Question

Show that the linearization of \(f(x)=(1+x)^{k}\) at \(x=0\) is \(L(x)=1+k x\)

Short answer

The linearization of the function \(f(x)=(1+x)^{k}\) at \(x=0\) is \(L(x)=1+k x\).

Step-by-step solution

Calculate the function value at x=0

First, let's calculate the function value at \(x=0\). We replace \(x\) with 0 in the function \(f(x)=(1+x)^{k}\), this gives us \(f(0)=(1+0)^{k}=1\).

Calculate the derivative of the function

Now, we calculate the derivative of the function \(f(x)=(1+x)^{k}\). The derivative is given by the power rule \(\frac{d}{dx}x^{n}=n x^{n-1}\). In this case, \(n=k\) and we have \(\frac{d}{dx}(1+x)^{k}=k(1+x)^{k-1}\).

Calculate the derivative value at x=0

Next, we calculate the value of the derivative at \(x=0\). We replace \(x\) with 0 in the derivative, this gives us \(f'(0)=k(1+0)^{k-1}=k\).

Formulate the linearization

Finally, the linearization \(L(x)\) of the function \(f(x)\) at \(x=0\) is given by the equation \(L(x)=f(0)+f'(0)(x-0)\). Therefore, substituting the previously calculated values, we have \(L(x)=1+k x\).

Key concepts

Calculus

Calculus is a branch of mathematics that deals with the study of change and motion. It can be divided into two major fields: differential calculus and integral calculus. Differential calculus concerns the rate at which quantities change, while integral calculus focuses on the accumulation of quantities. The core idea of calculus is to approach problems involving change by breaking them down into infinitesimally small pieces, and then to solve these simpler problems to arrive at an overall conclusion.

One of the foundational operations in calculus is differentiation, which is used to calculate the rate of change or the slope of a curve at any point. The opposite operation, integration, is used to find areas, volumes, central points, and many useful things. Calculus has widespread applications in sciences and engineering and is typically introduced in secondary education.

Power Rule for Derivatives

One of the most commonly used rules for differentiation in calculus is the power rule. This rule simplifies the process of taking derivatives of polynomial functions where the exponent is a real number. The power rule states that the derivative of a monomial term with the form \(x^n\) is \(nx^{n-1}\).

To implement this rule, simply multiply the exponent by the coefficient of the term and subtract one from the exponent. For example, the derivative of \(x^3\) is \(3x^2\). This rule is vital for calculating derivatives in a quick and efficient manner and lays the groundwork for analyzing the behavior of functions, such as their growth rates and concavity.

Tangent Approximation

Tangent approximation, otherwise known as linearization, is a method used in calculus to approximate the value of a function near a point. It uses the tangent line at a specific point on a curve to approximate the function's values near that point. The equation of the tangent line to a function \(f(x)\) at a point \(x = a\) is given by \(L(x) = f(a) + f'(a)(x - a)\), where \(L(x)\) is the linear approximation and \(f'(a)\) is the derivative of the function at the point \(a\).

This technique is extremely useful in situations where calculating the exact value of functions is complex or impossible. By using the tangent line, which is a straight line, approximation becomes much simpler—providing valuable insights into the behavior of functions in the vicinity of the point of tangency.

Differential Calculus

Differential calculus is a subfield of calculus concerned with the concept of a derivative, which represents an instantaneous rate of change. It's about understanding how one quantity changes in relation to another. Finding a derivative is fundamentally about finding the slope of the function at any point, which is analogous to the steepness of the line tangent to the function at that point.

The process of differentiation, which is at the heart of differential calculus, enables us to calculate not only the slope of the tangent but also to determine the maximum and minimum values of functions, points of inflection, and the motion of objects under the influence of forces. Mastery of differential calculus is essential for solving problems in physics, engineering, economics, and beyond, making it an indispensable tool in the sciences.