Question

Suppose that \(f\) is differentiable for all \(x\) and consider the function $$D(x)=\frac{f(x+0.01)-f(x)}{0.01}.$$ For the following functions, graph \(D\) on the given interval, and explain why the graph appears as it does. What is the relationship between the functions \(f\) and \(D ?\) $$f(x)=\sin x \text { on }[-\pi, \pi]$$

Short answer

Answer: The relationship between \(f(x) = \sin x\) and \(D(x)\) is that \(D(x)\) approximates the derivative of \(f(x)\), scaled by a factor of \(\frac{1}{0.01}\). The graph of \(D(x)\) on the interval \([-π, π]\) is a rescaled version of the derivative of \(f(x)\), which is \(\cos x\), with its amplitude increased by a factor of 100.

Step-by-step solution

Differentiate the function f(x)

Find the derivative of the function \(f(x) = \sin x\), which is \(\frac{d}{dx}(f(x)) = f'(x) = \cos x\).

Define the function D(x)

According to the problem, we have the function \(D(x)=\frac{f(x+0.01)-f(x)}{0.01}\). Now let's plug in the values of \(f(x)\) and \(f'(x)\) that we obtained in Step 1: $$D(x)=\frac{f(x+0.01)-f(x)}{0.01}\approx\frac{f'(x)}{0.01}$$ Since \(f'(x) = \cos x\), we have: $$D(x) \approx \frac{\cos x}{0.01}$$

Graph D(x) on the interval [-π, π]

Graph the function \(D(x) \approx \frac{\cos x}{0.01}\) on the interval \([-\pi, \pi]\). This graph will appear as a sinusoidal function with amplitude increased by a scaling factor of \(\frac{1}{0.01} = 100\). Therefore, the graph will range from \(-100\) to \(100\) and have a period of \(2\pi\).

Explain the graph and deduce the relationship between f(x) and D(x)

The graph of \(D(x)\) is a rescaled version of the derivative of \(f(x)\), which is \(\cos x\). The scaling is due to the \(\frac{1}{0.01}\) factor. The relationship between \(f(x) = \sin x\) and \(D(x)\) is that \(D(x)\) approximates the derivative of \(f(x)\), scaled by a factor of \(\frac{1}{0.01}\). Since the derivative of \(\sin x\) is \(\cos x\), \(D(x)\) can be seen as an approximation of \(\cos x\) with its amplitude increased by a factor of 100.

Key concepts

Derivative

In calculus, a "derivative" is a fundamental concept that measures how a function changes as its input changes. It's like asking, "How does this function move?" The derivative tells us the rate at which a function's value changes at any given point. For the function \( f(x) = \sin x \), the derivative is \( f'(x) = \cos x \). This expression tells us that the rate of change of the sine function is determined by the cosine function.

Calculating the derivative involves differentiating the original function. When you differentiate \( \sin x \), you find that at each point \( x \), the steepness of the sine curve is determined by the cosine of \( x \). This makes sense since the sine and cosine functions are just phase-shifted versions of each other, cycling through values as you move along the x-axis. Understanding how derivatives function helps you understand how graphs change direction, slope, and shape.

• Rate of Change: The derivative provides the rate of change of the function.
• Function Behavior: By knowing the derivative, one can understand growth, shrinkage, and trends of the original function.
• Applications: Derivatives are used in optimizing functions, understanding motion, and in many real-world problems.

Sine Function

The sine function, denoted as \( \sin x \), is one of the basic trigonometric functions that relates the angle of a right triangle to the length of its opposite side divided by the hypotenuse. This function is periodic, which means it continues to repeat its values in regular intervals. For the sine function, this period is \( 2\pi \), which means it completes a full cycle every \( 2\pi \) units.

In mathematics, \( \sin x \) oscillates between \(-1\) and \(1\), taking on these maximum and minimum values at certain points along its graph. Importantly, the sine function has a smooth, wave-like pattern which becomes evident when graphed on the interval \([-\pi, \pi]\). When working with the sine function, it is critical to understand its properties of periodicity, symmetry, and range.

• Periodicity: The sine function repeats every \( 2\pi \).
• Range: Values in the range of \(-1\) to \(1\).
• Symmetry: It is an odd function, meaning \( \sin(-x) = -\sin(x) \).

Graphing Functions

Creating graphs of mathematical functions provides a visual understanding of the function's behavior over different intervals. For the function \( D(x) = \frac{f(x+0.01)-f(x)}{0.01} \), graphing involves understanding its relationship with \( f(x) = \sin x \). The given exercise concerns graphing \( D(x) \) over the interval \([-\pi, \pi]\).

This function \( D(x) \) behaves like \( \cos x \) but is stretched vertically due to the division by the small constant \( 0.01 \). This generates a graph that has the same frequency as \( \cos x \) but oscillates between \(-100\) and \(100\) due to scaling. Visualizing \( D(x) \) on a graph helps us see how it models the rapid changes of the sine function over small changes in \( x \), as it's essentially an approximation of the sine function's derivative scaled by 100.

• Scaling: Small changes in \( x \) cause large changes in \( D(x) \), leading to a scaled appearance.
• Frequency: Same frequency as \( \cos x \), highlighting its periodic nature.
• Amplitude: Dramatically increased due to the factor of \( \frac{1}{0.01} \).