Question

Numerically approximate the value of \(f^{\prime}(x)\) at the indicated \(x\) value. \(f(x)=\cos x\) at \(x=\pi\)

Short answer

The approximate value of \( f'(\pi) = 0.005 \).

Step-by-step solution

Understanding the Derivative

The derivative of a function, \( f'(x) \), represents the rate of change of the function at any given point. We are asked to find \( f'(x) \) for \( f(x) = \cos x \) at \( x = \pi \).

Formula for Numerical Differentiation

The numerical differentiation formula we'll use is the forward difference approximation given by \( f'(x) \approx \frac{f(x+h) - f(x)}{h} \), where \( h \) is a small increment. Often \( h \) is chosen as 0.01 or smaller for better accuracy.

Choosing a Small h Value

To approximate \( f'(x) \) accurately, we choose \( h = 0.01 \). This small step size allows us to calculate the slope of the function at \( x = \pi \).

Calculate \( f(x+h) \) and \( f(x) \)

Calculate \( f(x+h) = \cos(x + h) = \cos(\pi + 0.01) \) and \( f(x) = \cos(\pi) \). The cosine of \( \pi \) is -1, and you need a calculator for \( \cos(\pi + 0.01) \), which is approximately -0.99995.

Apply Forward Difference Approximation

Now, substitute the values into the formula: \[ f'(\pi) \approx \frac{\cos(\pi + 0.01) - \cos(\pi)}{0.01} = \frac{-0.99995 - (-1)}{0.01} = \frac{0.00005}{0.01} = 0.005 \]

Interpret the Result

The approximate value of the derivative \( f'(\pi) \) is 0.005. This represents the slope of the tangent to the curve at \( x = \pi \), implying a nearly horizontal tangent line since it's close to 0.

Key concepts

Understanding Derivatives

In calculus, the derivative of a function is a fundamental concept used to understand the rate at which the function changes. For any given function, like \( f(x) = \cos x \), the derivative, denoted as \( f'(x) \), represents how the function's output changes as the input \(x\) changes. This is a core part of understanding how a function behaves and reacts to changes in its input values.

The derivative is essentially the slope of the tangent line to the function at any specific point on its graph. Think of it like measuring the steepness of a hill. If the hill is very steep, the rate of change is high, and thus the derivative is large. If the hill flattens out, the rate of change decreases, and so does the derivative.

To calculate the derivative at a specific point \( x = \pi \) for \( f(x) = \cos x \), numerical methods such as forward difference approximation can be employed when analytical methods are cumbersome or impossible.

Forward Difference Approximation

Numerical differentiation is a technique used to estimate the derivative of a function, and one of the simplest methods is the forward difference approximation. This approach is particularly helpful when you have complex functions or lack a closed-form expression for the derivative.

The forward difference approximation formula is expressed as:

• \( f'(x) \approx \frac{f(x+h) - f(x)}{h} \)

Here, \( h \) represents a very small increment, and it's crucial because the smaller \( h \) is, the more accurate the approximation will be. Typically, \( h \) is selected as a small value like 0.01 or smaller.

When applying this to find the derivative of \( f(x) = \cos x \) at \( x = \pi \), you calculate both \( f(\pi + h) \) and \( f(\pi) \). Then, apply the formula to achieve an approximation. This straightforward calculation is a powerful tool in understanding the behavior of functions without extensive calculus.

Connecting with Rate of Change

The rate of change is another way to describe the concept of the derivative, providing tangible insights into how much a function's value will increase or decrease as its input changes. In the context of functions like \( f(x) = \cos x \), the rate of change at any point can describe how sharply or gently the cosine wave rises or falls.

With the approximation \( f'(\pi) \approx 0.005 \), this tells us that around \( x = \pi \), the change in the cosine function as you move a small step right from \( \pi \) is quite small, reflecting a nearly flat slope. When the derivative or rate of change is close to zero, it generally indicates a point where the function peaks or drops within its cycle.

Understanding the rate of change equips you with the ability to predict region-specific behaviors of functions, a vital aspect in fields that rely heavily on modeling real-world scenarios using mathematical functions.