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Magazine Chapter 7: Problem 2
In Problems 1-5, find the linear approximation of \(f(x)\) at \(x=0\). $$ f(x)=\sin (3 x) $$
Short Answer
Expert verified
The linear approximation of \(f(x) = \sin(3x)\) at \(x=0\) is \(L(x) = 3x\).
Step by step solution
01
Understand Linear Approximation
Linear approximation of a function at a point is given by the formula: \(L(x) = f(a) + f'(a)(x - a)\). This is essentially the equation of the tangent line at the point \(x = a\). Here, we are asked to find the linear approximation at \(x = 0\).
02
Identify Function and Point of Approximation
The function given is \(f(x) = \sin(3x)\). We need to find the linear approximation at \(x = 0\), which means we need to find \(f(0)\) and \(f'(0)\).
03
Calculate \(f(0)\)
Substitute \(x = 0\) into the function: \(f(0) = \sin(3 \times 0) = \sin(0) = 0\). So, \(f(0) = 0\).
04
Compute the Derivative \(f'(x)\)
To find the linear approximation, compute the derivative \(f'(x)\). The derivative of \(f(x) = \sin(3x)\) is \(f'(x) = 3\cos(3x)\).
05
Calculate \(f'(0)\)
Now substitute \(x = 0\) into the derivative: \(f'(0) = 3\cos(3 \times 0) = 3\cos(0) = 3 \times 1 = 3\).
06
Write the Linear Approximation
Substitute \(f(0)\) and \(f'(0)\) into the linear approximation formula: \(L(x) = f(0) + f'(0)(x - 0) = 0 + 3x = 3x\). Thus, the linear approximation of \(f(x)\) at \(x=0\) is \(L(x) = 3x\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Calculus
Calculus is a branch of mathematics that focuses on the study of change and motion. It is essential for understanding a wide range of phenomena in science and engineering. Calculus is typically divided into two main subfields:
- Differential Calculus: This area deals with the concept of the derivative, which represents the rate of change of a function. It helps in analyzing curves and slopes of lines through the calculation of derivatives.
- Integral Calculus: This subfield deals with the calculation of integrals, which represent the accumulation of quantities and can determine areas under curves.
Derivative
The derivative of a function is a fundamental concept in differential calculus. It expresses how a function changes as its input changes.
- Definition: The derivative of a function \(f(x)\) at a particular point \(x = a\) is the slope of the tangent line to the curve at that point. It is denoted as \(f'(a)\) or \(\frac{df}{dx}(a)\).
- Calculation: To find the derivative of a function, you apply differentiation rules such as the power rule, product rule, quotient rule, and chain rule. For example, in the function we worked with, \(f(x) = \sin(3x)\), the derivative was found to be \(f'(x) = 3\cos(3x)\).
- Application: Derivatives are crucial for finding linear approximations, as they provide the slope needed for the equation of the tangent line.
Tangent Line
The tangent line to a curve at a particular point is a straight line that best approximates the curve at that point. It touches the curve but doesn鈥檛 cut through it, meaning it just "grazes" the surface of the curve.
- Equation: The equation of a tangent line at a point \((a, f(a))\) on the curve \(y = f(x)\) is given by \(L(x) = f(a) + f'(a)(x - a)\). This is our linear approximation formula.
- Importance: The tangent line provides a linear model of the function's behavior near the point \(x = a\). This is valuable for making predictions and performing calculations close to that point.
- Example: In the given problem, the tangent line at \(x=0\) was calculated using \(f(0)\) and \(f'(0)\), resulting in a simple linear equation: \(L(x) = 3x\).
Sine Function
The sine function, denoted as \(\sin(x)\), is a crucial component of trigonometry and appears frequently in calculus.
- Characteristics: Sine is a periodic function with a period of \(2\pi\), meaning it repeats its values in regular intervals. When graphed, it produces a smooth, wave-like pattern.
- Sine in Calculus: The sine function is smooth and continuous, making it differentiable everywhere. Its derivative is another trigonometric function: the cosine function. Specifically, \(\frac{d}{dx} [\sin(x)] = \cos(x)\).
- Example: In our exercise, we dealt with \(f(x) = \sin(3x)\), which is a transformed sine function. The derivative, \(f'(x) = 3\cos(3x)\), where '3' is due to the chain rule, captures how quickly the sine function changes at any given point.
