Problem 17 - Calculus Early Transcendentals

Question

Determine whether the following statements are true and give an explanation or counterexample. a. If the function \(f\) is differentiable for all values of \(x\), then \(f\) is continuous for all values of \(x\). b. The function \(f(x)=|x+1|\) is continuous for all \(x\), but not differentiable for all \(x\). c. It is possible for the domain of \(f\) to be \((a, b)\) and the domain of \(f^{\prime}\) to be \([a, b]\).

Step-by-Step Solution

Answer

Short Answer: Statement a is true because differentiability implies continuity as the existence of a derivative for all values of x means that the function is continuous for all values of x. Statement b is true because the function \(f(x) = |x+1|\) is continuous for all values of x but not differentiable at \(x=-1\). Statement c is false because the domain of \(f^{\prime}\) cannot be a closed interval \([a, b]\) if the domain of the function \(f\) is an open interval \((a, b)\).

Step 1: Statement a: Differentiability implies Continuity

If a function \(f\) is differentiable for all values of \(x\), this means that its derivative, \(f'(x)\), exists for all values of \(x\). Now, a function is continuous at a point if its limit exists and is equal to the value of the function at that point. We know that a function being differentiable at a point implies that it is continuous at that point. Since \(f'(x)\) exists for all values of \(x\), it implies that the function \(f\) is continuous for all values of \(x\). So, the statement is true.

Step 2: Statement b: Analyzing Function \(f(x) = |x + 1|\)

First, let's check if this function is continuous for all values of \(x\). The function is a combination of simple continuous functions: \(x + 1\) and taking the absolute value. As the composition of continuous functions is continuous, \(f(x) = |x + 1|\) is continuous for all values of \(x\). Now, let's look at the differentiability of the function. We need to find the derivative of the function, which is the limit of the difference quotient. The function \(f(x) = |x + 1|\) can be broken into two functions: \[f(x) = \begin{cases} x + 1, & x \geq -1 \\ -(x + 1), & x < -1 \end{cases}\] Now, we can find the derivative of each part: \[f'(x) = \begin{cases} 1, & x > -1 \\ -1, & x < -1 \end{cases}\] However, at the point \(x=-1\), the function has a sharp turn (change in direction) and does not have a unique tangent, which means that the derivative does not exist at \(x = -1\). Therefore, this function is not differentiable for all values of \(x\). So, the statement is true.

Step 3: Statement c: Domains of \(f\) and \(f'\)

For this statement, we have to determine if it is possible for the domain of \(f\) to be \((a, b)\) and the domain of \(f^{\prime}\) to be \([a, b]\). Domain of \(f^{\prime}\) represents the set of points where the derivative of the function exists. If the derivative exists on the interval \((a, b)\), then applying the mean value theorem would imply that the derivative should also exist at the end-points of the interval, i.e., \(a\) and \(b\). However, since the domain of the function, \(f\) is given to be an open interval \((a, b)\), we cannot evaluate the function on the end-points, and thus the derivative cannot exist at the end-points. Therefore, it is not possible for the domain of \(f^{\prime}\) to be a closed interval \([a, b]\), if the domain of the function, \(f\) is the open interval \((a, b)\). So, the statement is false.

Key Concepts

Function Analysis

Function analysis is a crucial aspect of calculus where we delve deeper into understanding the nature and behavior of functions. It involves:

• Determining the continuity of a function.
• Finding points of differentiability.
• Analyzing how these characteristics influence the graph of a function.

When a function is both continuous and differentiable, it signifies smooth and predictable behavior on its graph without any jumps or cusps. However, points of discontinuity or non-differentiability can reveal sharp turns or breaks in the graph, providing insight into more complex behaviors. Continuous functions, like simple linear functions, often exhibit stable and reliable outcomes, whereas points of non-differentiability, like that found in the function \(f(x) = |x + 1|\) at \(x = -1\), highlight changes in direction or abrupt shifts.

Domain and Range

Every function has a domain and a range, which essentially outline where the function is defined and the possible outputs it can produce, respectively.

• Domain: This refers to all the input values \(x\) for which the function is defined.
• Range: This indicates all possible output values \(f(x)\) that result from plugging domain values into the function.

When examining statements relating to differentiability and domains, it's important to understand the constraints of the domain. For instance, the domain of the function \(f(x) = |x + 1|\) is all real numbers because it is defined for every possible value of \(x\). However, as discussed in the exercise solution, the derivative \(f'(x)\) does not exist at \(x = -1\), which highlights that the domain of the derivative can be different from the domain of the original function. Understanding these differences is crucial for accurate analysis and application of functions in calculus.

Mean Value Theorem

The Mean Value Theorem (MVT) is a fundamental principle in calculus that relates the derivative of a function to the overall behavior of the function within an interval. It essentially states:

• If a function \(f\) is continuous on the closed interval \([a, b]\) and differentiable on the open interval \((a, b)\), then there exists at least one point \(c\) in \((a, b)\) such that \(f'(c) = \frac{f(b) - f(a)}{b - a}\).

This implies that there is at least one point where the instantaneous rate of change (the slope of the tangent) matches the average rate of change over the interval. The Mean Value Theorem is profoundly useful for understanding how derivatives can describe the behavior over intervals, informing us about speed, acceleration, growth rates, and more in practical problems. However, it also highlights that even if a function is differentiable over an open interval, the behavior at the endpoints, which might be included in the domain of \(f\), cannot be guaranteed unless specifically continuous and differentiable.