A student sketches and asserts that the tangent line at the origin is horizontal…

MATHEMATICS · Single correct · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

A student sketches y=arcsin⁡(2x)y=\arcsin(2x) and asserts that the tangent line at the origin is horizontal because the graph crosses the xx-axis there. Which of the following best evaluates the student’s claim? (a) The claim is correct; since the graph crosses the xx-axis at (0,0)(0,0), we have y′(0)=0y'(0)=0, so the tangent is horizontal. (b) The claim is incorrect; although the curve passes through (0,0)(0,0), the derivative y′=21−4x2y'=\dfrac{2}{\sqrt{1-4x^{2}}} equals 22 at x=0x=0, so the tangent is not horizontal. (c) The claim is incorrect; the derivative y′=21−4x2y'=\dfrac{2}{\sqrt{1-4x^{2}}} is undefined at x=0x=0, so the tangent is vertical rather than horizontal. (d) The claim is incorrect; using y′=11+(2x)2⋅2y'=\dfrac{1}{\sqrt{1+(2x)^{2}}}\cdot 2, we find y′(0)=1y'(0)=1, so the tangent has slope 11, not 00."

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