Problem 4 - Calculus with Concepts in Calculus

Question

Approximate the value of \(f\) at the given point. $$ f(x, y)=\sin \pi x y ;(-1.97,2.005) $$

Step-by-Step Solution

Answer

The approximate value of \( f(-1.97, 2.005) \) is 0.

Step 1: Identify the Function and Point

We are given the function \( f(x, y) = \sin(\pi x y) \) and the point \((-1.97, 2.005)\). Our goal is to approximate \( f(-1.97, 2.005) \).

Step 2: Calculate \(\pi x y\) for the Point

Insert the given values into the expression \(\pi x y\): \[\pi (-1.97) \times 2.005 = -\pi \times (1.97 \times 2.005) = -3.95515\pi\]

Step 3: Use Approximations for Constants

Use approximations for easier calculations. Assume \(\pi \approx 3.14\). Then,\[-3.95515 \times 3.14 \approx -12.4142\]

Step 4: Use Sine Function for Approximation

Now, find \(\sin(-12.4142)\). Since this is a difficult calculation without a calculator, note that sine function values range between -1 and 1, and small perturbations do not greatly change their values, so approximate using periodic properties of sine function:\[\sin(-12.4142) \approx \sin(2\pi) = 0\]

Step 5: Conclude with the Approximate Value

After all the calculations and approximations:\[ f(-1.97, 2.005) \approx 0 \]

Key Concepts

Trigonometric Functions

Trigonometric functions are a fundamental concept in mathematics, especially in topics like multivariable calculus. These functions, which include sine, cosine, and tangent, are used to relate the angles and sides of triangles. In the context of multivariable functions like \( f(x, y) = \sin(\pi x y) \), they help in understanding how values oscillate between -1 and 1 due to their cyclical nature.

• The sine function, \( \sin(x) \), is a periodic function with a period of \(2\pi\), which means it repeats its values every \(2\pi\) units.
• In multivariable calculus, trigonometric functions are often used to model periodic phenomena.

Understanding these functions' behavior is crucial when dealing with periodic processes or wave patterns in real-world applications. When approximating values of such functions, especially for complex inputs, recognizing their repetitive behavior simplifies calculations. This can often mean approximating by aligning with known values, like \(\sin(2\pi) = 0\) in our example.

Approximation Methods

In calculus, and particularly multivariable calculus, approximation methods are vital tools. They help us estimate the values of functions at difficult points to calculate exactly. The exercise with the function \( f(x, y) = \sin(\pi x y) \) demonstrates such a technique.

• Approximating \(\pi\) as 3.14 is a common method for simplifying calculations.
• By approximating \(-3.95515\pi\) as \(-12.4142\), you simplify the calculation for the sine function.
• Since directly calculating \(\sin(-12.4142)\) is complex, using known values of the sine function at periodic intervals helps simplify this.

These methods, while they involve simplifying assumptions, provide efficient ways to handle computations without complex tools like calculators. They are especially useful in real-world applications where exact values are often unnecessary or unobtainable.

Periodic Properties

The periodic properties of functions like sine and cosine play a significant role in their application in multivariable calculus. Understanding periodicity simplifies calculations, especially when dealing with large inputs. In the given solution, recognizing that the sine function is periodic every \(2\pi\) allowed for an easier analysis of the problem.

• Even complex values, when examined through their periodic nature, often cycle back to simpler, known values.
• Sine has a period of \(2\pi\), so any multiple of \(2\pi\) results in the sine value cycling back to zero, e.g., \(\sin(2\pi) = 0\) or \(\sin(4\pi) = 0\).
• Recognizing the periodic interval can often reduce lengthy calculations, giving accurate approximations quickly.

By using periodic properties, we step into an elegant aspect of mathematics where problems become easier to handle by leveraging symmetry and repeatability. This is crucial for solving many practical problems efficiently and with minimal computational effort.