{Use of Tech} Difference quotients Suppose f is differentiable for all x and consider the function D(x) = 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 on [−π,π]

Verified step by step guidance

1. Understand the concept of the difference quotient: The difference quotient D(x) = (f(x+0.01) - f(x)) / 0.01 is an approximation of the derivative of f at x. It measures the average rate of change of the function f over a small interval [x, x+0.01].
2. Recognize the function f(x) = sin(x): The sine function is periodic with a period of 2π and is differentiable everywhere. Its derivative, f'(x), is cos(x).
3. Graph D(x) over the interval [-π, π]: To graph D(x), calculate the difference quotient for several values of x within the interval [-π, π]. This will give you an approximation of the derivative f'(x) = cos(x) at each point.
4. Analyze the graph of D(x): The graph of D(x) should closely resemble the graph of cos(x) over the interval [-π, π]. This is because D(x) approximates the derivative of f(x), which is cos(x). The approximation is quite accurate due to the small increment (0.01) used in the difference quotient.
5. Explain the relationship between f and D: The function D(x) is an approximation of the derivative of f(x). As the increment in the difference quotient approaches zero, D(x) becomes a better approximation of f'(x). For f(x) = sin(x), D(x) approximates cos(x), which is the exact derivative of sin(x).

Key Concepts

Differentiability

A function is differentiable at a point if it has a defined derivative at that point, meaning the function's graph has a tangent line at that point. This implies that the function is continuous and does not have any sharp corners or vertical tangents. Differentiability is crucial for understanding how functions change and is foundational for concepts like the difference quotient.

Difference Quotient

The difference quotient is a formula used to approximate the derivative of a function at a point. It is defined as D(x) = (f(x + h) - f(x)) / h, where h is a small increment. In this case, h is set to 0.01, allowing us to analyze the average rate of change of the function f over a small interval, which approaches the instantaneous rate of change as h approaches zero.

Graphing D(x)

Graphing the function D(x) allows us to visualize how the average rate of change of f(x) behaves over the interval. The shape of the graph of D will reflect the slope of the tangent line to f(x) at various points, illustrating how the derivative of f relates to the values of D. This relationship helps in understanding the behavior of f and its rate of change across the specified interval.