Question

Find the linear approximation to the given functions at the specified points. Plot the function and its linear approximation over the indicated interval. $$ f(x)=x^{2} \text { at } a=2,[0,3] $$

Short answer

The linear approximation is \( L(x) = 4x - 4 \).

Step-by-step solution

Understand the Linear Approximation

The linear approximation of a function \( f(x) \) at a certain point \( a \) is given by the equation of the tangent line at that point. The tangent line can be expressed as \( L(x) = f(a) + f'(a)(x-a) \). Here, \( f(a) \) is the function value at \( x = a \), and \( f'(a) \) is the derivative of the function at \( x = a \).

Calculate the Function Value at a

First, calculate \( f(a) \) for \( f(x) = x^2 \) at \( a = 2 \).\[ f(2) = 2^2 = 4 \]So, \( f(a) = 4 \).

Find the Derivative of the Function

Calculate the derivative \( f'(x) \) of the function \( f(x) = x^2 \).\[ f'(x) = 2x \]

Evaluate the Derivative at a

Now, find \( f'(a) \) by substituting \( x = 2 \) into the derivative.\[ f'(2) = 2 imes 2 = 4 \]

Formulate the Linear Approximation

With \( f(a) = 4 \) and \( f'(a) = 4 \), substitute into the linear approximation formula:\[ L(x) = 4 + 4(x - 2) \]Simplify to get:\[ L(x) = 4 + 4x - 8 = 4x - 4 \]

Plot the Function and Linear Approximation

To complete the task, plot both \( f(x) = x^2 \) and its linear approximation \( L(x) = 4x - 4 \) over the interval \([0,3]\). You should observe that \( L(x) \) approximates \( f(x) \) very closely around the point \( x=2 \).

Key concepts

Calculus

Calculus is a branch of mathematics focused on studying change. It primarily deals with functions, rates of change, and the accumulation of quantities. At the heart of calculus are two fundamental concepts:

• Differentiation: This refers to finding the derivative of a function, representing the rate at which the function's value changes as its input changes.
• Integration: This involves finding the total accumulation of a quantity, often representing areas under curves.

In the context of linear approximation, calculus helps us understand how we can use the tangent line to approximate the function's value near a specific point.
By applying differentiation, we can determine how steeply a function is rising or falling, which is crucial to accurate linear approximations.

Differentiation

Differentiation is a core concept of calculus that describes computing the derivative of a function. A derivative is a formula that tells us how the function is changing at any point. This process allows us to determine the slope of the tangent line at any given point on a curve.
Taking our example of the function \( f(x) = x^2 \), differentiation helps us find \( f'(x) \), which symbolizes the instantaneous rate of change of \( f(x) \).
In our specific exercise, the derivative \( f'(x) \) was calculated as \( 2x \), leading to a value of \( 4 \) when evaluated at \( x = 2 \). This derivative value is crucial for forming the linear approximation, as it determines the slope of the tangent line.

Tangent Line

A tangent line is a straight line that touches a curve at a single point without crossing it near that point. It represents the best linear approximation of a function at a given point.

• The equation of the tangent line can be given by the formula: \( L(x) = f(a) + f'(a)(x-a) \).
• This can be understood as the function value at a specific point, adjusted by the product of the derivative at that point and the distance from the point \( a \).

In the example, the tangent line at \( x = 2 \) for the function \( f(x) = x^2 \) was represented as \( L(x) = 4x - 4 \).
This line closely mirrors the behavior of the function around \( x = 2 \), allowing us to use it as a reliable approximation over short intervals.

Function Approximation

Function approximation is a significant concept in mathematics, particularly useful when dealing with functions that are complex or difficult to solve analytically.
Linear approximation is one straightforward method of approximating a function near a point using the line tangent to the function at that point.

• The closer the point \( x \) is to \( a \), the more accurate the approximation \( L(x) \) will be.
• This method assumes the curve is nearly straight, which holds true over small intervals.

By approximating \( f(x) \) with \( L(x) \), we simplify calculations, especially over defined short ranges.
In our exercise, \( L(x) = 4x - 4 \) provides a near-exact prediction of \( f(x) = x^2 \) near \( x = 2 \), showcasing the effectiveness of linear approximation in function approximation.