Problem 50 - University Calculus: Early Transcendentals

Question

a. Find the derivative \(f^{\prime}(x)\) of the given function \(y=f(x)\) b. Graph \(y=f(x)\) and \(y=f^{\prime}(x)\) side by side using separate sets of coordinate axes, and answer the following questions. c. For what values of \(x\), if any, is \(f^{\prime}\) positive? Zero? Negative? d. Over what intervals of \(x\) -values, if any, does the function \(y=f(x)\) increase as \(x\) increases? Decrease as \(x\) increases? How is this related to what you found in part (c)? (We will say more about this relationship in Section \(4.3 .\) ) $$y=-1 / x$$

Step-by-Step Solution

Answer

Derivative is \( f'(x) = \frac{1}{x^{2}} \), always positive; \( y = -1/x \) increases on \((-\infty, 0)\) and \((0, \infty)\).

Step 1: State the Function

The given function is \( y = f(x) = -\frac{1}{x} \).

Step 2: Find the Derivative

To find the derivative of \( y = -\frac{1}{x} \), we rewrite the function as \( y = -x^{-1} \). The derivative is then found using the power rule: \( f'(x) = \frac{d}{dx}(-x^{-1}) = x^{-2} = \frac{1}{x^{2}} \). Thus, \( f'(x) = \frac{1}{x^{2}} \).

Step 3: Graph the Functions

Graph the function \( y = -\frac{1}{x} \) and its derivative \( y = \frac{1}{x^{2}} \) on separate coordinate axes. The graph of \( y = -\frac{1}{x} \) is a hyperbola, and the graph of \( y = \frac{1}{x^{2}} \) is a parabola opening upwards.

Step 4: Analyze the Derivative Sign

Since \( f'(x) = \frac{1}{x^{2}} \), it is always positive except when \( x = 0 \), where it is undefined. Therefore, \( f'(x) > 0 \) for all \( x eq 0 \).

Step 5: Determine Intervals of Increase and Decrease

Since \( f'(x) > 0 \) for all \( x eq 0 \), the original function \( y = -\frac{1}{x} \) is increasing on the intervals \( (-\infty, 0) \) and \( (0, \infty) \), as this is where the derivative is positive. The function does not decrease, which matches the results from the derivative analysis in part (c).

Key Concepts

Derivatives

The concept of derivatives is a fundamental idea in calculus, essentially measuring how a function changes as its input changes. When we talk about the derivative of a function, we are referring to the rate at which the function's value is changing with respect to change in the input. For the given function

• The function is given as \( y = -\frac{1}{x} \).
• To find its derivative, we can rewrite the function as \( y = -x^{-1} \).
• Using the power rule, which states that \( \frac{d}{dx} x^n = nx^{n-1} \), we differentiate the function.
• Applying the power rule here, the derivative is calculated as \( f'(x) = \frac{d}{dx}(-x^{-1}) = x^{-2} = \frac{1}{x^{2}} \).

This derivative indicates how the function \( y = -\frac{1}{x} \) behaves as \( x \) changes. Notably, this derivative \( f'(x) \) is positive for all \( x eq 0 \), signifying that the function consistently increases at every point, except where it is undefined.

Graphing functions

Graphing functions and their derivatives provide a visual representation of the behavior of both the original function and its rate of change. When you graph

• The original function \( y = -\frac{1}{x} \), you will see it is a hyperbola, which shows that as \( x \) approaches zero from either direction, \( y \) sharply decreases or increases - illustrating vertical asymptotes.
• The derivative \( y = \frac{1}{x^2} \) is a parabola that opens upwards, illustrating that the rate of change is never negative; it’s always heading upward.

By examining these graphs together, you can identify where the function is increasing or decreasing based on the positive or negative value of the derivative, though for this particular function, \( f'(x) \) remains positive. Thus the insights gained from the graph of the derivative stands in line with our calculations under this topic: \( y = -\frac{1}{x} \) consistently increases.

Intervals of increase and decrease

One helpful application of the derivative is determining intervals of increase and decrease of the function based on the sign of the derivative.

• If the derivative \( f'(x) \) is positive over an interval, then the function \( y = f(x) \) increases over that interval.
• If the derivative is negative, the function decreases across that range.

Since \( f'(x) = \frac{1}{x^2} \), it remains positive except where undefined at \( x = 0 \). Thus, the graph of \( y = -\frac{1}{x} \) shows that the function increases on the intervals \(( -\infty, 0)\) and \(( 0, \infty) \).
This observation aligns perfectly with the mathematical analysis done earlier. Intervals of increase correspond with positive derivative values, proving that understanding and interpreting derivative behavior lead to insights about the function's behavior over different sections of the graph. Consequently, the relationship explored here can directly help predict the function's behavior without needing intensive calculations.