MATHEMATICS · Single correct · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
A student states, “Because g(0)=0 for g(x)=2x3+7x, the slope of the tangent line to the graph of g at x=0 must also be 0.” Which of the following best evaluates the student’s claim?
(a) The claim is correct since g(0)=0 means the graph crosses the x-axis, and graphs have a horizontal tangent whenever they cross the x-axis.
(b) The claim is incorrect because the slope depends on g′(0)=7, which is found by differentiating g, not by evaluating g itself.
(c) The claim is correct because g′(x)=6x2, so g′(0)=0.
(d) The claim cannot yet be evaluated; determining the slope at x=0 would require the coordinates of a second nearby point.