91Ó°ÊÓ

In calculus, the first derivative of a function, denoted as \( f'(x) \), helps us understand how the function's output changes as its input changes. We use the power rule to differentiate functions like \( f(x) = x^2(3-x) \), which, when expanded, yields \( f(x) = 3x^2 - x^3 \). By applying the power rule, we find the first derivative to be \( f'(x) = 6x - 3x^2 \).
This first derivative is essential for locating the slopes of tangent lines at any point on the curve, which in turn helps us identify critical points and intervals where the function is increasing or decreasing.

• When \( f'(x) > 0 \), the function is increasing.
• When \( f'(x) < 0 \), the function is decreasing.

Finding where \( f'(x) = 0 \) helps locate critical points where the function either reaches a peak (local maximum) or a trough (local minimum).
Understanding the sign of the first derivative in different intervals allows us to see where the graphed function moves upward or downward on the chart.

The second derivative, \( f''(x) \), gives us insight into the curvature and concavity of a function's graph. It is basically the derivative of the first derivative, providing information about how the slope is changing. For our function, the second derivative is obtained from \( f'(x) = 6x - 3x^2 \) by differentiating again, which results in \( f''(x) = 6 - 6x \).
This second derivative helps you understand not only the rate at which the slope of the curve is changing but also the type of curvature (concave up or down) the graph has at any given point.

• If \( f''(x) > 0 \), the graph is concave up like a smile.
• If \( f''(x) < 0 \), it is concave down like a frown.

This tells us where the curve's direction changes, pertaining to whether it bends upwards or downwards, crucial for sketching accurate graphs.

Critical points are the values of \( x \) at which the first derivative is zero or undefined, indicating potential local maxima or minima in the function. For our function, \( f'(x) = 6x - 3x^2 \), setting this equal to zero gives the equation \( 6x - 3x^2 = 0 \). Solving this by factoring out \( 3x \), we find \( 3x(2-x) = 0 \), resulting in critical points \( x = 0 \) and \( x = 2 \).
Examining these points helps pinpoint where the function has changes in direction from increasing to decreasing (or vice versa). At these points, the graph of the function pauses momentarily before switching to a new trend. Examining behavior around these critical points, we determine whether each is a peak, valley, or potentially a point of inflection.

Inflection points occur where the graph of a function changes its concavity: from concave up to concave down, or vice versa. These are found by testing where the second derivative equals zero or is undefined. For \( f''(x) = 6-6x \), we set this to zero, resulting in \( 6 = 6x \) or \( x = 1 \).
This point at \( x = 1 \) is crucial as it signifies a shift in the function's curvature from being concave up to concave down, marking a transition between bending upwards like a bowl to bending downwards like an arch.

• To confirm, test intervals around the inflection point. If sign changes, an inflection point is present.

Inflection points are important as they indicate turning points in the slope's rate of change, affecting how the graph continues onwards.

Concavity tells you how the graph of the function curves and is identified using the second derivative. By computing \( f''(x) = 6 - 6x \), we determine from the sign of \( f''(x) \) whether the graph is concave up or down in different intervals of \( x \).
If \( f''(x) > 0 \) in an interval, the graph is concave up, forming a cup shape where the function appears like it holds water. Conversely, if \( f''(x) < 0 \), the graph is concave down, resembling an upturned bowl. In our example,

• for \( x < 1, \ f''(x) > 0 \), showing concavity upwards.
• for \( x > 1, \ f''(x) < 0 \), showing concavity downwards.

Understanding concavity is key to visualizing and sketching the graph's overall structure, indicating how and where the graph curves and changes direction. Each concavity interval gives insight into the overall shape and behavior of the curve across different sections of its domain.