Question

Find \(d f\) and \(\Delta f\). (a) If \(f(x, y)=x^{2}+3 y\), express \(d f\) in terms of \(d x\) and \(d y\). (b) For \(f(x, y)=x^{2}+3 y\), integrate from \((x, y)=\) \((1,1)\) to \((x, y)=(3,3)\) to obtain \(\Delta f\).

Short answer

For (a), \( df = 2x \, dx + 3 \, dy \). For (b), \( \Delta f = 14 \).

Step-by-step solution

Differentiate f with respect to x and y

Given the function is \[ f(x, y) = x^2 + 3y \] First, differentiate with respect to x to get: \[ \frac{\partial f}{\partial x} = 2x \] Then differentiate with respect to y to get: \[ \frac{\partial f}{\partial y} = 3 \]

Express total differential df

Now, use the total differential formula: \[ df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy \] Substitute the partial derivatives from Step 1: \[ df = 2x \, dx + 3 \, dy \]

Set up the integral for Δf

The expression for \( Δf \), which is the change in the function, can be determined by integrating the differential from the initial point (1,1) to the final point (3,3). Therefore, we set up the integral: \[ \Delta f = \int_{(1,1)}^{(3,3)} df = \int_{(1,1)}^{(3,3)} (2x \, dx + 3 \, dy) \]

Integrate with respect to x

Integrate the first term \( 2x \, dx \) from 1 to 3: \[ \int_1^3 2x \, dx = [x^2]_1^3 = 3^2 - 1^2 = 9 - 1 = 8 \]

Integrate with respect to y

Integrate the second term \( 3 \, dy \) from 1 to 3: \[ \int_1^3 3 \, dy = [3y]_1^3 = 3(3 - 1) = 3 \times 2 = 6 \]

Sum the integrals to find Δf

Add the results from Step 4 and Step 5: \[ \Delta f = 8 + 6 = 14 \]

Key concepts

partial derivatives

Partial derivatives are a fundamental concept in differential calculus. They measure how a function changes as each variable changes while keeping other variables constant. For instance, if we have a function of two variables, such as \( f(x, y) = x^2 + 3y \), we find the partial derivative with respect to \( x \) (notated as \( \frac{\partial f}{\partial x} \)) and with respect to \( y \) (notated as \( \frac{\partial f}{\partial y} \)).

Here’s how you do it:

• To compute \( \frac{\partial f}{\partial x} \), treat \( y \) as a constant and differentiate the function with respect to \( x \). In our example, \( \frac{\partial f}{\partial x} = 2x \).
• To compute \( \frac{\partial f}{\partial y} \), treat \( x \) as a constant and differentiate the function with respect to \( y \). For our function, \( \frac{\partial f}{\partial y} = 3 \).

Understanding partial derivatives is crucial because they help to explore how functions behave in multi-dimensional spaces.

total differential

The total differential, denoted as \( df \), provides a linear approximation of how a function changes with all its variables. It’s found by summing up the differential changes of each variable's partial derivatives.

For our function \( f(x, y) = x^2 + 3y \), the formula for the total differential is: \[ df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy \] By substituting our partial derivatives, we get: \[ df = 2x \, dx + 3 \, dy \] This equation tells us how small changes in \( x \) and \( y \) will affect the function \( f(x, y) \).

The total differential is a powerful tool for approximating changes in multi-variable functions. By understanding it, you can predict how these tiny variations influence the entire system.

definite integration

Integration helps to find the total accumulation of quantities, such as areas under curves. Definite integration, in particular, calculates the total change over a specific interval.

Given \[ \Delta f = \int_{(1,1)}^{(3,3)} ( 2x \, dx + 3 \, dy ) \], we split the integral into two parts:

1. Integrate with respect to \( x \): \[ \int_1^3 2x \, dx = [x^2]_1^3 = 3^2 - 1^2 = 9 - 1 = 8 \]
2. Integrate with respect to \( y \): \[ \int_1^3 3 \, dy = [3y]_1^3 = 3(3 - 1) = 6 \]

By summing the results, we find the complete change in the function from the initial to the final point: \[ \Delta f = 8 + 6 = 14 \]

Definite integration is essential for evaluating how functions evolve over intervals, making them vital in not just mathematics, but physics, engineering, and beyond.

mathematical problem-solving

Mathematical problem-solving involves applying appropriate concepts and methods to derive solutions efficiently. The exercise above is a great example involving multiple steps:

• Differentiating to find partial derivatives.
• Using these derivatives to express the total differential.
• Setting up definite integrals to find the change \( \Delta f \).

To solve such problems successfully:

1. **Understand the problem**: Identify what's given and what needs to be found.
2. **Choose the right tools**: Know when to apply differentiation, integration, and other techniques.
3. **Keep the process organized**: Break the problem into manageable steps.

With practice and exposure to various problems, you’ll develop a strong intuition for mathematical problem-solving. Always recheck your work to avoid mistakes, and don’t hesitate to go over fundamental concepts again whenever necessary.