AP Calculus AB Implicit Differentiation Test with Point Values (PDF)

AP Calculus AB math test: Implicit Differentiation – Maths PDF

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

This test checks implicit differentiation, slopes of curves, tangent lines, horizontal and vertical tangents and second derivatives (AP Calculus AB, 3.2). It has 8 questions worth 20 points in total. Allow 45 minutes, work without notes, and show all work, since justification earns points on the AP exam.

Time and points

About this test

Grade: AP Calculus AB
Chapter: Implicit Differentiation
Standards: AP CALC AB 3.2
Time: 45 minutes
Total: 20 points

Implicit test: questions

Question 1 – Implicit derivative [3 pts]

Find \(\frac{dy}{dx}\) for \(x^{3}+3xy+y^{2}=11\).

Question 2 – Slope on an ellipse [2 pts]

The ellipse x² + 4y² = 16.

The point \((2,\sqrt{3})\) lies on \(x^{2}+4y^{2}=16\). Find the slope of the tangent there and write its equation.

Question 3 – Chain rule in y [2 pts]

Find \(\frac{dy}{dx}\) for \(\cos y=x^{2}y\).

Question 4 – Horizontal and vertical tangents [3 pts]

The circle x² + y² = 9.

For the circle \(x^{2}+y^{2}=9\), find \(\frac{dy}{dx}\) and the points where the tangent is horizontal or vertical.

Question 5 – Tangent line [3 pts]

Find the tangent line to \(x^{2}+xy=2\) at \((1,1)\).

Question 6 – Second derivative [2 pts]

For \(x^{2}+y^{2}=25\), show that \(\frac{d^{2}y}{dx^{2}}=-\frac{25}{y^{3}}\).

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

Part of the curve x² + xy = 2.

The curve \(x^{2}+xy=2\) passes through \((1,1)\). Find \(\frac{dy}{dx}\) at that point.

Question 8 – Ferris wheel [3 pts]

A seat on a Ferris wheel moves on the circle \(x^{2}+y^{2}=64\) (feet, origin at the center). (a) Find \(\frac{dy}{dx}\). (1 point) (b) Find the slope of the path at the point \((4\sqrt{3},4)\). (1 point) (c) What is the slope at the top, \((0,8)\)? (1 point)

Key facts behind this implicit test

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

Time yourself: the test is designed for an average student in 45 minutes. Answer the questions you are sure of first, then come back to the longer ones and check your units.

All AP Calculus AB chapters: AP Calculus AB math.

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