Problem 45 - Thomas Calculus

Question

In Exercises \(45-48\) 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 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 Chapter \(4 . )\) $$ y=-x^{2} $$

Step-by-Step Solution

Answer

Derivative is \( f'(x) = -2x \); \( f'(x) \) is zero at \( x = 0 \), positive when \( x < 0 \), and negative when \( x > 0 \).

Step 1: Find the Derivative

Start with the given function \( y = f(x) = -x^2 \). To find the derivative \( f'(x) \), use the power rule for derivatives, which states: \( \frac{d}{dx} [x^n] = nx^{n-1} \). Apply this to \( -x^2 \):\[ f'(x) = \frac{d}{dx} (-x^2) = -2x^{2-1} = -2x \].Thus, the derivative of the function is \( f'(x) = -2x \).

Step 2: Graph the Functions

Graph \( y = f(x) = -x^2 \), which is a downward-opening parabola with vertex at the origin. Next, graph the derivative \( y = f'(x) = -2x \), which is a straight line with slope \(-2\) passing through the origin. Ensure each function is on separate axes but side by side to compare them directly.

Step 3: Finding Zeroes of f'(x)

To find where \( f'(x) \) is zero, solve \( -2x = 0 \) for \( x \). Divide both sides by \(-2\) to get \( x = 0 \). So, \( f'(x) \) is zero at \( x = 0 \).

Step 4: Identifying Where f'(x) is Positive or Negative

For \( f'(x) = -2x \), determine when the derivative is positive or negative:- \( f'(x) = -2x > 0 \) when \( x < 0 \).- \( f'(x) = -2x < 0 \) when \( x > 0 \).

Step 5: Determine Intervals of Increase and Decrease in f(x)

When \( f'(x) > 0 \), the function \( y = f(x) \) is increasing, and when \( f'(x) < 0 \), it is decreasing:- \( y = f(x) = -x^2 \) increases as \( x \) increases for \( x < 0 \), because \( f'(x) > 0 \) in this interval.- \( y = f(x) = -x^2 \) decreases as \( x \) increases for \( x > 0 \), since \( f'(x) < 0 \) in this interval.This behavior corresponds directly to the sign of \( f'(x) \).

Key Concepts

Power Rule for Derivatives

The power rule is a fundamental tool for finding the derivative of a function. It is particularly useful when dealing with functions of the form \( x^n \). The rule states: if \( y = x^n \), then the derivative \( \frac{d}{dx}[x^n] = nx^{n-1} \).

So, for the function \( y = -x^2 \), applying the power rule means bringing down the exponent 2, multiplying by the coefficient (which is -1 in this case), and reducing the exponent by 1. Hence, we derive:

• \( f'(x) = \frac{d}{dx} [-x^2] = -2x \).

By understanding the power rule, we can easily find derivatives of polynomial functions. Knowing derivatives helps in analyzing the behavior of functions, such as increasing or decreasing trends, and points of inflection.

This straightforward method is key in calculus, allowing us to deduce complex changes in functions smoothly.

Graphing Functions

Graphing a function and its derivative provides a visual insight into the function's behavior.

• The function \( y = -x^2 \) is a downward-opening parabola with its vertex at the origin. This shape is characteristic of a quadratic with a negative leading coefficient.
• On the other hand, its derivative \( f'(x) = -2x \) is a straight line passing through the origin, with a negative slope of -2.

Graphing them side by side helps in directly comparing the original function to how it changes:

• The slope of the tangent (given by the derivative) tells us how steep the parabola is at any point on its curve.
• When graphing these, ensure each graph is on separate axes to clearly see how the behavior of the derivative influences the graph of the original function.

This visualization between a function and its derivative aids in understanding the nature of the changes occurring in the function.

Intervals of Increase and Decrease

The intervals of increase and decrease tell us about the behavior of a function as \( x \) changes. If \( f'(x) > 0 \), the function \( y = f(x) \) is increasing. Conversely, if \( f'(x) < 0 \), it is decreasing.

For the function \( y = -x^2 \):

• When \( x < 0 \), \( f'(x) = -2x > 0 \). This means the function is increasing in the interval from negative infinity up to zero.
• When \( x > 0 \), \( f'(x) = -2x < 0 \). Thus, the function is decreasing for positive values of \( x \).

By identifying where the derivative equals zero (at \( x = 0 \)), we note that the graph reaches a vertex point at this location, which is often a maximum for parabolas like \( y = -x^2 \).

This relationship is crucial in calculus for understanding where a function reaches its peaks and troughs, and it connects directly to where the derivative changes sign.