Problem 45 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use a graphing calculator or a computer to do Problems \(45-48\) Let \(f(x)=\sin x+\cos (x / 2)\) on the interval \(I=(-2,7)\). (a) Draw the graph of \(f\) on \(I .\) (b) Use this graph to estimate where \(f^{\prime}(x)<0\) on \(I\). (c) Use this graph to estimate where \(f^{\prime \prime}(x)<0\) on \(I\). (d) Plot the graph of \(f^{\prime}\) to confirm your answer to part (b). (e) Plot the graph of \(f^{\prime \prime}\) to confirm your answer to part (c).

Step-by-Step Solution

Answer

Graph shows intervals: \(f'(x) < 0\) on corresponding tangent slopes; \(f''(x) < 0\) on downward concavity.

Step 1: Graph the Function

Using a graphing calculator or computer software, plot the graph of the function \(f(x) = \sin x + \cos (x/2)\) over the interval \((-2, 7)\). Ensure the graph has enough resolution to clearly see the detail of the function's behavior over this interval.

Step 2: Analyze f'(x) from the Graph

By observing the graph from Step 1, estimate the intervals where the slope of the tangent (i.e., the derivative \(f'(x)\)) is less than zero. Look for intervals where the function curves downwards, indicating negative slopes.

Step 3: Analyze f''(x) from the Graph

From the graph, estimate where the concavity of \(f(x)\) is downward, indicating that \(f''(x) < 0\). Look for intervals where the graph is curving concave down.

Step 4: Plot f'(x)

Using the same graphing tool, plot the derivative \(f'(x) = \frac{d}{dx} [\sin x + \cos (x/2)]\) over the interval \((-2, 7)\). Confirm the estimates from Step 2 by checking where the graph is below the x-axis (negative values).

Step 5: Plot f''(x)

Similarly, plot the second derivative \(f''(x) = \frac{d^2}{dx^2} [\sin x + \cos (x/2)]\) over the interval \((-2, 7)\). Confirm the estimates from Step 3 by identifying intervals where the graph is below the x-axis.

Key Concepts

Graphical Analysis

Graphical analysis is pivotal for understanding the behavior of functions without relying solely on algebraic manipulation. Using a graphing calculator or computer software, you can visually interpret the function \(f(x) = \sin x + \cos (x/2)\) over the interval \((-2, 7)\). This provides a clear picture of its behavior, showing peaks, troughs, and inflection points.
Graphs allow us to see where the function increases or decreases and helps identify critical points like maxima or minima. Through graphical analysis, identifying where the function's derivative or second derivative changes can become more intuitive.
When you draw the graph, pay attention to:

• Where the function crosses the x-axis.
• Peaks and valleys, which hint at where the derivative might be zero.
• Overall upward or downward trends across the interval.

Derivative Estimation

Derivative estimation involves gauging the rate of change of \(f(x)\) from its graph. The first derivative \(f'(x)\) signifies where the function is increasing or decreasing. When \(f'(x) < 0\), the function is decreasing.
On the graph of \(f(x)\), observe where the graph slopes downward as this indicates a negative derivative.
To estimate \(f'(x)\) from the graph, you can:

• Look for sections where the graph moves downwards as you move from left to right.
• Approximate where the tangent line would have a negative slope.

This visual analysis helps predict where the derivative is less than zero. For instance, you might observe such behavior over specific segment(s) of \((-2, 7)\). Confirming these intervals with a more precise graph of the derivative can solidify your estimates.

Second Derivative

The second derivative, \(f''(x)\), provides insights into the concavity and the rate of change of \(f'(x)\). Understanding \(f''(x)\) is key to analyzing the function's curvature—whether it's concave up or concave down.
If \(f''(x) < 0\), the curve is concave down, indicating that slopes are increasingly negative (or less positive). These points often signal the presence of maxima or points of inflection depending on \(f'(x)\) behavior around them.
You can identify where \(f''(x) < 0\) by:

• Noticing sections of the graph that curve downward like a frown.
• Checking where these sections further change from up to down or vice versa.

This helps estimate intervals of downward concavity, which can be confirmed with a graph of \(f''(x)\).

Concavity Analysis

Concavity analysis delves into whether the graph curves upwards or downwards at different intervals, as identified by the second derivative \(f''(x)\). Analyzing concavity not only confirms specific features of the graph but also predicts the behavior of critical points.
Typically, if \(f''(x) > 0\), the graph is concave up like a cup, indicating points might be minima. If \(f''(x) < 0\), the curve is concave down like a frown. This helps pinpoint potential maxima or inflection points where the concavity changes.
Approaching concavity analysis might involve:

• Pinpointing sections where the slope changes direction.
• Looking for points where curving direction changes sharply—these often correspond with \(f'(x)\) crossing zero or being at extremes.

Confirming your findings by plotting \(f''(x)\) not only verifies concavity estimates but also improves your predictive insights into the function's behavior.