Problem 18 - Calculus Concepts

Question

Cattle Prices The figure shows cattle prices (for choice 450 -pound steer calves) from October 1994 through May \(1995 .\) \(p(m)\) dollars (Source: Based on data from the National Cattleman's Association) a. For which input value does the derivative fail to exist? Explain. b. Sketch a slope graph of \(n\). Label both axes

Step-by-Step Solution

Answer

The derivative fails at sharp turns in the graph; sketch the slope graph to show continuity and discontinuities.

Step 1: Understanding the Graph

First, we need to identify the behavior of the graph of cattle prices over time. The graph plots price against months from October 1994 to May 1995. Observe any sharp turns or discontinuities in the graph as these can be places where the derivative fails to exist.

Step 2: Identifying Non-Differentiable Points

The derivative does not exist at points where the graph has a vertical tangent, a cusp, or a discontinuity. Look at the graph to see if there is a sharp corner or a point where the graph is not smooth.

Step 3: Evaluate Each Month for Non-differentiability

Examine each month starting from October 1994 to May 1995 on the graph. Identify any month where the slope of the tangent to the curve is not defined or the graph has a sharp corner.

Step 4: Sketching a Slope Graph

Create a graph with the x-axis representing the months (from October 1994 to May 1995) and the y-axis representing the slope of the function, or the derivative value at each point. Plot the slope values observed at each month, connecting them smoothly except for any points where the derivative does not exist.

Step 5: Labeling the Slope Graph

Label the horizontal axis as 'Months' and the vertical axis as 'Slope (derivative of p(m))'. Mark any point where the derivative does not exist with an asterisk or other marker. Ensure each axis is clearly labeled.

Key Concepts

Derivatives

In calculus, the derivative of a function measures how the function's output changes with respect to changes in the input. It is essentially the rate of change at any given point on a curve.

• If a function is differentiable at a point, it means there exists a tangent line with a defined slope at that point.
• To find the derivative of the cattle prices graph, you would look for the slope of the tangent line at various points representing different months.

When analyzing the given graph of cattle prices, each month's price can be seen as a point on a curve. The derivative provides insights into how the price fluctuates over that time period.

Graph Analysis

Graph analysis involves examining and interpreting the visual data represented on a chart to make informed conclusions. For the cattle prices graph:

• Consider the overall direction in which the data points move. This can indicate price increases or decreases over time.
• Look for patterns such as linear trends, periodic cycles, or anomalies.

Graph analysis is crucial for identifying whether the graph is smooth and differentiable, as well as for spotting any sharp changes or discontinuities that may affect the derivative.

Non-Differentiability

Non-differentiability refers to points on a graph where the derivative does not exist. This can occur due to several reasons:

• Sharp corners: These are points where the graph has a sudden change in direction, resulting in an undefined tangent slope.
• Vertical tangents: When the tangent at a point is vertical, the slope is undefined, leading to non-differentiability.
• Discontinuities: Breaks or jumps in the graph where there is a gap between values can also cause the derivative to be undefined.

To find points of non-differentiability on the cattle prices graph, look for any abrupt changes or breaks during the observed months.

Slope Graphs

A slope graph, also known as a derivative graph, provides a visualization of how steep the curve is at any given point, effectively plotting the derivative of the original graph.

• The x-axis represents time or months, which in this case, runs from October 1994 to May 1995.
• The y-axis corresponds to the slope of the price function, indicating how quickly the cattle prices are changing month to month.

To create a slope graph based on the cattle prices data: - Start by estimating the slope at each month on the original graph. - Mark these values on the slope graph accordingly. - Pay special attention to points where the derivative does not exist and denote them clearly, such as with an asterisk. This visual tool helps in understanding the dynamic changes in price trends over the observed period.