Question

Linear approximation and the second derivative Draw the graph of a function \(f\) such that \(f(1)=f^{\prime}(1)=f^{\prime \prime}(1)=1\) Draw the linear approximation to the function at the point (1,1). Now draw the graph of another function \(g\) such that \(g(1)=g^{\prime}(1)=1\) and \(g^{\prime \prime}(1)=10 .\) (It is not possible to represent the second derivative exactly, but your graphs should reflect the fact that \(f^{\prime \prime}(1)\) is relatively small compared to \(g^{\prime \prime}(1) .\) ) Now suppose linear approximations are used to approximate \(f(1.1)\) and \(g(1.1)\) a. Which function has the more accurate linear approximation near \(x=1\) and why? b. Explain why the error in the linear approximation to \(f\) near a point \(a\) is proportional to the magnitude of \(f^{\prime \prime}(a)\)

Short answer

Answer: The function \(f\) will have a more accurate linear approximation near \(x=1\) because its second derivative at \(x=1\) is smaller (\(f''(1) = 1\)) compared to the function \(g\) (\(g''(1)=10\)). The error in linear approximation is proportional to the magnitude of the second derivative, and since \(f''(1)\) has a smaller magnitude, its linear approximation will be more accurate near \(x=1\).

Step-by-step solution

Understand the given information

Function \(f\): - \(f(1) = f'(1) = f''(1) = 1\) Function \(g\): - \(g(1) = g'(1) = 1\) - \(g''(1) = 10\) Our main goal is to: - Draw the graphs of both functions - Draw the linear approximations for both at the point (1,1) - Understand which function will have a more accurate linear approximation near \(x=1\) - Explain why the error in the linear approximation to \(f\) near a point \(a\) is proportional to the magnitude of \(f''(a)\).

Draw the graphs for f and g

As we don't have explicit formulas for functions \(f\) and \(g\), we can't draw their exact graphs. However, we can use their given properties to obtain some qualitative understanding of their behavior. For function \(f\), we know that: - \(f(1) = 1\) (the graph of \(f\) passes through the point (1,1)) - \(f'(1) = 1\) (the slope of \(f\) at point (1,1) is 1) - \(f''(1) = 1\) (the concavity of \(f\) at point (1,1) is upward) Similarly, for function \(g\), we know that: - \(g(1) = 1\) (the graph of \(g\) passes through the point (1,1)) - \(g'(1) = 1\) (the slope of \(g\) at point (1,1) is 1) - \(g''(1) = 10\) (the concavity of \(g\) at point (1,1) is upward but with a larger magnitude compared to \(f\)) Based on this information, we can draw qualitatively the graphs for functions \(f\) and \(g\).

Draw the linear approximations for f and g

To draw the linear approximations for both functions at the point (1,1), we need to find each linear approximation's equation. Since both functions share the same slope and point, their linear approximations will be the same. Linear approximation of \(f\) at point (1,1) is given by: - \(L_f(x) = f'(1)(x-1) + f(1) = 1(x-1) + 1 = x\) Linear approximation of \(g\) at point (1,1) is given by: - \(L_g(x) = g'(1)(x-1) + g(1) = 1(x-1) + 1 = x\) So, both approximations are the same, i.e., \(L_f(x) = L_g(x) = x\).

More accurate linear approximation near x=1

Now we need to determine which function has the more accurate linear approximation near \(x=1\). The accuracy of the linear approximation depends on how well it approximates the function and how closely the second derivative is to zero. In our case, the function \(f\) has a smaller second derivative (\(f''(1) = 1\)) compared to the function \(g\) (\(g''(1)=10\)). Therefore, the linear approximation of function \(f\) is more accurate near \(x=1\).

Error in linear approximation is proportional to the magnitude of the second derivative

The error in the linear approximation to a function, say \(f\), near a point \(a\), depends on the difference between the function's value and its linear approximation. The second derivative, \(f''(a)\), is a measure of the curvature of the function. The larger the magnitude of \(f''(a)\), the higher the curvature of the function, and thus the greater the difference between the function and its linear approximation. Consequently, the error in the linear approximation is proportional to the magnitude of the second derivative, \(f''(a)\).

Key concepts

Second Derivative

The concept of the second derivative is fundamental to understanding the graph behavior of functions. It provides deep insight into the curvature and concavity of a function's graph. Formally, the second derivative of a function at a point, denoted as \( f''(a) \), measures how the rate of change of the function's slope is changing near that point.

For instance, if \( f''(a) > 0 \), the function is considered to be concave up at \( a \), which means that the graph is curving upwards like the shape of a cup. Conversely, if \( f''(a) < 0 \), the graph is concave down, like an upside-down cup. When the second derivative is close to zero, the graph is relatively flat because the rate at which the slope is changing is minimal. In the context of the exercise, function \( f \) has a smaller second derivative at \( x = 1 \) than function \( g \), indicating that \( f \)'s graph has a gentle curve, while \( g \) has a much steeper curvature due to its larger second derivative.

Function Graph Behavior

Understanding the behavior of a function's graph is pivotal when it comes to approximating its values near a specific point. The slope of the tangent line at a point on the graph represents the first derivative of the function. More importantly, the curvature or concavity of the graph around that point guides us about the function's behavior.

Looking at the behavior of functions \( f \) and \( g \) given in the exercise, both functions have the same value and slope at \( x = 1 \), but their second derivatives are distinctly different. This difference in the second derivatives will reflect in the shapes of their graphs. \( f \) will show a milder curve, while \( g \), with a second derivative of 10, will depict a much sharper curve. Consequently, the curvature can impact how well a linear approximation can predict values close to the point of tangency because a straight line cannot capture the turn taken by the function.

Error in Linear Approximation

When employing linear approximation to estimate the value of a function near a specific point, we inevitably introduce some error. This error arises because we are using a straight line (the tangent) to approximate the values of a function that may not be straight. The degree of curvature, which is indicated by the magnitude of the second derivative, significantly contributes to the error size.

In essence, the error in the linear approximation is directly tied to how sharply the function is curving away from the tangent line. If the second derivative—or the curvature—at the point of tangency is high, the function's graph will stray more from the tangent line, leading to a greater approximation error. This is why, in the exercise, the function \( g \), with a larger second derivative at point \( x=1 \), is less accurately approximated by the linear function compared to \( f \). The higher the second derivative, the larger the discrepancy between the linear approximation and the function's true value, especially as one moves away from the point of tangency.