Problem 50 - Calculus Early Transcendentals: Pearson New International Edition

Question

Give an example of a function \(f\) that is continuous on \([0,1]\), differentiable on \((0,1)\), and not differentiable on \([0,1]\), and has a tangent line at every point of \([0,1]\).

Step-by-Step Solution

Answer

\(f(x) = \sqrt{x}\) satisfies all given conditions.

Step 1: Understanding the Requirements

We need to find a function \(f\) that meets these criteria: \(f\) is continuous on \([0,1]\), differentiable on \((0,1)\), not differentiable on \([0,1]\), and yet has a tangent line at every point in \([0,1]\). A tangent line exists where the derivative is defined (including one-sided derivatives).

Step 2: Choosing the Right Function

A suitable candidate is the function \( f(x) = \sqrt{x} \). This function is continuous over the interval \([0,1]\) and differentiable on the interval \((0,1)\).

Step 3: Checking Differentiability on the Open Interval

Calculate the derivative of \(f(x) = \sqrt{x}\): \( f'(x) = \frac{1}{2\sqrt{x}} \). This derivative exists for all \(x \in (0,1)\), confirming differentiability on this interval.

Step 4: Analyzing Differentiability at the Endpoints

The derivative \( f'(x) = \frac{1}{2\sqrt{x}} \) at \(x=0\) approaches infinity, implying \(f(x)\) is not differentiable at \(x=0\). At \(x=1\), the derivative is defined, giving \(f'(1) = \frac{1}{2}\), so \(f\) is differentiable at \(x=1\).

Step 5: Confirming Tangent Line Existence

Despite \(f\) being non-differentiable at \(x=0\), we can approach it with a one-sided tangent. The slope from the right (as \(x\to0^+\)) matches the derivative trend. Therefore, a tangent line, though vertical, exists at \(x=0\), defined by the limit.

Step 6: Conclusion

The function \(f(x) = \sqrt{x}\) meets all the criteria: it is continuous on \([0,1]\), differentiable on \((0,1)\), not differentiable at \(x=0\) but differentiable at \(x=1\), and has a tangent line at every point on \([0,1]\).

Key Concepts

continuous functions

A continuous function is a type of function that does not have any abrupt changes or breaks in its graph. For a function to be continuous, its graph must be unbroken over its entire domain.
One important property of continuous functions is that small changes in the input result in small changes in the output.

• This means we can draw the graph of the function without lifting our pencil from the paper.
• Mathematically, a function \(f\) is continuous at a point \(a\) if the following holds: \(\lim_{{x \to a}} f(x) = f(a)\).
• Continuous functions are defined for every point in their domain, including intervals like \([0,1]\).

For the function \(f(x) = \sqrt{x}\), it remains continuous on \([0,1]\) as there are no breaks or jumps in its graph over this range.

tangent line

A tangent line is a straight line that touches a curve at a single point without crossing it at that point. This concept is fundamental in the study of curves and calculus.
The slope of a tangent line at a point on a function represents the instantaneous rate of change of the function at that specific point.

• Mathematically, the slope of a tangent line to the curve \(y = f(x)\) at a point \(x = a\) is given by the derivative \(f'(a)\).
• At \(x=0\), the function \(f(x) = \sqrt{x}\) has a vertical tangent since the derivative tends to infinity.
• Other than endpoints involving non-differentiable points, tangent lines can have a slope that is not defined as a real number.

Thus, finding a tangent line involves determining where the derivative exists, considering one-sided derivatives if necessary.

open interval

An open interval in mathematics represents a set of numbers between two endpoints, but without including the endpoints themselves. It is denoted using parentheses, such as \((a, b)\), indicating the numbers that are strictly between \(a\) and \(b\).
This concept is significant when discussing differentiability because many functions are differentiable over open intervals.

• When we say a function is differentiable on an open interval, like \((0, 1)\) for \(f(x) = \sqrt{x}\), it means the derivative exists for every point within these bounds.
• The endpoints are not included in this definition, which simplifies how we discuss functions near the boundaries.

An open interval allows us to focus on the behavior of functions without worrying about the complexity that endpoints can introduce.

derivative

A derivative is a measure of how a function changes as its input changes. It is a critical concept in calculus, capturing the idea of the rate of change or slope of the function.
Derivatives provide insight into many aspects of the behavior of functions.

• The derivative of a function \(f\) at a point \(x\) is denoted as \(f'(x)\) or \(\frac{df}{dx}\).
• For \(f(x) = \sqrt{x}\), the derivative is given by \(f'(x) = \frac{1}{2\sqrt{x}}\).
• This derivative illustrates that \(f\) is differentiable on the open interval \((0,1)\).

However, note that at the endpoint \(x=0\), \(f'(x)\) approaches infinity, illustrating a point where the derivative does not exist in the traditional sense, but we can still discuss a tangent line in terms of its limit behavior.