Problem 55 - University Calculus: Early Transcendentals

Question

Does knowing that a function \(f(x)\) is differentiable at \(x=x_{0}\) tell you anything about the differentiability of the function \(-f\) at \(x=x_{0} ?\) Give reasons for your answer.

Step-by-Step Solution

Answer

Yes, if \(f(x)\) is differentiable at \(x_0\), so is \(-f(x)\). The derivative of \(-f\) at \(x_0\) is \(-f'(x_0)\).

Step 1: Understand Differentiability

A function \( f(x) \) is said to be differentiable at \( x_0 \) if the derivative \( f'(x_0) \) exists. This means the limit \( \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h} \) exists.

Step 2: Consider the Function \(-f(x)\)

If \( f(x) \) is differentiable at \( x_0 \), then \(-f(x)\) is the function "negative \( f \)" which means each outcome of \( f(x) \) is multiplied by \(-1\). The function \(-f(x)\) reflects \( f(x) \) across the x-axis.

Step 3: Discover the Derivative of \(-f(x)\)

The derivative of \(-f(x)\) at \( x_0 \) is computed as \((-f)'(x_0) = \lim_{h \to 0} \frac{-f(x_0 + h) + f(x_0)}{h} = -\lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h} = -f'(x_0)\).

Step 4: Conclude Differentiability

Since \( f'(x_0) \) exists by the assumption that \( f \) is differentiable at \( x_0 \), \(-f'(x_0) \) also exists. Therefore, \(-f(x)\) is differentiable at \( x_0 \). The differentiability of \( f(x) \) ensures the differentiability of \(-f(x)\) as well.

Key Concepts

Derivative

The derivative of a function is a fundamental concept in calculus, which essentially measures how a function changes as its input changes. More formally, for a function \( f(x) \), the derivative at a point \( x_0 \) is the limit:\[ f'(x_0) = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}\]This limit represents the instantaneous rate of change of the function at \( x_0 \). If the limit exists, we say the function is differentiable at that point.

• This gives us the slope of the tangent line to the curve at \( x_0 \).
• A function can have a derivative at a particular point even if it is not differentiable elsewhere.

When considering the derivative of a function that is reflected over the x-axis, essentially the transformation to \(-f(x)\), it directly affects the derivative by flipping the sign.
This intriguing property demonstrates how the geometry of a function is deeply connected to its calculus properties.

Limits

Limits are pivotal in the realm of calculus as they form the foundation for defining derivatives. When we talk about the limit in the context of derivatives, it is about understanding what the slopes of secant lines approach as the interval between two points becomes infinitesimally small.
The limit is evaluated by:\[ \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}\]This expression captures how the difference quotient behaves as \( h \), the distance between two points, goes to zero.

• The existence of this limit is crucial for the derivative to exist at a point.
• Without limits, the concept of instantaneous rate of change wouldn't be possible to define mathematically.

Using limits, we can understand different types of behavior in functions, such as continuity and differentiability. They allow us to peek into instantaneous actions rather than just average behaviors.

Function Reflection

Reflecting a function over the x-axis is a simple yet significant transformation. When we take a function \( f(x) \) and consider \(-f(x)\), each output value is multiplied by \(-1\). This reflection inverts the graph across the x-axis, thereby reversing the direction of all the function's outputs.

• This reflection doesn't affect the differentiability of the function.
• If \( f(x) \) is differentiable at \( x_0 \), so is \(-f(x)\), because the derivative of \(-f(x)\) is simply \(-f'(x)\).

You can visualize this as flipping every point of the function downwards, creating a mirror image in terms of position but keeping the smoothness and differentiability intact.
Function reflections provide an interesting look at symmetries and transformations, which can be valuable when considering real-world applications in physics and engineering.