Question

{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]

Accepted Answer

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]. 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). 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. 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. 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).

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Differentiability

Difference Quotient

Graphing D(x)