Question

In Exercises \(1-4,\) find a first-degree polynomial function \(P_{1}\) whose value and slope agree with the value and slope of \(f\) at \(x=c .\) Use a graphing utility to graph \(f\) and \(P_{1} .\) What is \(P_{1}\) called? $$ f(x)=\sec x, \quad c=\frac{\pi}{4} $$

Short answer

\( P_{1}(x) = \sqrt{2} + \sqrt{2}(x - \frac{\pi}{4}) \). \( P_{1} \) is called the linear approximation or linearization of \( f \) at \( x = \frac{\pi}{4} \).

Step-by-step solution

Calculate f(c)

Calculate \( f(c) \) by substituting the given value of \( c = \frac{\pi}{4} \) into the function \( f(x) = \sec x \). This will give the y-coordinate for the point at \( x = c \). Here, \( \sec(\frac{\pi}{4}) = \sqrt{2} \). So, \( f(c) = \sqrt{2} \)

Calculate the derivative of f(x)

Since the slope of a function at a given point is given by the derivative of the function at that point, we need to calculate the derivative of \( f(x) = \sec x \). Now, \( f'(x) = \sec x \tan x \).

Calculate f'(c)

Next, substitute \( c = \frac{\pi}{4} \) into the derivative of \( f(x) \) to obtain the slope at \( x = c \). \( f'(\frac{\pi}{4}) = \sec(\frac{\pi}{4}) \tan(\frac{\pi}{4}) = \sqrt{2} \). So, the slope of the function \( f(x) \) at \( x = c \) is \( \sqrt{2} \)

Calculate \( P_{1} \)

The equation of the first degree polynomial \( P_{1} \) which has the same value as \( f(x) \) at \( x = c \) and the same slope is given by \( P_{1}(x) = f(c) + f'(c)(x - c) \). Substitute \( f(c) = \sqrt{2} \) and \( f'(c) = \sqrt{2} \) into the equation. This gives \( P_{1}(x) = \sqrt{2} + \sqrt{2}(x - \frac{\pi}{4}) \).

Naming \( P_{1} \)

The polynomial \( P_{1} \) is called the linear approximation or linearization of \( f \) at \( x = c \).

Key concepts

First-Degree Polynomial Function

A first-degree polynomial function, often referred to as a linear function, is a mathematical expression that has no exponents higher than one and is typically written in the form of \( P(x) = mx + b \), where \( m \) represents the slope of the line and \( b \) is the y-intercept.

In the context of calculus, the first-degree polynomial function can serve as a linear approximation of more complex functions around a certain point \( x = c \). This approximation is particularly useful because it simplifies calculations and provides an estimation of the function's behavior in a small interval around \( c \). When the value of the first-degree polynomial and its derivative at \( c \) match those of the function it's approximating, we achieve the best linear approximation, known as linearization.

The key to constructing a linear approximation is to find both the value of the function at \( c \) and the slope of the tangent line at that point, and then use the point-slope form of the linear equation: \( P_1(x) = f(c) + f'(c)(x - c) \). This linear model can be a powerful tool for understanding and approximating complex functions in mathematics.

Derivative of Secant Function

When dealing with trigonometric functions, the derivative provides us with crucial information about the function's rate of change. For the secant function, \( f(x) = \text{sec}\thinspace x \), its derivative is somewhat unique compared to the basic trigonometric functions we are introduced to early in calculus.

The derivative of the secant function is given by \( f'(x) = \text{sec}\thinspace x \tan\thinspace x \). This result is derived from the quotient rule applied to the secant function, which is the reciprocal of the cosine function.

Knowing the derivative of the secant function allows us to determine the angle of the slope of the tangent line to the graph of the sec function at any given point. This slope is essential for linear approximations as it tells us how steeply the function is rising or falling at the point of tangency. Using this derivative reflects how intricate function behaviors can be understood and estimated using their linear counterparts.

Slope of Tangent Line

The slope of a tangent line is a foundational concept in calculus and a critical part of understanding how functions behave. It refers to the steepness or incline of the line that lightly touches, or 'tangents', a curve at a given point. The slope is the visual representation of the derivative at that specific point.

Mathematically, if we have a function \( f(x) \), and we want to find the slope of the tangent line at a point \( c \), we calculate \( f'(c) \), the derivative of the function evaluated at \( c \). This slope tells us the rate of change of the function's value at approximately that point. It gives an idea of whether the function is increasing or decreasing, and at what rate, precisely at the point where the tangent line touches the curve.

In practical application, as in the given exercise for the function \( f(x) = \text{sec}\thinspace x \), once we know the slope at \( x = \frac{\text{pi}}{4} \), we can construct our linear model, or our linear approximation, which can be graphed alongside the original function to observe their similarities and differences close to the point of tangency. The construction of the linear approximation relies closely on understanding and accurately calculating the slope of this tangent line.