Math Problem Statement
Solution
We are tasked with finding the equation of the tangent to the graph of the function at the origin, i.e., the point .
Steps to find the tangent equation:
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Differentiate the function: The equation of the function is . We need to differentiate it with respect to to find the slope of the tangent line.
Using the chain rule: Simplifying:
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Evaluate the derivative at the origin: To find the slope of the tangent line at the origin , substitute into the derivative: So, the slope of the tangent line at the origin is .
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Equation of the tangent line: The general equation of a tangent line at a point is given by: Here, , , and . Therefore, the equation of the tangent line at the origin is:
Final Answer:
The equation of the tangent line to the graph of at the origin is:
Would you like further clarification or details on any part of the solution?
Here are five related questions:
- How is the derivative of the inverse sine function derived?
- What is the significance of the chain rule in this problem?
- How can you determine if a tangent line is horizontal at a point?
- Can we find the tangent at any other point on this function?
- How does the arc sine function behave for values of close to ?
Tip: Always verify the domain of inverse trigonometric functions to avoid evaluating them at points where they are undefined.
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Math Problem Analysis
Mathematical Concepts
Calculus
Differentiation
Inverse Trigonometric Functions
Formulas
dy/dx = 1 / 2√(1 - (x^2/4))
Theorems
Chain Rule
Suitable Grade Level
Grades 11-12