Math Problem Statement
Выберите все верные утверждения.
Solution
Let's go through each statement to determine which ones are correct. I will analyze each of them in detail.
-
. Then .
- Since and have periodic derivatives, the derivatives of repeat every four terms as follows:
- Therefore, , which does not equal . This statement is incorrect.
- Since and have periodic derivatives, the derivatives of repeat every four terms as follows:
-
. Then .
- implies that , so any higher derivatives, including , are also zero. This statement is correct.
-
. Then .
- , , and .
- The third derivative is , not . This statement is incorrect.
-
. Then .
- .
- This statement is correct.
-
. Then .
- For , each derivative introduces a factor of .
- , so . This statement is correct.
-
. Then .
- and for (4x
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Math Problem Analysis
Mathematical Concepts
Differentiation
Higher Order Derivatives
Trigonometric Functions
Polynomial Functions
Exponential Functions
Formulas
Derivative of trigonometric functions: sin(x), cos(x)
Derivative of power functions: d/dx (x^n) = n*x^(n-1)
Exponential function differentiation: d/dx (a^x) = a^x ln(a)
Theorems
Rules for Differentiation
Periodicity of Trigonometric Functions
Suitable Grade Level
University level (Calculus)