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Magazine Chapter 6: Problem 5
Show in detail that the linear approximation of \(\sin x\) at \(x=0\) is \(L(x)=x\) and the linear approximation of \(\cos x\) at \(x=0\) is \(L(x)=1\)
Short Answer
Expert verified
At \(x=0\), \(L_{\sin}(x)=x\) and \(L_{\cos}(x)=1\).
Step by step solution
01
Understanding Linear Approximation
Linear approximation, also called linearization, of a function at a point is given by the formula \(L(x) = f(a) + f'(a)(x-a)\), where \(a\) is the point of approximation, \(f(a)\) is the function value at \(a\), and \(f'(a)\) is the derivative of the function at \(a\).
02
Setup for \(\sin x\)
To find the linear approximation of \(\sin x\) at \(x=0\), set \(a = 0\). The formula becomes \(L(x) = \sin(0) + \cos(0)(x-0)\).
03
Calculate \(\sin(0)\) and \(\cos(0)\)
\(\sin(0) = 0\) and \(\cos(0) = 1\). Use these values to update the linear approximation formula: \(L(x) = 0 + 1 \, x = x\).
04
Verify Result for \(\sin x\)
The linear approximation of \(\sin x\) at \(x=0\) is \(L(x) = x\). The terms fit the requirements as \(x\) becomes the approximation.
05
Setup for \(\cos x\)
To find the linear approximation of \(\cos x\) at \(x=0\), set \(a = 0\). The formula becomes \(L(x) = \cos(0) + (-\sin(0))(x-0)\).
06
Calculate \(\cos(0)\) and \(-\sin(0)\)
\(\cos(0) = 1\) and \(\sin(0) = 0\), thus \(-\sin(0) = 0\). Substitute these into the formula: \(L(x) = 1 + 0 \, x = 1\).
07
Verify Result for \(\cos x\)
The linear approximation of \(\cos x\) at \(x=0\) is \(L(x) = 1\). The terms again show the expected outcome.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Derivative
When we explore derivatives, we are delving into one of the core concepts of calculus. Think of a derivative as a way to capture how a function changes as its input changes.
This concept becomes crucial when performing a linear approximation because it provides the "slope" for our linearization formula. In our exercise, we used derivatives of the \(\sin x\) and \(\cos x\) functions at \(x=0\) to find their linear approximations.
- It's like finding the slope of a line tangent to the curve of the function at any given point.
- For a function like \(f(x)\), its derivative is denoted as \(f'(x)\) or \(\frac{df}{dx}\).
- The derivative tells us the rate of change of the function with respect to its variable.
This concept becomes crucial when performing a linear approximation because it provides the "slope" for our linearization formula. In our exercise, we used derivatives of the \(\sin x\) and \(\cos x\) functions at \(x=0\) to find their linear approximations.
Sin and Cos Functions
The sine and cosine functions, \(\sin x\) and \(\cos x\), are fundamental trigonometric functions. They describe the relationship between angles and sides of a right triangle and extend to describe repetitive oscillations, like waves.
In the linear approximation task, understanding \(\sin x\) and \(\cos x\) and their behavior at \(x=0\) provides the foundational values needed for approximation.
- \(\sin(0)\) is 0 because when an angle is 0, the opposite side compared to the hypotenuse is also 0.
- \(\cos(0)\) is 1 because the adjacent side is the same length as the hypotenuse at an angle of 0.
- Besides triangles, these functions are incredibly useful in physics, engineering, and any field that deals with cyclical patterns.
In the linear approximation task, understanding \(\sin x\) and \(\cos x\) and their behavior at \(x=0\) provides the foundational values needed for approximation.
Taylor Series
A Taylor Series is a powerful mathematical tool used to represent a function as an infinite sum of terms calculated from its derivatives at a single point. This concept helps approximate functions using polynomials.
Taylor Series allow approaches for complex functions using simpler polynomial expressions, and they are behind linear approximations for functions like \(\sin x\) and \(\cos x\).
- In our case, the linear approximation is essentially the first term of a Taylor Series expansion.
- The formula for Taylor Series is \(f(x) \approx f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \ldots\)
- For linear approximation, we simply use \(f(a) + f'(a)(x-a)\), which is the first-order Taylor polynomial.
Taylor Series allow approaches for complex functions using simpler polynomial expressions, and they are behind linear approximations for functions like \(\sin x\) and \(\cos x\).
Approximation
Approximation involves finding values or solutions that are close to the exact ones. In mathematics, and especially in calculus, approximation provides a way to simplify complex problems.
By approximating \(\sin x\) as \(x\) and \(\cos x\) as 1 at \(x = 0\), we get easy-to-use linear forms of these periodic functions.
- Linear approximation is a method used to estimate the value of a function near a specific point using its tangent line or linear function.
- It helps in computational efficiency and offers a good initial guess for more complex problems.
- While not exact, this method gives us results that are sufficiently close for practical purposes, especially when changes are small.
By approximating \(\sin x\) as \(x\) and \(\cos x\) as 1 at \(x = 0\), we get easy-to-use linear forms of these periodic functions.
Calculus
Calculus is the broad field of mathematics that deals with continuous change, using derivatives and integrals as fundamental tools. Derivatives measure rates of change, while integrals quantify accumulation.
The exercise with linear approximations of \(\sin x\) and \(\cos x\) leverages the concepts and methods foundational to calculus, reinforcing its practical application scope.
- Calculus helps us understand and describe the motion, growth, and varying quantities in mathematical terms.
- The tangent line problem and area under a curve are classical calculus problems that leverage derivatives and integrals, respectively.
- In essence, calculus expands our ability to solve practical problems and model real-world phenomena, from physics to economics.
The exercise with linear approximations of \(\sin x\) and \(\cos x\) leverages the concepts and methods foundational to calculus, reinforcing its practical application scope.
