Question

Multiple Choice What is the linearization of \(f(x)=e^{x}\) at \(x=1 ?\) (A) \(y=e \quad\) (B) \(y=e x \quad\) (C) \(y=e^{x}\) \((\mathbf{D}) y=x-e \quad\) (\mathbf{E} ) ~ \(y=e(x-1)\)

Short answer

The linearization of \(f(x)=e^{x}\) at \(x=1\) is \(y = e(x - 1)\)

Step-by-step solution

Find the Derivative

The first step is to find the derivative of the function \(f(x) = e^{x}\). Since the derivative of \(e^{x}\) is \(e^{x}\), the derivative of the function at \(x=1 (f'(1))\) is \(e^{1}\), which equals \(e\).

Create the Linearization Formula

With the derivative in hand, the linearization \(L(x)\) of the function \(f(x) = e^{x}\) at a point \(x=a\) is given by the equation \(L(x) = f(a) + f'(a)(x-a)\). Substitute \(a=1\), \(f(1)=e^{1}=e\), and \(f'(1)=e\) into the linearization equation to get \(L(x) = e + e(x - 1)\). This equals \(ex - e\), which can also be written as \(y = e(x - 1)\).

Compare to Given Options

Comparing the linear function obtained in the previous step to the answer options in the question, it is clear that the correct answer is (E) \(y = e(x - 1)\)

Key concepts

Differentiation

Understanding the process of differentiation is crucial in identifying how a function changes at any given point. Differentiation is the action of computing a derivative. The derivative of a function provides us with the rate at which the function's value is changing at any point. For instance, if we're dealing with acceleration in physics, the process can tell us how quickly the velocity of an object is changing over time.

In mathematical terms, given a function f(x), the derivative f'(x) is the limit of the average rate of change of the function over an interval, as the interval's length approaches zero. Applying this to exponential functions like e^x, where e is Euler's number, the derivative yields the same exponential function, that is, f'(x) = e^x. In the context of the provided exercise, when differentiating f(x) = e^x, we get f'(x) = e^x, which at x=1 gives us f'(1) = e.

Expanding the knowledge of differentiation not only helps us solve textbook exercises but also enhances our understanding of the dynamics of various natural and scientific phenomena.

Tangent Line Approximation

The tangent line approximation, also known as linearization, is a method for approximating the value of a function near a particular point using the tangent line at that point. This concept is extremely helpful in situations where the function itself is too complex for straightforward evaluation.

For a differentiable function f(x), the tangent line at x=a can be represented as L(x) = f(a) + f'(a)(x - a). Here, f(a) is the value of the function at x=a, and f'(a) is the value of the derivative (or the slope of the tangent line) at the same point. This linear equation provides a simple approximation for f(x) when x is near a, making it a powerful tool for estimations and simplifying complex expressions.

Applying this to the example in the exercise, at x=1 for the function f(x) = e^x, we would obtain the linear approximation L(x) = e + e(x - 1), which simplifies to y = e(x - 1), matching answer option (E). This approximation is particularly useful for quick calculations and insights into the behavior of the function near the point of tangency.

Exponential Functions

An exponential function is one of the most important concepts in mathematics, characterized by a constant base raised to a variable exponent. The general form is f(x) = b^x where b is a positive real number. Among these functions, the most notable is e^x, where e (approximately 2.71828) is Euler's number, a fundamental constant in mathematics.

The special property of e^x is that it is its own derivative, meaning that the rate of change of this function at any point is equal to the value of the function at that point. This property makes e^x incredibly important in various scientific calculations, including growth processes, radioactive decay, and compound interest in finance.

In the context of the exercise, recognizing that f(x) = e^x is an exponential function allows us to apply the concepts of differentiation and tangent line approximation effectively to find the linearization at x=1. Understanding the unique characteristics of exponential functions can greatly enhance one's ability to solve a wide range of real-world problems.