Question

How can linear approximation be used to approximate the change in \(y=f(x)\) given a change in \(x ?\)

Short answer

Question: Use linear approximation to approximate the change in y for the function \(f(x) = x^2\) when \(x = 3\) changes by \(\Delta x = -0.1\). Answer: To approximate the change in y using linear approximation, follow the steps: Step 1: Find the derivative of the function f(x) The derivative of \(f(x) = x^2\) is \(f'(x) = 2x\). Step 2: Evaluate the derivative at the point x = 3 At x = 3, the slope of the tangent line is \(f'(3) = 2(3) = 6\). Step 3: Find the equation of the tangent line The tangent line has the form: \(y = mx + b\). At x = 3, the slope (m) is 6, and the point is (3, f(3)) = (3, 9). Now, we can find the y-intercept (b): 9 = 6(3) + b 9 = 18 + b b = -9 So, the equation of the tangent line is \(y = 6x - 9\). Step 4: Use the tangent line equation to approximate the change in y, given a change in x Given a change in x, \(\Delta x = -0.1\), we need to find the change in y (\(\Delta y\)). 1. Plug \(x_1 = 3\) into the tangent line equation to find the initial y value, \(y_1\): \(y_1 = 6(3) - 9 = 9\) 2. Plug \(x_2 = 3 - 0.1 = 2.9\) into the tangent line equation to find the new y value, \(y_2\): \(y_2 = 6(2.9) - 9 = 8.4\) 3. The change in y, \(\Delta y \approx y_2 - y_1 = 8.4 - 9 = -0.6\). Thus, for the function \(f(x) = x^2\), when x changes by \(-0.1\) at \(x = 3\), the approximate change in y using linear approximation is \(-0.6\).

Step-by-step solution

Find the derivative of the function f(x)

To use the linear approximation, we need to find the equation of the tangent line at a given point. The tangent line is determined by the slope, which is the derivative of the function f(x). Depending on the given function, we may use the power rule, chain rule, or other derivative rules to find the derivative.

Evaluate the derivative at the point x=a

Once we have the derivative of the function, we need to evaluate it at the given point x=a to find the slope of the tangent line. This will give us the change in y, with respect to x, at the point x = a.

Find the equation of the tangent line

Now that we have the slope of the tangent line, we can find its equation. The tangent line equation will be in the form \(y = mx + b\), where m is the slope and b is the y-intercept. We already know the slope (m) from evaluating the derivative in Step 2. We can find b by substituting the given point (a, f(a)) into the equation and solving for b.

Use the tangent line equation to approximate the change in y, given a change in x

With the tangent line equation in hand, we can use it to approximate the change in y, given a change in x. Since the tangent line is a linear function, we can easily find the change in y by plugging in the changed value of x into the equation. Let's suppose that we have an initial x value, \(x_1\), and we are given a change, \(\Delta x\). The new x value, \(x_2\), will be \(x_1 + \Delta x\). Now, we can use the tangent line equation to approximate the corresponding change in y: 1. Plug \(x_1\) into the tangent line equation to find the initial y value, \(y_1\). 2. Plug \(x_2\) into the tangent line equation to find the new y value, \(y_2\). 3. The change in y, \(\Delta y \approx y_2 - y_1\). The linear approximation allows us to calculate the change in y for a given change in x, using the tangent line equation.

Key concepts

Tangent Line Equation

In mathematics, the tangent line to a curve at a given point represents the best linear approximation of the function at that point. To find the tangent line equation, we need to focus on two main parts: the slope of the line and the y-intercept.
First, we determine the slope, often represented as \(m\), which is derived from the function's derivative. The derivative provides a measure of how the function changes at any point, fundamentally describing the slope of the tangent. Next, we need to find the y-intercept \(b\), which is crucial to forming the complete equation of the line.
Combining these, the tangent line equation takes the form \(y = mx + b\). This equation allows us to approximate the function near the given point by entering values close to the known \(x\). The tangent line serves as a simple linear approximation, which is very handy when working with swift calculations.

Derivative

Derivatives play a crucial role in calculus and are foundational for understanding linear approximations. In simple terms, the derivative of a function at a particular point provides the rate at which the function is changing at that point.

• It gives the slope of the tangent line to the curve of the function at that specific point.
• We notate the derivative of a function \(f(x)\) as \(f'(x)\) or \(\frac{df(x)}{dx}\).

To apply linear approximation, one must first calculate the derivative of the function. This derivative is then evaluated at the point of interest to find the instantaneous rate of change, which we can use when constructing the tangent line equation. Understanding the concept of derivatives helps in accurately estimating the values of a function for small changes in \(x\), leading to a better grasp of linear approximations.

Chain Rule

In calculus, the chain rule is an essential technique used to find the derivative of composite functions. A composite function is a function made up of other functions, often in the form of \(f(g(x))\). When performing linear approximations for complex functions, the chain rule becomes very handy.
The chain rule states that to differentiate a composite function, you take the derivative of the outer function evaluated at the inner function, and multiply it by the derivative of the inner function.
If \(y = f(g(x))\), then using the chain rule, the derivative \( \frac{dy}{dx} \) can be calculated as: \[ \frac{dy}{dx} = f'(g(x)) \times g'(x) \]This rule ensures we account for the rate of change within nested functions, which aids in finding accurate derivatives crucial for determining tangent lines. Having a solid grasp of the chain rule allows students to handle more complicated functions that arise in linear approximation problems.

Power Rule

The power rule is one of the simplest derivative rules, but it's extremely powerful when finding derivatives of polynomials. This rule makes it easy to calculate how functions behave at a particular interval, especially for linear approximations.
For a function of the form \(x^n\), where \(n\) is any real number, the power rule states that the derivative is given by:\[ \frac{d}{dx} x^n = nx^{n-1} \]This means you multiply the power by the coefficient, and then reduce the power by one. The power rule quickly gives the slope or rate of change of polynomial functions, which is integral for constructing the tangent line equation. It simplifies the process of finding derivatives, which are vital for linear approximations, making calculations less daunting.
Understanding and applying the power rule is a fundamental skill that helps in breaking down more complicated functions into manageable parts.