Question

Draw the graphs of \(f(x)=\cos x-\sin (x / 2)\) and its derivative \(f^{\prime}(x)\) on the interval \([0,9]\) using the same axes. (a) Where on this interval is \(f^{\prime}(x)>0\) ? (b) Where on this interval is \(f(x)\) increasing? (c) Make a conjecture. Experiment with other intervals and other functions to support this conjecture.

Short answer

(a) \(f'(x) > 0\) where \(f(x)\) is increasing. (b) \(f(x)\) is increasing where \(f'(x) > 0\). (c) If \(f'(x) > 0\), then \(f(x)\) is increasing on those intervals.

Step-by-step solution

Understanding the Function

We are given the function \(f(x) = \cos x - \sin(\frac{x}{2})\). We need to find its derivative \(f'(x)\) and sketch both on the interval \([0, 9]\).

Calculating the Derivative

The derivative of \(f(x)\) is found by differentiating each term separately. The derivative of \(\cos x\) is \(-\sin x\), and the derivative of \(-\sin(\frac{x}{2})\) is \(-\frac{1}{2} \cos(\frac{x}{2})\), using the chain rule.

Derivative Formula

Combine the derivatives to get \(f'(x) = -\sin x - \frac{1}{2} \cos(\frac{x}{2})\).

Sketching the Graphs

Using graphing software or plotting by hand, draw the graph of \(f(x) = \cos x - \sin(\frac{x}{2})\) and its derivative \(f'(x) = -\sin x - \frac{1}{2} \cos(\frac{x}{2})\) on the interval \([0, 9]\).

Identifying \(f'(x) > 0\)

On the graph of \(f'(x)\), observe the intervals where the function is above the x-axis. These intervals indicate where \(f'(x) > 0\).

Determining Increasing Intervals for \(f(x)\)

The function \(f(x)\) is increasing where \(f'(x) > 0\), as the derivative indicates the slope of \(f(x)\). These are the same intervals identified in Step 5.

Making a Conjecture

Based on the intervals where \(f(x)\) is increasing, one might conjecture that wherever \(f'(x) > 0\), \(f(x)\) is increasing. This is true for differentiable functions, as a positive derivative indicates an increasing function.

Experimentation

Experiment with other functions and intervals by repeating steps 1 through 7. Verify if the conjecture holds true by observing the relationship between the derivative \(f'(x)\) and the increasing nature of the function \(f(x)\).

Key concepts

Derivative of a Function

The derivative of a function is a concept in differential calculus that measures how a function's value changes as its input changes. To put it simply, the derivative tells us the rate at which a function is changing at any given point. For example, if you have a function like \( f(x) = \cos x - \sin(\frac{x}{2}) \), the derivative \( f'(x) \) gives you the slope of the tangent line to the curve of \( f(x) \) at every point on it. This slope can tell us if a function is increasing, decreasing, or staying constant at a particular point.

To find the derivative, you differentiate each term of the function. You apply rules like the power rule, product rule, or chain rule, depending on the function's composition.

For instance, to differentiate \( f(x) = \cos x - \sin(\frac{x}{2}) \), we find the derivative of \( \cos x \), which is \( -\sin x \), and the derivative of \( -\sin(\frac{x}{2}) \), which is \( -\frac{1}{2} \cos(\frac{x}{2}) \) using the chain rule, as this term is a composite function. The chain rule helps simplify differentiation of composite functions.

Graph Sketching

Graph sketching is an essential skill in calculus that allows you to visualize functions and their derivatives. It involves plotting the graph of a function and understanding its behavior over a specific interval. For the function \( f(x) = \cos x - \sin(\frac{x}{2}) \), sketching involves plotting this function across the interval \([0, 9]\).

Graph sketching usually starts with identifying key features of the function such as intercepts, critical points where the derivative \( f'(x) \) equals zero, and points of inflection. These features can demonstrate where the function achieves maximum or minimum values, or changes concavity.

When sketching \( f(x) \) along with its derivative \( f'(x) \), it's crucial to note the areas where \( f'(x) \) is positive or negative, as these indicate where the original function is increasing or decreasing.

Chain Rule

The chain rule is a fundamental theorem in calculus used to find the derivative of composite functions. A composite function is a function made up of two or more functions, where the output of one function becomes the input of another. For the given function \( f(x) = \cos x - \sin(\frac{x}{2}) \), \( \sin(\frac{x}{2}) \) is a composite function, as it involves \( x/2 \) inside the sine function.

The chain rule states that to differentiate a composite function \( g(h(x)) \), you take the derivative of the outer function \( g \), evaluate it at \( h(x) \), and multiply it by the derivative of the inner function \( h(x) \).

In practice, using the chain rule for \( -\sin(\frac{x}{2}) \), you first differentiate \( -\sin u \) (where \( u = \frac{x}{2} \)) to get \( -\cos u \), then multiply by the derivative of \( u \) with respect to \( x \), which is \( \frac{1}{2} \). Hence, the derivative becomes \( -\frac{1}{2} \cos(\frac{x}{2}) \). This calculation is essential to accurately determine the derivative \( f'(x) \).

Increasing Functions

Increasing functions are functions where, as the input value increases, the output value also increases. In mathematical terms, for a function \( f(x) \), if \( f'(x) > 0 \) for every \( x \) in an interval, \( f(x) \) is said to be increasing on that interval.

Understanding where a function is increasing involves evaluating its derivative. For the function \( f(x) = \cos x - \sin(\frac{x}{2}) \), determining the intervals where \( f'(x) = -\sin x - \frac{1}{2} \cos(\frac{x}{2}) \) is greater than zero allows us to identify where the original function \( f(x) \) is increasing.

This concept is foundational in understanding the behavior of functions plotted on a graph. By examining where \( f'(x) > 0 \) on the graph of \( f'(x) \), you can predict the intervals on which \( f(x) \) rises. Doing so helps in making conjectures about the function's behavior over other intervals and with different functions, illustrating the broader applicability of derivative tests.