Problem 52 - Calculus

Question

Decide if the statements in Problems are true or false. Give an explanation for your answer. The linear approximation to \(f(x)\) near \(x=-1\) shows that if \(f(-1)=g(-1)\) and \(f^{\prime}(-1) < g^{\prime}(-1),\) then \(f(x) < g(x)\) for all \(x\) sufficiently close to -1 (but not equal to -1 ).

Step-by-Step Solution

Answer

The statement is true for \( x > -1 \) sufficiently close to -1.

Step 1: Understanding the Problem

We need to compare the functions \( f(x) \) and \( g(x) \) using their linear approximations around the point \( x = -1 \). The given conditions are \( f(-1) = g(-1) \) and \( f'(-1) < g'(-1) \). We need to determine if \( f(x) < g(x) \) holds true for \( x \) close to -1 but not equal to -1.

Step 2: Linear Approximation Formula

The linear approximation of a function \( f(x) \) near a point \( a \) is given by \( f(x) \approx f(a) + f'(a)(x-a) \). Using this formula, we can approximate \( f(x) \) and \( g(x) \) near \( x=-1 \).

Step 3: Computing Linear Approximations

For \( f(x) \) near \( x = -1 \), the linear approximation is \( f(x) \approx f(-1) + f'(-1)(x+1) \). Similarly, for \( g(x) \), it is \( g(x) \approx g(-1) + g'(-1)(x+1) \). Since \( f(-1) = g(-1) \), these simplify to \( f(x) \approx f(-1) + f'(-1)(x+1) \) and \( g(x) \approx f(-1) + g'(-1)(x+1) \).

Step 4: Comparing Approximations

Since \( f(-1) = g(-1) \), we focus on the derivative terms. Given \( f'(-1) < g'(-1) \), it follows that \( f'(-1)(x+1) < g'(-1)(x+1) \) for \( x + 1 > 0 \). This implies that \( f(x) < g(x) \) for \( x > -1 \) when \( x \) is close to -1.

Step 5: Conclusion

The statement is true. For \( x \) sufficiently close to but greater than -1, \( f(x) < g(x) \) due to the fact that \( f'(-1) < g'(-1) \). If \( x < -1 \), the inequality direction might change since \( x+1 \) becomes negative. Therefore, the condition holds in an interval right after \( x = -1 \).

Key Concepts

Derivative Comparison

In calculus, derivatives help us understand how functions behave as their inputs change. For two functions, comparing their derivatives at a specific point gives insights into which function is increasing faster around that point.
This is especially useful when analyzing linear approximations.When the derivative of a function, say \(f(x)\), at a point \(a\) is less than the derivative of another function, \(g(x)\), at the same point, it tells us that near \(a\), \(f(x)\) increases or decreases at a slower rate compared to \(g(x)\).

• If \(f'(-1) < g'(-1)\), then for values \(x\) slightly larger than \(-1\), \(f(x)\) grows less rapidly or decreases more slowly compared to \(g(x)\). Thus, \(f(x) < g(x)\) in this region.

Noting this comparison helps determine which function overtakes the other just by examining their derivatives at a single point.

Function Inequality

Understanding function inequality is a way of determining how one function outpaces another over a certain range. In our case, we know that two functions, \(f(x)\) and \(g(x)\), start from the same point at \(x = -1\) since \(f(-1) = g(-1)\).
Yet, we need to inspect their behavior near this point.
The inequality \(f(x) < g(x)\) suggests that \(f\) stays below \(g\) when close to \(-1\) and within an interval to the right.

• Given \(f'(-1) < g'(-1)\), this implies that immediately to the right of \(-1\), \(f(x)\) will remain below \(g(x)\), thanks to \(g\) having a steeper (or more positive) slope.

This inequality illustrates that a difference in rates, captured by the derivative, dictates the relationship between the functions near our point of interest.

Approximation Near a Point

Linear approximation allows us to estimate the value of a function using its tangent line at a given point. This is particularly useful for making predictions about function behavior near that point.
In our exercise, we explored how \(f(x)\) and \(g(x)\) can be approximated using linear functions around \(x = -1\).The formula for this is straightforward: \[ f(x) \approx f(a) + f'(a)(x-a) \]This means for \(f(x)\) near \(-1\) we have:\[ f(x) \approx f(-1) + f'(-1)(x+1) \]And similarly for \(g(x)\), since they share the same initial value, but different slopes.

• The linear approximation highlights how both functions behave similarly at the point but diverge as you move away, based on their respective slopes.
• This technique provides a way to predict and compare the proximity of different functions at specific intervals.

By using linear approximation, we get valuable insights into the functions' behavior in a very simplified yet effective manner.