Apentix/AP Calculus AB/Unit 5: Analytical Applications of Differentiation/Question

AP Calculus ABhardmcq1 pt

At which points does the curve $x^2+xy+y^2=7$ have a vertical tangent line?

A.

\left(-2\sqrt{\dfrac{7}{3}},\sqrt{\dfrac{7}{3}}\right)

and

\left(2\sqrt{\dfrac{7}{3}},-\sqrt{\dfrac{7}{3}}\right)

B.

(\sqrt{7},0)

and

(-\sqrt{7},0)

C.

(0,\sqrt{7})

and

(0,-\sqrt{7})

D.The curve has no vertical tangent lines.

Explanation

Core Concept

Implicit differentiation gives $y'=-(2x+y)/(x+2y)$ . Vertical tangents occur where $x+2y=0$ , so $x=-2y$ . Substituting gives $3y^2=7$ , so $y=\pm\sqrt{7/3}$ and $x=\mp 2\sqrt{7/3}$ .

Correct Answer

A $\left(-2\sqrt{\dfrac{7}{3}},\sqrt{\dfrac{7}{3}}\right)$ and $\left(2\sqrt{\dfrac{7}{3}},-\sqrt{\dfrac{7}{3}}\right)$

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