Problem 78
Question
Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. \(f(x)=\cos ^{-1} x\) is a decreasing function.
Step-by-Step Solution
Verified Answer
The statement is true. The function \(f(x) = \cos^{-1} x\) is a decreasing function because its derivative, \(f'(x) = -\frac{1}{\sqrt{1-x^2}}\), is always negative for \(-1 < x < 1\), which is the domain of the function. Since the derivative is negative, the function is decreasing over its entire domain.
1Step 1: Find the derivative of the function
In order to determine if a function is decreasing, we must first find its derivative. The function we are given is \(f(x) = \cos^{-1} x\). Differentiating this function with respect to x, we get:
\[f'(x) = -\frac{1}{\sqrt{1-x^2}}\]
2Step 2: Analyze the sign of the derivative
Now that we have the derivative, we can analyze its sign to determine if the function is decreasing. The derivative is \(f'(x) = -\frac{1}{\sqrt{1-x^2}}\).
Since the function under the square root, \(1-x^2\), is always positive or equal to 0 for \(-1 \leq x \leq 1\), the square root is also positive or zero. Moreover, since we are multiplying this positive quantity by -1, the derivative will always be negative for \(-1 < x < 1\). The function \(\cos^{-1} x\) is defined only for \(-1 \leq x \leq 1\).
3Step 3: Conclude if the function is increasing or decreasing
We have found that the derivative, \(f'(x) = -\frac{1}{\sqrt{1-x^2}}\), is always negative for \(-1 < x < 1\). Since the function \(\cos^{-1} x\) is defined only for \(-1 \leq x \leq 1\), and the derivative is negative within this domain, we can conclude the statement is true, and the function \(f(x) = \cos^{-1} x\) is a decreasing function.
Key Concepts
Derivative of Inverse Trigonometric FunctionsFunction AnalysisMonotonicity in Calculus
Derivative of Inverse Trigonometric Functions
Understanding the derivative of inverse trigonometric functions is pivotal in function analysis, particularly when we need to determine the increasing or decreasing trend of a function. Inverse trigonometric functions, including \( \cos^{-1} x \), have specific derivatives that can be applied when analyzing their behavior.
For instance, the derivative of \( \cos^{-1} x \) is \( f'(x) = -\frac{1}{\sqrt{1-x^2}} \). This derivative is crucial in determining the function's rate of change. If we unpack this, for every tiny increment of \( x \), the function \( \cos^{-1} x \) decreases at a rate given by its derivative. Since \( 1-x^2 \) is always positive within the domain of the function, \( f'(x) \) will be negative, implying a decreasing trend.
To compute the derivatives of inverse trigonometric functions correctly, it's important to remember their standard forms. For effective learning, practicing with diverse examples ensures thorough understanding and ability to apply these derivatives in various contexts.
For instance, the derivative of \( \cos^{-1} x \) is \( f'(x) = -\frac{1}{\sqrt{1-x^2}} \). This derivative is crucial in determining the function's rate of change. If we unpack this, for every tiny increment of \( x \), the function \( \cos^{-1} x \) decreases at a rate given by its derivative. Since \( 1-x^2 \) is always positive within the domain of the function, \( f'(x) \) will be negative, implying a decreasing trend.
To compute the derivatives of inverse trigonometric functions correctly, it's important to remember their standard forms. For effective learning, practicing with diverse examples ensures thorough understanding and ability to apply these derivatives in various contexts.
Function Analysis
Function analysis combines multiple calculus concepts to understand a function’s behavior. By analyzing the function \( f(x) = \cos^{-1} x \) through its derivative, one can gain insight into its monotonicity—whether the function is increasing or decreasing over its domain.
The analysis process involves two key steps. First, find the derivative of the function, which gives us the rate of change regarding \( x \). Second, assess the sign of the derivative to infer the function’s behavior. A negative derivative, as in our \( \cos^{-1} x \) example, suggests a consistent decrease within its domain. Properly interpreting the derivative’s sign across the function’s domain provides a clear picture of the function's nature.
For students tackling function analysis, it’s advisable to sketch the function, noting its domain and range. This visual approach, combined with derivative sign analysis, can greatly enhance understanding of the function’s overall behavior.
The analysis process involves two key steps. First, find the derivative of the function, which gives us the rate of change regarding \( x \). Second, assess the sign of the derivative to infer the function’s behavior. A negative derivative, as in our \( \cos^{-1} x \) example, suggests a consistent decrease within its domain. Properly interpreting the derivative’s sign across the function’s domain provides a clear picture of the function's nature.
For students tackling function analysis, it’s advisable to sketch the function, noting its domain and range. This visual approach, combined with derivative sign analysis, can greatly enhance understanding of the function’s overall behavior.
Monotonicity in Calculus
Monotonicity in calculus refers to the unchanging direction of a function’s trend, whether it is solely increasing or decreasing. The concept of monotonicity helps us describe how functions behave over specific intervals and is a fundamental part of calculus.
When examining a function like \( f(x) = \cos^{-1} x \), determining monotonicity involves looking at the sign of the derivative throughout the function’s domain. If the derivative is consistently negative or positive over an interval, the function is considered to be strictly decreasing or increasing, respectively. In the case of \( f(x) = \cos^{-1} x \), its derivative \( f'(x) = -\frac{1}{\sqrt{1-x^2}} \) reveals that it is strictly decreasing over \( -1 < x < 1 \), since the derivative never changes its negative sign.
To strengthen comprehension in this area, practice with varying functions and intervals is beneficial, as it hones the skill of identifying patterns of change. Furthermore, understanding the implications of monotonicity on function graphs aids visual learners in recognizing these trends intuitively.
When examining a function like \( f(x) = \cos^{-1} x \), determining monotonicity involves looking at the sign of the derivative throughout the function’s domain. If the derivative is consistently negative or positive over an interval, the function is considered to be strictly decreasing or increasing, respectively. In the case of \( f(x) = \cos^{-1} x \), its derivative \( f'(x) = -\frac{1}{\sqrt{1-x^2}} \) reveals that it is strictly decreasing over \( -1 < x < 1 \), since the derivative never changes its negative sign.
To strengthen comprehension in this area, practice with varying functions and intervals is beneficial, as it hones the skill of identifying patterns of change. Furthermore, understanding the implications of monotonicity on function graphs aids visual learners in recognizing these trends intuitively.
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Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false.
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