62–65. {Use of Tech} Graphing f and f'
b. Compute and graph f'.
f(x) = (x−1) sin^−1 x on [−1,1]

검증된 단계별 안내

First, understand the function f(x) = (x - 1) * sin^(-1)(x). This function is defined on the interval [-1, 1] because the inverse sine function, sin^(-1)(x), is only defined for x in [-1, 1].

To find the derivative f'(x), apply the product rule. The product rule states that if you have a function h(x) = u(x) * v(x), then h'(x) = u'(x) * v(x) + u(x) * v'(x). Here, let u(x) = x - 1 and v(x) = sin^(-1)(x).

Calculate the derivatives: u'(x) = 1 and v'(x) = 1 / sqrt(1 - x^2). The derivative of sin^(-1)(x) is 1 / sqrt(1 - x^2).

Apply the product rule: f'(x) = (x - 1) * (1 / sqrt(1 - x^2)) + 1 * sin^(-1)(x). Simplify this expression to get the final form of f'(x).

Use graphing technology to plot both f(x) and f'(x) on the interval [-1, 1]. Observe the behavior of the function and its derivative, noting any critical points or changes in concavity.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In this context, computing the derivative f' of the function f(x) = (x−1) sin^−1 x will provide insights into the function's behavior, such as its increasing or decreasing nature.

Graphing Functions

Graphing a function involves plotting its output values against its input values on a coordinate plane. For the function f(x) = (x−1) sin^−1 x, this means calculating f(x) for various x values within the interval [-1, 1] and representing these points visually. Understanding how to graph both f and its derivative f' helps in analyzing the function's characteristics, such as local maxima and minima.

Inverse Sine Function

The inverse sine function, denoted as sin^−1 x or arcsin x, is the function that returns the angle whose sine is x. It is defined for x in the range [-1, 1], producing outputs in the range [-π/2, π/2]. In the given function f(x), the presence of sin^−1 x means that the behavior of f will be influenced by the properties of the inverse sine function, particularly its shape and limits within the specified interval.

62–65. {Use of Tech} Graphing f and f'b. Compute and graph f'.f(x) = (x−1) sin^−1 x on [−1,1]