AP Calculus AB Implicit Differentiation Test Solutions and Scoring (PDF)

AP Calculus AB math test solutions: Implicit Differentiation – Maths PDF

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This implicit test answers for AP Calculus AB covers implicit differentiation (Common Core AP CALC AB 3.2), with a free PDF to print.

Use this answer key to score the implicit differentiation test out of 20 points. Each solution shows the work and a scoring guide that says how the points are awarded. Give partial credit for a correct setup, a correct derivative or integral, and a correct final statement with units where they apply.

AP Calculus AB · Common Core State Standards: AP CALC AB 3.2

Implicit test answers: worked solutions

Question 1 – Implicit derivative [3 pts]

Differentiate: \(3x^{2}+3y+3xy^{\prime}+2yy^{\prime}=0\).

Collect: \(y^{\prime}(3x+2y)=-(3x^{2}+3y)\).

Final answer: \(\frac{dy}{dx}=-\frac{3x^{2}+3y}{3x+2y}\).

Scoring: 1 point for each term differentiated correctly (product and chain); 1 point for solving.

Question 2 – Slope on an ellipse [2 pts]

\(2x+8yy^{\prime}=0\), so \(y^{\prime}=-\frac{x}{4y}\). At the point: \(-\frac{2}{4\sqrt{3}}=-\frac{1}{2\sqrt{3}}\).

Tangent: \(y-\sqrt{3}=-\frac{1}{2\sqrt{3}}(x-2)\).

The ellipse x² + 4y² = 16.

Scoring: 1 point for the derivative and the slope; 1 point for the equation.

Question 3 – Chain rule in y [2 pts]

\(-\sin y\cdot y^{\prime}=2xy+x^{2}y^{\prime}\). Collect: \(y^{\prime}(-\sin y-x^{2})=2xy\).

Final answer: \(\frac{dy}{dx}=-\frac{2xy}{x^{2}+\sin y}\).

Scoring: 1 point for the differentiation; 1 point for solving.

Question 4 – Horizontal and vertical tangents [3 pts]

\(y^{\prime}=-\frac{x}{y}\). Horizontal: \(x=0\), so \((0,3)\) and \((0,-3)\). Vertical: \(y=0\), so \((3,0)\) and \((-3,0)\).

The circle x² + y² = 9.

Scoring: 1 point for the derivative; 1 point for the horizontal points; 1 point for the vertical points.

Question 5 – Tangent line [3 pts]

Differentiate: \(2x+y+xy^{\prime}=0\), so \(y^{\prime}=-\frac{2x+y}{x}\). At \((1,1)\): \(-3\).

Tangent: \(y-1=-3(x-1)\). Final answer: \(y=-3x+4\).

Scoring: 1 point for the derivative; 1 point for the slope; 1 point for the equation.

Question 6 – Second derivative [2 pts]

\(y^{\prime}=-\frac{x}{y}\). Then \(y^{\prime\prime}=-\frac{y-xy^{\prime}}{y^{2}}=-\frac{y+\frac{x^{2}}{y}}{y^{2}}=-\frac{y^{2}+x^{2}}{y^{3}}=-\frac{25}{y^{3}}\).

Scoring: 1 point for the quotient rule and substituting y′; 1 point for using the original equation.

Question 7 – Implicit slope with a product [2 pts]

\(2x+y+xy^{\prime}=0\) gives \(y^{\prime}=-\frac{2x+y}{x}\). At \((1,1)\): \(-3\).

Part of the curve x² + xy = 2.

Scoring: 1 point for the product rule term; 1 point for the slope.

Question 8 – Ferris wheel [3 pts]

\(\frac{dy}{dx}=-\frac{x}{y}\).
\(-\frac{4\sqrt{3}}{4}=-\sqrt{3}\).
\(-\frac{0}{8}=0\): the path is horizontal at the top.

Scoring: 1 point per part.

Total score

Total

20 points

Key facts behind this implicit test answers

Key facts

Differentiate every term with respect to x; a y term gets a factor dy/dx.
Use the product rule for terms such as xy, and the chain rule for sin y or e^y.
The derivative may contain both x and y; substitute a point on the curve to get a slope.
Horizontal tangents: numerator zero. Vertical tangents: denominator zero.

How to use this implicit test answers

Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.

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