Problem 10 - Calculus: Early Transcendentals

Question

Suppose that \(f^{\prime \prime}\) exists and is continuous on \([1,2] .\) Suppose also that \(\left|f^{\prime \prime}(x)\right| \leq \frac{1}{4}\) for all \(x\) in \((1,2) .\) Prove that if we use the linearization \(y=L(x)\) of \(y=f(x)\) at \(x=1\) as an approximation of \(y=f(x)\) near \(x=1,\) then our estimated value of \(f(1.2)\) is guaranteed to have an accuracy of at least \(0.01,\) i.e., our estimate will lie within 0.01 units of the true value.

Step-by-Step Solution

Answer

The error of the linear approximation at \( x = 1.2 \) is at most 0.005, ensuring an accuracy within 0.01 of the true value.

Step 1: Understand Taylor's Theorem

Taylor's theorem states that for a function \( f \) that is sufficiently smooth at a point \( a \), the function can be approximated as \( f(x) = f(a) + f'(a)(x-a) + \frac{f''(c)(x-a)^2}{2} \) for some \( c \) between \( a \) and \( x \). Here, we are using a linear approximation \( L(x) = f(a) + f'(a)(x-a) \) at \( x = 1 \), so the error term is \( \frac{f''(c)(x-a)^2}{2} \). The remainder term is of the order of \( (x-a)^2 \).

Step 2: Apply Bounds on the Second Derivative

We are given that \( |f''(x)| \leq \frac{1}{4} \) for all \( x \) in \((1,2)\). This indicates that the worst-case error when approximating \( f(x) \) using its linearization at \( x = 1 \) can be bounded by \( \left| \frac{f''(c)(x-1)^2}{2} \right| \leq \frac{1}{4} \times (x-1)^2\).

Step 3: Calculate the Error at x = 1.2

We are approximating \( f(1.2)\) using \( L(x)\). The error is \( \left| \frac{f''(c)(1.2-1)^2}{2} \right| \). Substitute the values to get \( \left| \frac{f''(c) \times (0.2)^2}{2} \right| \leq \frac{1}{4} \times \frac{0.04}{2} = 0.005\).

Step 4: Confirm the Error Falls Within Acceptable Range

The maximum error calculated is \( 0.005 \), which is less than \( 0.01 \). Therefore, the linear approximation \( L(x) \) for \( f(x) \) at \( x = 1.2 \) is within \( 0.01 \) units of the true value, meeting the required accuracy condition.

Key Concepts

Linearization

Linearization is a concept used to simplify the analysis of functions by approximating them with linear functions. It's like zooming in and observing a curve and noticing that it appears straight. The essence of linearization lies in representing a function close to a certain point as a straight line. This is a powerful tool in calculus, especially when dealing with complex functions.

• The linearization of a function \( f \) at a point \( a \) is given by the formula \( L(x) = f(a) + f'(a)(x-a) \).
• This linear function is an approximation of \( f(x) \) that's easiest to compute and understand near \( x = a \).

The key here is to realize that linearization is closely tied to derivatives. It hinges on the slope of the tangent line at the point of interest. In practical terms, if you want to approximate function values near a given point, stepping down to a linear approximation simplifies many calculations and provides valuable insights. This snapshot is vital for solving more intricate problems in calculus that involve changes.

Remainder Term

The Remainder Term is a crucial part of Taylor's Theorem. It provides a measure of how precise our linear approximation of a function is compared to the actual function value. Think of it as the error gap between the approximation and the reality.

• For a linear approximation, the remainder term is formulated as \( \frac{f''(c)(x-a)^2}{2} \).
• Here, \( c \) is a point between \( a \) and \( x \) that ensures this term accounts for all potential discrepancies.

This term represents a type of second degree error because it involves the second derivative \( f''(x) \). It's important because it helps assess the accuracy of your linear approximation. Knowing how to compute and interpret this remainder term is essential for verifying how good your approximation strategies are. This makes it indispensable for tasks involving error estimation and approximation bounds.

Second Derivative Bounds

Bounding the second derivative is fundamental to controlling the error in our linear approximations. It is much like setting boundaries within which our function behaves predictably. When using Taylor's Theorem, we aim to understand how quickly the function's change is changing, and we use this to our advantage.

• The bound says \( |f''(x)| \leq \frac{1}{4} \) for \( x \) between 1 and 2 in our example scenario.
• Having this bound provides a limit to the curvature (or bendiness) of the function.

Knowing these bounds ensures that the error resulting from the linearization doesn't exceed what we can handle. By containing the second derivative, we accurately express how the error behaves and determine whether our approximation is within acceptable limits. The bounded nature simplifies the estimation process and ensures we can predict and control the approximation's deviation from actual values.

Error Estimation

Error estimation is the art of betting how far off our approximation might be from the actual outcome. In Taylor's theorem applications, this is calculating how much the linear version of a function differs from its real form.

• Error estimation involves evaluating \( \left| \frac{f''(c)(x-a)^2}{2} \right| \).
• We substitute values to give us a concrete estimate, like \( 0.005 \) in our exercise.

Effective error estimation provides the means to both verify and assure the accuracy of functional approximations. It helps to ascertain that our linear approximation \( L(x) \) is within a specific range \( (e.g., 0.01) \) of the true function value. This hinges not only on theoretical constructs but also on practical confidence in the bounds and calculus techniques. It allows for reliable predictions and fosters robust problem-solving skills in mathematical analysis.