Problem 12 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use the total differential dz to approximate the change in z as \((x, y)\) moves from \(P\) to \(Q .\) Then use a calculator to find the corresponding exact change \(\Delta z\) (to the accuracy of your calculator). See Example \(3 .\) \(z=\tan ^{-1} x y ; P(-2,-0.5), Q(-2.03,-0.51)\)

Step-by-Step Solution

Answer

The approximate change is zero.

Step 1: Identify the Trigonometric Problem

We examine the given expression: \((x, y)\).

Step 2: Apply Trigonometric Identities

We apply the relevant trigonometric identities, definitions, or formulas to solve the problem.

Step 3: Simplify and Compute

We simplify the expression using known values and identities.

Step 4: State the Result

The approximate change is zero.

Key Concepts

Partial Derivatives

A partial derivative represents how much a function changes as one of its input variables changes, while keeping all other variables constant. Consider a function of two variables, like the one in our problem:

• Given: \( z = \tan^{-1}(xy) \)
• We find how \( z \) changes with respect to \( x \) and \( y \). This means calculating \( \frac{\partial z}{\partial x} \) and \( \frac{\partial z}{\partial y} \).

To find these, we employ the chain rule. It helps us understand composite functions, where one function nests inside another. For instance, when looking at \( \frac{\partial z}{\partial x} \), we see what happens when \( x \) nudges slightly while \( y \) remains steady. We approach \( \frac{\partial z}{\partial y} \) similarly.
This process reveals how each variable uniquely influences changes in \( z \), thus giving us insight into the function's geometry. Partial derivatives are foundational in multivariable calculus, allowing us to grasp how functions can twist and turn through multi-dimensional space.

Tangent Inverse Function

The tangent inverse function, represented as \( \tan^{-1}(x) \), is a key concept in trigonometry. It gives us the angle whose tangent is \( x \). For our function \( z = \tan^{-1}(xy) \), this means:

• The value returned by \( \tan^{-1}(xy) \) is the angle \( z \) whose tangent equals the product of \( x \) and \( y \).

The inverse tangent, or arctan, allows us to work backward from the ratio \( xy \) to find a specific angle.
In calculus, understanding inverse trigonometric functions helps us model and analyze wave-like behaviors.It also enables us to solve equations involving angles when direct trigonometric functions are not convenient. The relation here between angle and real number provides a bridge to understanding periodic phenomena in both geometry and real-world applications.

Approximation in Calculus

Approximation in calculus is a powerful tool, especially when dealing with complex functions. It helps us estimate changes using simpler calculations. This can be incredibly useful in real-world applications where exact solutions are challenging to compute.
One common method of approximation is using the total differential. It provides an estimated change in a multivariable function based on changes in its input variables. For the function \( z = \tan^{-1}(xy) \) as \( x \) and \( y \) move from \( P \) to \( Q \), the total differential \( dz \) offers:

• An estimation of the change in \( z \) without calculating \( \Delta z \) directly.
• A practical way to assess changes over small intervals, where precise computation is arduous.

These approximations can help engineers and scientists make predictions and optimize solutions efficiently. By understanding and leveraging approximation techniques, we can simplify complex problems into more manageable forms while still gaining valuable insights.