness. This structured approach helps in systematically solving calculus problems.