Math Problem Statement

An equation for a tangent to the graph of y = arcsin(x/2) at the origin is

Solution

We are tasked with finding the equation of the tangent to the graph of the function y=arcsin(x2)y = \arcsin\left(\frac{x}{2}\right) at the origin, i.e., the point (0,0)(0, 0).

Steps to find the tangent equation:

  1. Differentiate the function: The equation of the function is y=arcsin(x2)y = \arcsin\left(\frac{x}{2}\right). We need to differentiate it with respect to xx to find the slope of the tangent line.

    Using the chain rule: dydx=11(x2)212\frac{dy}{dx} = \frac{1}{\sqrt{1 - \left(\frac{x}{2}\right)^2}} \cdot \frac{1}{2} Simplifying: dydx=121x24\frac{dy}{dx} = \frac{1}{2\sqrt{1 - \frac{x^2}{4}}}

  2. Evaluate the derivative at the origin: To find the slope of the tangent line at the origin x=0x = 0, substitute x=0x = 0 into the derivative: dydxx=0=121024=12\frac{dy}{dx} \bigg|_{x = 0} = \frac{1}{2\sqrt{1 - \frac{0^2}{4}}} = \frac{1}{2} So, the slope of the tangent line at the origin is 12\frac{1}{2}.

  3. Equation of the tangent line: The general equation of a tangent line at a point (x0,y0)(x_0, y_0) is given by: yy0=m(xx0)y - y_0 = m(x - x_0) Here, m=12m = \frac{1}{2}, x0=0x_0 = 0, and y0=0y_0 = 0. Therefore, the equation of the tangent line at the origin is: y=12xy = \frac{1}{2}x

Final Answer:

The equation of the tangent line to the graph of y=arcsin(x2)y = \arcsin\left(\frac{x}{2}\right) at the origin is: y=12xy = \frac{1}{2}x

Would you like further clarification or details on any part of the solution?

Here are five related questions:

  1. How is the derivative of the inverse sine function derived?
  2. What is the significance of the chain rule in this problem?
  3. How can you determine if a tangent line is horizontal at a point?
  4. Can we find the tangent at any other point on this function?
  5. How does the arc sine function behave for values of xx close to 22?

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