Find F'(π) If F(x) = Sin(4x): Calculus Solution

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Hey guys! Today, we're diving into a fun little calculus problem. We need to find the value of the derivative of the function f(x) = sin(4x), specifically at the point x = π. Buckle up; it's going to be a smooth ride!

Understanding the Problem

Before we jump into solving, let's break down what we're actually trying to do. We have a function, f(x) = sin(4x). The derivative, f'(x), represents the instantaneous rate of change of this function at any given point x. In simpler terms, it tells us how much the function's value is changing as x changes. We want to find the value of this rate of change specifically when x is equal to π.

So, our mission is clear: first, find the derivative f'(x), and then plug in x = π to get our answer. Let's get to it!

Step 1: Finding the Derivative f'(x)

To find the derivative of f(x) = sin(4x), we'll need to use the chain rule. The chain rule is a fundamental concept in calculus that helps us differentiate composite functions – functions that are made up of other functions. In our case, sin(4x) is a composite function because it's the sine function applied to the function 4x.

The chain rule states that if you have a composite function f(g(x)), then its derivative is f'(g(x)) * g'(x). In other words, you take the derivative of the outer function (with the inner function left as is) and then multiply by the derivative of the inner function.

Let's apply this to our function, f(x) = sin(4x). Here, the outer function is sin(u) and the inner function is u = 4x. So, we have:

  • Outer function: sin(u), derivative is cos(u)
  • Inner function: 4x, derivative is 4

Using the chain rule, we get:

f'(x) = cos(4x) * 4 = 4cos(4x)

Alright! We've found the derivative. Now, let's move on to the next step.

Step 2: Evaluating f'(π)

Now that we have f'(x) = 4cos(4x), we can find f'(π) by simply plugging in x = π into the derivative:

f'(π) = 4cos(4π)

Now, we need to figure out what cos(4π) is. Remember your unit circle! cos(θ) represents the x-coordinate of a point on the unit circle at an angle of θ.

4π is just two full rotations around the unit circle (since 2π is one full rotation). So, cos(4π) is the same as cos(0), which is 1. Therefore:

f'(π) = 4 * 1 = 4

And that's it! We've found that the value of the derivative of f(x) = sin(4x) at x = π is 4.

Wrapping Up

So, to recap, we wanted to find the value of f'(π) for the function f(x) = sin(4x). We used the chain rule to find the derivative f'(x) = 4cos(4x), and then we plugged in x = π to get f'(π) = 4.

Understanding the chain rule is crucial for tackling many calculus problems, so make sure you're comfortable with it. And always remember your unit circle values – they come in handy more often than you might think!

Why This Matters: Real-World Applications

Okay, so we solved a calculus problem. But why is this important? Well, derivatives have countless applications in the real world. They're used in physics to calculate velocities and accelerations, in engineering to optimize designs, in economics to model growth and change, and even in computer graphics to create realistic animations.

Let's think about a simple example: Suppose f(x) = sin(4x) represents the position of a particle oscillating back and forth. The derivative, f'(x) = 4cos(4x), then represents the velocity of that particle. Knowing the velocity at a specific time (like x = π) can be crucial for understanding the particle's motion.

Derivatives in Physics:

In physics, derivatives are the backbone of understanding motion. Velocity, acceleration, and jerk (the rate of change of acceleration) are all derivatives of position with respect to time. For example:

  • Velocity: The first derivative of position with respect to time.
  • Acceleration: The second derivative of position with respect to time (or the first derivative of velocity).
  • Jerk: The third derivative of position with respect to time (or the first derivative of acceleration).

These concepts are used to model everything from the trajectory of a baseball to the movement of planets.

Derivatives in Engineering:

Engineers use derivatives to optimize designs and processes. For instance, they might use derivatives to find the maximum load a bridge can support or the minimum amount of material needed to build a container.

  • Optimization: Derivatives help find maximum and minimum values, crucial for optimizing designs.
  • Rate of Change Analysis: Understanding how one variable changes with respect to another helps in designing efficient systems.

Derivatives in Economics:

In economics, derivatives are used to model economic growth, predict market trends, and optimize resource allocation.

  • Marginal Analysis: Economists use derivatives to analyze marginal cost, marginal revenue, and marginal profit.
  • Growth Models: Derivatives help model and predict economic growth rates.

Derivatives in Computer Graphics:

Derivatives are used in computer graphics to create smooth animations and realistic lighting effects.

  • Animation: Derivatives help create smooth, natural-looking movements.
  • Lighting and Shading: Derivatives are used to calculate how light reflects off surfaces, creating realistic images.

Practice Problems

Want to test your understanding? Try these practice problems:

  1. Find f'(π/2) if f(x) = cos(2x).
  2. Find g'(0) if g(x) = sin(3x).
  3. Find h'(π/4) if h(x) = tan(x) (Hint: The derivative of tan(x) is sec²(x)).

Work through these problems, and you'll solidify your understanding of derivatives and the chain rule. Remember to break down each problem into steps, identify the inner and outer functions, and apply the chain rule carefully.

That's all for today, folks! Keep practicing, and you'll become a calculus pro in no time. Happy calculating!