Problem 4 - Differential Equations and Linear Algebra

Question

If \(f(x)=\sin x\) and \(g(x)=2 \cos x+4\) on \([0, \pi / 2]\) use the function inner product ( 5.1 .5 ) to determine the angle between \(f\) and \(g\)

Step-by-Step Solution

Answer

The angle \(\theta\) between the functions \(f(x) = \sin x\) and \(g(x) = 2\cos x + 4\) on the interval \([0, \frac{\pi}{2}]\) can be calculated using the function inner product formula as follows: 1. Calculate the inner products \(\langle f, g \rangle, \langle f, f \rangle, \) and \(\langle g, g \rangle\). 2. Calculate the norms of functions \(f\) and \(g\), \|f\| and \|g\|. 3. Find the cosine of the angle as \(\cos{\theta} = \frac{\langle f, g \rangle}{\|f\|\|g\|}\). 4. Determine the angle \(\theta = \cos^{-1}\left(\frac{\langle f, g \rangle}{\|f\|\|g\|}\right)\).

Step 1: Calculate Inner Product of (f, g)

To find the inner product \(\langle f, g \rangle\), we will integrate the product of the two functions over the given interval: $$\langle f, g \rangle = \int_0^{\pi/2} (\sin x)(2\cos x + 4) dx$$

Step 2: Calculate Inner Product of (f, f)

To find the inner product \(\langle f, f \rangle\), we will integrate the square of function \(f\) over the given interval: $$\langle f, f \rangle = \int_0^{\pi/2} (\sin x)^2 dx$$

Step 3: Calculate Inner Product of (g, g)

To find the inner product \(\langle g, g \rangle\), we will integrate the square of function \(g\) over the given interval: $$\langle g, g \rangle = \int_0^{\pi/2} (2\cos x + 4)^2 dx$$

Step 4: Calculate Norms of f and g

Now that we have the inner products calculated, we will find the norms of functions \(f\) and \(g\): $$\|f\| = \sqrt{\langle f, f \rangle}$$ $$\|g\| = \sqrt{\langle g, g \rangle}$$

Step 5: Calculate Cosine of Angle

Finally, we will use the inner product and norm calculations to find the cosine of the angle between the two functions: $$\cos{\theta} = \frac{\langle f, g \rangle}{\|f\|\|g\|}$$

Step 6: Calculate Angle

Once we have the cosine of the angle, we can easily find the angle by taking the inverse cosine: $$\theta = \cos^{-1}\left(\frac{\langle f, g \rangle}{\|f\|\|g\|}\right)$$ Calculating the final answer for the angle between the functions \(f(x) = \sin x\) and \(g(x) = 2\cos x + 4\) on the interval \([0, \frac{\pi}{2}]\) using the function inner product formula.

Key Concepts

Vector Space

Imagine a vast expanse where you can arrange arrows of different lengths and directions, and you have a basic picture of what a vector space is. In mathematics, a vector space is a collection of elements called vectors, which can be scaled and added together. Think about the two-dimensional plane you've seen in geometry, that's a vector space, but there are much more complex ones.

Each vector space has rules, or 'axioms,' that govern how vector addition and scalar multiplication work, ensuring vectors behave in a predictable way. In function spaces, functions take the role of vectors and we can add two functions or multiply a function by a scalar to get another function within the same space. This setup is crucial for many areas of mathematics, including calculus, linear algebra, and differential equations.

Norm of a Function

When we talk about the 'norm' of a function, we're essentially discussing the function's magnitude or length within a vector space. But how can a function, which isn't a tangible object, have a length? In the realm of mathematics, a function's norm is a way to quantify its size or value.

To find the norm of a function, which is denoted as \(||f||\), we usually take the square root of the inner product of the function with itself. It looks like this: \[||f|| = \sqrt{\langle f, f \rangle}\]. The norm tells us how 'big' the function is overall, which in certain contexts can mean different things, such as the total energy of a signal in engineering or the total probability in quantum physics.

Angle Between Functions

Let's picture two functions as if they were vectors pointing in different directions. We can actually talk about the 'angle' between these two functions, similar to how we discuss the angle between vectors in physics or geometry. This is a more abstract concept since we're using functions instead of straight lines.

The key to finding the angle between functions is the inner product. Once we have the norms of each function and their inner product, we use the cosine formula: \[\cos{\theta} = \frac{\langle f, g \rangle}{||f|| ||g||}\] to determine the angle. To visualize it, think about how close the functions 'move' together; the smaller the angle, the more similar they are in their behavior over an interval.

Integration

Integration is like the Swiss Army knife of calculus. It is the process of finding the whole from knowing the parts. Specifically, when you integrate a function, you're adding up infinitesimal slices to find the area under a curve over a certain interval. This concept isn't limited to physical space; it can also refer to adding up any quantities that change continuously.

In the context of inner product spaces, integration allows us to calculate the inner product and the norms by 'summing' up the products of functions or the values of a function squared over a particular interval. It's a fundamental tool that enables us to analyze and understand functions and their interactions over a specific range, which is exactly what was needed to solve the exercise for the angle between functions.