Problem 40 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the linear approximation to the given functions at the specified points. Plot the function and its linear approximation over the indicated interval. $$ F(x)=3 x+4 \text { at } a=3,[0,6] $$

Step-by-Step Solution

Answer

The linear approximation of \( F(x) = 3x + 4 \) at \( x = 3 \) is \( L(x) = 3x + 4 \).

Step 1: Understand Linear Approximation

Linear approximation is the process of approximating a function by a tangent line at a given point. The formula for the linear approximation of a function \( f(x) \) at a point \( a \) is given by \( L(x) = f(a) + f'(a)(x-a) \).

Step 2: Evaluate the Function at \( a \)

Calculate \( F(a) \) for \( a = 3 \). Since \( F(x) = 3x + 4 \), we have \( F(3) = 3(3) + 4 = 13 \).

Step 3: Find the Derivative

Find the derivative of \( F(x) \). The derivative of \( F(x) = 3x + 4 \) is \( F'(x) = 3 \).

Step 4: Evaluate the Derivative at \( a \)

Evaluate \( F'(a) \) at \( a = 3 \). Since the derivative is constant, \( F'(3) = 3 \).

Step 5: Write the Linear Approximation Formula

Plug the values from Step 2 and Step 4 into the linear approximation formula: \( L(x) = F(3) + F'(3) (x - 3) = 13 + 3(x - 3) = 3x + 4 \).

Step 6: Plot the Function and its Linear Approximation

Plot \( F(x) = 3x + 4 \) and its linear approximation \( L(x) = 3x + 4 \) over the interval \([0,6]\). Note that because the function is linear, the approximation matches the function exactly.

Key Concepts

Derivative

To understand linear approximation, we first need to grasp the concept of the derivative. The derivative of a function at a point gives the slope of the tangent line to the function at that point. It tells us how fast the function's value is changing at any given moment.
In terms of notation, if we have a function \( F(x) = 3x + 4 \), its derivative, denoted \( F'(x) \), is found using basic rules of differentiation. For instance, since \( F(x) \) is a straightforward linear function, its derivative is simply the coefficient of \( x \), which is 3.
Calculating derivatives can become more complex with polynomial or non-linear functions, in which case differentiation rules such as the power rule, product rule, or chain rule are applied. However, in this example, because of the simplicity of the function, the derivative remains constant across its domain.

Tangent Line

A tangent line is a straight line that touches a curve at a single point, without crossing it. The slope of this line at the contact point is equal to the derivative of the function at that point.
When we talk about linear approximation, we are essentially using this tangent line to approximate the function. This is why linear approximations are also referred to as tangent line approximations.
In our example with the function \( F(x) = 3x + 4 \), because the function itself is linear, the tangent line and the function overlap entirely. Therefore, the linear approximation we calculated using the method \( L(x) = F(a) + F'(a)(x-a) \) resulted in the same equation as the original function.

Function Plotting

Plotting functions is a crucial skill for visualizing mathematical concepts. By plotting both the function and its linear approximation on a graph, we can see how well the approximation represents the function over a specific interval.
For our function \( F(x) = 3x + 4 \), the plot will show a straight line because it's a linear function. When plotted over the interval \([0,6]\), both the function and its linear approximation \( L(x) = 3x + 4 \) will appear as the same line on the graph.
Visualizing such plots helps to better understand the behavior of functions and their approximations, and reveals insights into why linear approximations work so well in linear cases. It also assists in identifying errors or mismatches in more complex functions and their linear approximations.

Calculus Concepts

Calculus is a vast field that deals with how things change, and linear approximation is one of its many tools. It helps in estimating the value of a function near a given point using the slope provided by the derivative.
This concept is particularly useful in mathematics and physics when dealing with complex systems where exact calculations are difficult. Linear approximations can simplify these systems into manageable problems that are easier to solve.
By breaking down the concept into steps—calculating the derivative, evaluating it, and using the tangent line—we simplify and make accessible the broader ideas of calculus. Understanding these basics lays the groundwork for delving into more advanced calculus topics that can handle even more dynamic changes and behaviors in functions.