Problem 45
Question
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
Verified Answer
Graph \(f(x)\) to identify intervals for decreasing trend (for \(f'<0\)) and concavity (for \(f''<0\)), and confirm by plotting \(f'\) and \(f''\).
1Step 1: Graph the Function
Use a graphing tool to plot the function \( f(x) = \sin x + \cos (\frac{x}{2}) \) over the interval \((-2, 7)\). Identify key features such as maximum, minimum points, and zero crossings. This visual representation will help us analyze the derivative sections later.
2Step 2: Estimate Where f'(x) < 0
Examine the graph of \(f(x)\). The derivative \(f'(x)\) is negative where the function \(f(x)\) is decreasing. Note the intervals where \(f\) slopes downward as this indicates \(f'(x) < 0\). This usually occurs between peaks and troughs.
3Step 3: Estimate Where f''(x) < 0
Look at the concavity of \(f(x)\) in the graph. The second derivative \(f''(x)\) is negative where \(f(x)\) is concave down. This will typically occur on intervals where \(f(x)\) appears to form a downward-facing bowl.
4Step 4: Plot the Graph of f'(x)
Derive \( f'(x) \) using differentiation: \( f'(x) = \cos x - \frac{1}{2} \sin (\frac{x}{2}) \). Plot \( f'(x) \) over the interval \((-2, 7)\) to confirm where it is negative, supporting your findings in Step 2. Note the intervals where the graph lies below the x-axis.
5Step 5: Plot the Graph of f''(x)
Derive \( f''(x) \) to check concavity: \( f''(x) = -\sin x - \frac{1}{4} \cos (\frac{x}{2}) \). Plot \( f''(x) \) over the same interval to confirm where it is negative, validating your assessment in Step 3. Identify sections where the graph is below the x-axis.
Key Concepts
DerivativeGraphing FunctionsConcavity
Derivative
Understanding derivatives is an essential part of calculus. A derivative, often denoted as \( f'(x) \), measures the rate at which a function changes at any given point. It is essentially the slope of the tangent line to a function at a point.
A positive derivative indicates that the function is increasing at that point, while a negative derivative suggests the function is decreasing. When assessing whether \( f'(x) < 0 \), we are identifying parts of the function where there is a downward slope. These are typically found between high and low points on the graph, known as maximum and minimum points, respectively.
To find \( f'(x) \) for the function \( f(x) = \sin x + \cos(\frac{x}{2}) \), we use differentiation rules. The derivative will be \( f'(x) = \cos x - \frac{1}{2} \sin(\frac{x}{2}) \). By examining the graph of \( f'(x) \), one can visually confirm where the function's slope is negative, supporting estimations made from the original function graph.
A positive derivative indicates that the function is increasing at that point, while a negative derivative suggests the function is decreasing. When assessing whether \( f'(x) < 0 \), we are identifying parts of the function where there is a downward slope. These are typically found between high and low points on the graph, known as maximum and minimum points, respectively.
To find \( f'(x) \) for the function \( f(x) = \sin x + \cos(\frac{x}{2}) \), we use differentiation rules. The derivative will be \( f'(x) = \cos x - \frac{1}{2} \sin(\frac{x}{2}) \). By examining the graph of \( f'(x) \), one can visually confirm where the function's slope is negative, supporting estimations made from the original function graph.
Graphing Functions
Graphing functions is a crucial part of understanding their behavior over an interval. The graph provides a visual tool that can depict changes, reveal maximum and minimum points, and show zero crossings.
To start, choosing the function \( f(x) = \sin x + \cos(\frac{x}{2}) \), we can use graphing tools to plot its behavior over the interval \((-2,7)\). Important features to identify include:
To start, choosing the function \( f(x) = \sin x + \cos(\frac{x}{2}) \), we can use graphing tools to plot its behavior over the interval \((-2,7)\). Important features to identify include:
- Zero crossings: Points where the graph intersects the x-axis.
- Local maxima and minima: Peaks and troughs indicating changes in direction.
- Increasing and decreasing intervals: Sections where the graph is moving upwards or downwards.
Concavity
The concept of concavity describes the direction in which a curve bends. This is determined by the second derivative \( f''(x) \) of a function. If \( f''(x) > 0 \), the function is concave up, resembling an upward-facing bowl, and if \( f''(x) < 0 \), the function is concave down, similar to a downward-facing bowl.
For the function \( f(x) = \sin x + \cos(\frac{x}{2}) \), the second derivative \( f''(x) = -\sin x - \frac{1}{4} \cos(\frac{x}{2}) \) gives insights into its concavity over the specified interval. By plotting \( f''(x) \), we can visually confirm sections of the graph where \( f(x) \) is concave down, indicated by negative values of \( f''(x) \).
Understanding concavity helps in predicting the overall shape of a graph and its behavior between the extreme points. It is a valuable tool in both analyzing graphs and solving calculus problems.
For the function \( f(x) = \sin x + \cos(\frac{x}{2}) \), the second derivative \( f''(x) = -\sin x - \frac{1}{4} \cos(\frac{x}{2}) \) gives insights into its concavity over the specified interval. By plotting \( f''(x) \), we can visually confirm sections of the graph where \( f(x) \) is concave down, indicated by negative values of \( f''(x) \).
Understanding concavity helps in predicting the overall shape of a graph and its behavior between the extreme points. It is a valuable tool in both analyzing graphs and solving calculus problems.
Other exercises in this chapter
Problem 45
In Problems 43-47, the graph of \(y=f(x)\) depends on a parameter \(c\). Using a \(C A S\), investigate how the extremum and inflection points depend on the val
View solution Problem 45
Consider \(f(x)=A x^{2}+B x+C\) with \(A>0\). Show that \(f(x) \geq 0\) for all \(x\) if and only if \(B^{2}-4 A C \leq 0\).
View solution Problem 45
Use the Mean Value Theorem to show that $$ |\sin x-\sin y| \leq|x-y| $$
View solution Problem 46
In Problems 43-47, the graph of \(y=f(x)\) depends on a parameter \(c\). Using a \(C A S\), investigate how the extremum and inflection points depend on the val
View solution