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AP®MathsCollege BoardCalculus ABExam QuestionsUnit 5: Analytical Applications of DifferentiationBehaviors of Implicit Relations

Behaviors of Implicit Relations (College Board AP® Calculus AB)

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Behaviors of Implicit Relations

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Behaviors of Implicit Relations

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1
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If y squared equals 4 x, what is fraction numerator d squared y over denominator d x squared end fraction ?

  • table row blank blank cell negative 4 over y cubed end cell end table

  • table row blank blank cell negative 2 over y end cell end table

  • table row blank blank cell 2 over y end cell end table

  • table row blank blank cell 4 over y cubed end cell end table

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2
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If ln open parentheses x close parentheses plus ln open parentheses y close parentheses equals 1, fraction numerator d squared y over denominator d x squared end fraction equals

  • negative x over y

  • negative y over x

  • fraction numerator x squared over denominator 2 y end fraction

  • fraction numerator 2 y over denominator x squared end fraction

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3
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Which of the following are the coordinates of the points on the curve x squared plus y cubed equals 9 where the tangent to the curve is vertical?

  • open parentheses 0 comma space cube root of 9 close parentheses only

  • open parentheses 0 comma space cube root of 9 close parentheses and open parentheses 0 comma space minus cube root of 9 close parentheses

  • open parentheses 3 comma space 0 close parentheses and open parentheses negative 3 comma space 0 close parentheses

  • open parentheses 3 comma space 0 close parentheses only

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4
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A curve has the equation x squared minus y squared equals 1.

Find the number of critical points on this curve.

  • 0

  • 1

  • 2

  • 4

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5
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If open parentheses x plus y close parentheses fraction numerator d y over denominator d x end fraction equals 3 x minus 2 y and x plus y not equal to 0, what is the value of fraction numerator d squared y over denominator d x squared end fraction at the point open parentheses 2 comma space 0 close parentheses ?

  • 9 over 2

  • negative 9 over 2

  • 3

  • negative 15 over 2

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1
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A curve goes through the point (4, 0). If the equation of the curve satisfies open parentheses x plus 3 y close parentheses fraction numerator d y over denominator d x end fraction equals 5 x minus 2 y, what is the value of fraction numerator d squared y over denominator d x squared end fraction at the point (4, 0)?

  • negative 85 over 4

  • 0

  • 5

  • 85 over 4

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2
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A curve has the equation 2 x squared minus 2 y squared minus 3 y equals 0. Find the number of critical points on this curve.

  • 0

  • 1

  • 2

  • 3

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3
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If x plus y squared equals 2 x squared, what is fraction numerator d squared y over denominator d x squared end fraction ?

  • table row blank blank cell fraction numerator 4 x plus 1 over denominator 2 y end fraction end cell end table

  • table row blank blank cell fraction numerator 4 x minus 1 over denominator 2 y end fraction end cell end table

  • table row blank blank cell fraction numerator 8 y squared plus open parentheses 4 x minus 1 close parentheses squared over denominator 4 y cubed end fraction end cell end table

  • table row blank blank cell fraction numerator 8 y squared minus open parentheses 4 x minus 1 close parentheses squared over denominator 4 y cubed end fraction end cell end table

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4
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Which of the following describes the behaviour of the normal line to the curve y equals sin open parentheses x y close parentheses at the point open parentheses 0 comma space 0 close parentheses ?

  • Horizontal

  • Vertical

  • Positive slope

  • Negative slope

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5
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What is an equation of the horizontal tangent line to the curve 2 x squared plus 3 y cubed equals 12 x y ?

  • x equals 0

  • y equals 0

  • x equals 6

  • y equals 6

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1
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A curve has the equation x y plus x cubed plus y equals 0, where x not equal to negative 1.

Find the number of critical points on the curve.

  • 0

  • 1

  • 2

  • 3

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2
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A curve has the equation y equals x e to the power of y. What is the value of the second derivative of the curve at the origin?

  • 0

  • 1

  • 2

  • -2

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3
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Given that a curve has the equation sin space x space cos space y space equals 1 half, what is an expression for fraction numerator d squared y over denominator d x squared end fraction?

  • cot space x space cot space y

  • negative csc squared x space cot space y space minus space csc squared y space cot space x

  • cot space y space open parentheses csc squared x space plus thin space cot squared x space csc squared y close parentheses

  • negative cot space y space open parentheses csc squared x space plus thin space cot squared x space csc squared y close parentheses

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4
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Which of the following is a critical point on the curve x cubed plus y cubed equals 3 x y ?

  • open parentheses 0 comma space 0 close parentheses

  • open parentheses cube root of 2 comma space cube root of 4 close parentheses

  • open parentheses cube root of 4 comma space cube root of 2 close parentheses

  • All of the above

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5
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What can be said is true about the point open parentheses negative 1 comma space minus 1 close parentheses on the curve x squared plus y squared minus 2 x y minus 4 y equals 4 ?

  • Local minimum

  • Local maximum

  • Critical point of inflection

  • Non-critical point of inflection

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