Problem 42 - 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. $$ g(x)=\sin ^{-1} x \text { at } a=0,[-1,1] $$

Step-by-Step Solution

Answer

The linear approximation of \( g(x) = \sin^{-1} x \) at \( x = 0 \) is \( L(x) = x \).

Step 1: Recall the formula for linear approximation

The linear approximation of a function \( f(x) \) at a point \( a \) is given by \( L(x) = f(a) + f'(a)(x - a) \). We will apply this formula to the given function \( g(x) = \sin^{-1} x \) at \( a = 0 \).

Step 2: Evaluate the function at the point a

Given \( g(x) = \sin^{-1} x \), we need to evaluate \( g(0) \). Since \( \sin^{-1}(0) = 0 \), we have \( g(0) = 0 \).

Step 3: Find the derivative of the function

The derivative of \( g(x) = \sin^{-1} x \) is \( g'(x) = \frac{1}{\sqrt{1-x^2}} \). Now, we evaluate \( g'(0) = \frac{1}{\sqrt{1-0^2}} = 1 \).

Step 4: Write the linear approximation

Using the linear approximation formula: \( L(x) = g(0) + g'(0)(x - 0) \), we find \( L(x) = 0 + 1(x - 0) = x \). The linear approximation of \( \sin^{-1}x \) at \( x = 0 \) is \( L(x) = x \).

Step 5: Plot the function and its linear approximation

To visualize the functions, plot \( g(x) = \sin^{-1} x \) and its linear approximation \( L(x) = x \) on the interval \([-1, 1]\). The plot should show how closely \( L(x) \) approximates \( g(x) \) near \( x=0 \).

Key Concepts

Inverse Functions

Inverse functions help us find the input of a function based on its output. Specifically, an inverse function reverses the effect of the original function. If you have a function, say, \( f(x) \), with an output \( y \), the inverse function, denoted \( f^{-1}(y) \), gives you back \( x \). In simpler terms, if \( f(x) = y \), then \( f^{-1}(y) = x \). This swapping of inputs and outputs is useful when solving for variables that aren't easily isolated. For example, in the exercise, the function \( g(x) = \sin^{-1} x \) illustrates how inverse functions work. Here, \( \sin^{-1} x \) is the inverse of the sine function, providing the angle whose sine is \( x \). Inverse functions can be identified on a graph as they are reflections of the original function across the line \( y = x \). This reflection property aids in visualizing and understanding how an inverse function operates in real contexts.

Derivative

Derivatives reveal the rate at which a function changes at any given point. They are fundamental in understanding the behavior of functions and are frequently used in linear approximations. A derivative gives us the slope of the tangent line to the function at a specific point, which is crucial for building linear models. For the function \( g(x) = \sin^{-1} x \), its derivative is \( g'(x) = \frac{1}{\sqrt{1-x^2}} \). At \( x = 0 \), this derivative equals 1, indicating that the slope of the tangent line at this point is 1.

• This tangent slope helps construct the linear approximation.
• Linear approximations utilize the derivative to ensure the curve of the line closely resembles the curve of the function at a given point.

Remember, the derivative can vary greatly, especially for non-linear functions like inverse trigonometric ones; hence its importance in capturing the function's behavior over different intervals.

Plotting Functions

Visualizing functions by plotting them provides insights into their behavior over specified intervals. In the given exercise, plotting \( g(x) = \sin^{-1} x \) and its linear approximation \( L(x) = x \) on the interval \([-1, 1]\) demonstrates the proximity of the linear approximation to the original function near the point \( a = 0 \). Attempting to understand this spatial relationship without visual aid can be challenging.

• By plotting, one can observe where the linear approximation holds closely to the original function (which is usually around the point of approximation \( a \)).
• It also highlights discrepancies further from this point, offering an understanding of approximation limitations.

Graphs are helpful tools for learners as visual cues often lead to better comprehension and retention. They take abstract mathematical concepts and make them concrete, showing a function's shape, its tangent, and how well an approximation represents it over an interval.