Questions: 17. Determine whether the statement is true or false. If f is periodic and f is differentiable, then f' is periodic. True False
18. Determine whether the statement is true or false. The most general antiderivative of f(x)=3 x^(-2) is F(x)=-3/x+C. True False
Transcript text: 17.
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DETAILS
MY NOTES
SCALC9 3.TF.017.
Determine whether the statement is true or false.
If $f$ is periodic and $f$ is differentiable, then $f^{\prime}$ is periodic.
True
False
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18.
[-/1 Points]
DETAILS
MY NOTES
SCALC9 3.TF.018.
Determine whether the statement is true or false.
The most general antiderivative of $f(x)=3 x^{-2}$ is $F(x)=-\frac{3}{x}+C$.
True
False
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Solution
Solution Steps
Solution Approach
For the first question, consider the properties of periodic functions and their derivatives. A function f is periodic if there exists a period T such that f(x+T)=f(x) for all x. Analyze whether the derivative f′ retains this periodicity.
For the second question, find the antiderivative of the function f(x)=3x−2. The antiderivative is found by integrating the function and adding a constant of integration C. Compare the result with the given expression to determine if it is correct.
Step 1: Analyze the Periodicity of the Derivative
To determine whether the statement "If f is periodic and f is differentiable, then f′ is periodic" is true, we consider the definition of periodic functions. A function f is periodic if there exists a period T such that f(x+T)=f(x) for all x.
For example, if we take f(x)=sin(x), which is periodic with period 2π, its derivative f′(x)=cos(x) is also periodic with the same period. However, if we consider a non-periodic function like f(x)=x, its derivative f′(x)=1 is constant and thus periodic with any period. Therefore, the statement is not universally true.
Step 2: Find the Antiderivative
Next, we analyze the function f(x)=3x−2 to find its most general antiderivative. The integral of f(x) is calculated as follows:
F(x)=∫3x−2dx=−x3+C
where C is the constant of integration. The given expression for the antiderivative is F(x)=−x3+C, which matches our calculated result.
Final Answer
The first statement is False, and the second statement is True.
Thus, the answers are:
For the first statement: \\(\boxed{\text{False}}\\)
For the second statement: \\(\boxed{\text{True}}\\)