Unveiling The Secrets Of Multivariable Calculus: A Deep Dive

Hey guys! Let's dive into the fascinating world of multivariable calculus. We're going to explore a cool function, f(x, y) = (e^x - 5x)cos(y), and uncover some of its secrets. We'll be focusing on how this function behaves, especially around a specific point. Our main goal is to find a vector that points in the direction where the function decreases the fastest at the point (4, 5). Sounds like fun, right?

Understanding the Function and the Surface

Alright, let's break down what we're dealing with. The function f(x, y) takes two inputs, x and y, and spits out a single output, z. We can think of this as a surface in 3D space, where the height of the surface at any point (x, y) is given by the value of f(x, y). Imagine a landscape, and this function defines the hills and valleys of that landscape. The given surface is defined by z = f(x, y). Now, a key concept here is the level curve. Picture slicing this 3D landscape horizontally with a plane. The intersection of this plane with the surface creates a curve. This curve is a level curve. All points on this curve have the same z-value. This value is constant for a specific level curve.

So, think of contour lines on a topographic map. Each contour line represents a level curve, and the height of the terrain (our z-value) is constant along any given contour line. We're going to use this concept to explore the behavior of our function and find the direction of its steepest descent. Understanding the surface means understanding how the value of z changes with changes in x and y. The function f(x, y) combines an exponential component (e^x - 5x) with a cosine component (cos(y)). The exponential part changes dramatically as x changes, while the cosine part oscillates between -1 and 1 as y changes. This means we'll have areas where f increases rapidly, decreases rapidly, or stays relatively constant, depending on the values of x and y. This understanding is crucial for finding the direction where the function decreases the most rapidly. We will need to compute the gradient of f. The gradient tells us the direction of the greatest increase of the function. The opposite of the gradient will then give us the direction of the greatest decrease.

Exploring the Level Curves

As we discussed, level curves are essential for understanding the function's behavior. They are curves in the xy-plane where f(x, y) has a constant value. Mathematically, for a constant c, a level curve is defined by the equation f(x, y) = c. To visualize this, imagine taking horizontal slices through our 3D surface. Each slice reveals a 2D curve, a level curve. The shape of these curves depends on the function f(x, y). For our specific function f(x, y) = (e^x - 5x)cos(y), the level curves will be complex due to the interplay of the exponential and cosine terms. However, they will help us understand how the function changes in the xy-plane. The gradient vector, which we'll calculate later, will always be perpendicular to these level curves. Understanding this perpendicularity is key to solving the problem.

The Gradient: Our Guiding Vector

Now, let's talk about the gradient. The gradient of a function is a vector that points in the direction of the function's greatest rate of increase. It's like a compass that tells you which way is uphill on our 3D landscape. For a function of two variables, f(x, y), the gradient is given by:

∇f(x, y) = (∂f/∂x, ∂f/∂y)

Where ∂f/∂x and ∂f/∂y are the partial derivatives of f with respect to x and y, respectively. The partial derivative ∂f/∂x tells us how f changes as x changes, keeping y constant. Similarly, ∂f/∂y tells us how f changes as y changes, keeping x constant. So, to find the gradient of our function, we need to calculate these partial derivatives. Then we'll evaluate the gradient at the point (4, 5). Finally, we'll find the direction of the greatest decrease, which is the negative of the gradient.

Let's get down to brass tacks and calculate those partial derivatives. Remember, our function is f(x, y) = (e^x - 5x)cos(y). Here are the steps:

Calculating Partial Derivatives

First, let's find ∂f/∂x. We treat y as a constant and differentiate with respect to x:

∂f/∂x = (d/dx)(e^x - 5x)cos(y) = (e^x - 5)cos(y)

Next, let's find ∂f/∂y. We treat x as a constant and differentiate with respect to y:

∂f/∂y = (e^x - 5x)(d/dy)cos(y) = -(e^x - 5x)sin(y)

So, the gradient of f(x, y) is:

∇*f(x, y) = ((e^x - 5)cos(y), -(e^x - 5x)sin(y))

Awesome, now that we have the gradient, let's evaluate it at the point (4, 5). This will give us the specific direction of the greatest increase at that point. Let's do it!

Evaluating the Gradient at (4, 5)

Now, let's plug in x = 4 and y = 5 into our gradient vector:

∇f(4, 5) = ((e^4 - 5)cos(5), -(e^4 - 54)sin(5))

Let's approximate this using a calculator:

e^4 ≈ 54.60 cos(5) ≈ 0.2837 sin(5) ≈ -0.9589

∇*f(4, 5) ≈ ((54.60 - 5)0.2837, -(54.60 - 20)(-0.9589))

∇*f(4, 5) ≈ (14.07, -33.56)

So, the gradient at (4, 5) is approximately (14.07, -33.56). This vector points in the direction of the greatest increase of f at the point (4, 5). But, we want the direction in which f decreases most rapidly.

The Direction of Fastest Decrease

Since the gradient points in the direction of the greatest increase, the direction of the greatest decrease is simply the negative of the gradient. This is because we want the vector that points in the opposite direction.

Therefore, to find the vector in the direction in which f decreases most rapidly at (4, 5), we take the negative of our calculated gradient:

-∇*f(4, 5) ≈ (-14.07, 33.56)

This vector (-14.07, 33.56) is perpendicular to the level curve of f through the point (4, 5) and points in the direction where f decreases most rapidly. This means, if you were standing at the point (4, 5) on the surface z = f(x, y), and you wanted to descend the fastest, you would move in this direction. This is our answer! The vector which is perpendicular to the level curve of f through the point (4,5) in the direction in which f decreases most rapidly is approximately (-14.07, 33.56). That's a wrap!

Final Answer

The vector that points in the direction where f decreases most rapidly at the point (4, 5) is approximately (-14.07, 33.56). This vector is perpendicular to the level curve and represents the direction of the steepest descent on the surface z = f(x, y). This means that if you start at the point (4, 5) and move along this vector, the value of the function f will decrease most quickly. This concept is fundamental in optimization problems in calculus, where we aim to find the maximum or minimum values of functions. Finding the gradient and its negative is a critical step in this process.

This exploration highlights the power of calculus in understanding and analyzing the behavior of functions in multivariable settings. By calculating the gradient and understanding its relationship to level curves, we can predict the direction of change and solve optimization problems. Keep up the good work and keep exploring!"