AP Calculus AB Derivatives of Inverse Functions Test with Point Values (PDF)

AP Calculus AB math test: Derivatives of Inverse Functions – Maths PDF

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This inverse derivatives test for AP Calculus AB covers derivatives of inverse functions (Common Core AP CALC AB 3.3, AP CALC AB 3.4, AP CALC AB 3.5, AP CALC AB 3.6), with a free PDF to print.

This test checks derivatives of inverse functions, derivatives of inverse trigonometric functions, choosing differentiation methods and higher-order derivatives (AP Calculus AB, 3.3, 3.4, 3.5, 3.6). 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: Derivatives of Inverse Functions
Standards: AP CALC AB 3.3, AP CALC AB 3.4, AP CALC AB 3.5, AP CALC AB 3.6
Time: 45 minutes
Total: 20 points

Inverse derivatives test: questions

Question 1 – Inverse from a table [3 pts]

Values of f and f′.

The table gives values of an invertible function \(f\). Find (a) \(\left(f^{-1}\right)^{\prime}(9)\) and (b) \(\left(f^{-1}\right)^{\prime}(15)\), and (c) the value of \(f^{-1}(4)\). (1 point each)

Question 2 – Inverse of a cubic [2 pts]

Graph of f(x) = x³ + 2x − 3.

Let \(f(x)=x^{3}+2x-3\). Find \(\left(f^{-1}\right)^{\prime}(0)\).

Question 3 – Inverse trig derivatives [2 pts]

Differentiate \(g(x)=\sin^{-1}x+\tan^{-1}(2x)\).

Question 4 – Inverse trig and a root [2 pts]

Differentiate \(h(x)=\tan^{-1}(\sqrt{x})\).

Question 5 – Second derivative [3 pts]

Find \(f^{\prime\prime}(x)\) for \(f(x)=x^{5}-2x^{3}+\sin x\), and then evaluate \(f^{\prime\prime}(0)\).

Question 6 – Acceleration [2 pts]

A runner’s position in meters is \(s(t)=2t^{3}-15t^{2}+36t\) after \(t\) seconds. Find the acceleration at \(t=3\) and the times when the velocity is zero.

Question 7 – Choosing rules [3 pts]

Name the best rule and differentiate: (a) \(e^{x}\cos x\) (b) \(\frac{\ln x}{x^{2}}\) (c) \((5x-1)^{4}\). (1 point each)

Question 8 – Inverse in context [3 pts]

Graphs of f and g = f⁻¹.

The volume of a cube-shaped planter is related to its side by \(f(x)=\frac{x^{3}}{8}\). Let \(g=f^{-1}\). (a) Find \(f(2)\) and \(f^{\prime}(2)\). (1 point) (b) Find \(g^{\prime}(1)\). (1 point) (c) Write the tangent line to \(g\) at \(x=1\). (1 point)

Key facts behind this inverse derivatives test

Key facts

(f⁻¹)′(b) = 1/f′(a) where f(a) = b; evaluate f′ at a, not at b.
d/dx sin⁻¹x = 1/√(1-x²), d/dx cos⁻¹x = -1/√(1-x²), d/dx tan⁻¹x = 1/(1+x²).
The second derivative is the derivative of the first derivative; acceleration is the second derivative of position.
Choose the rule by the outermost operation: sum, product, quotient or composition.

How to use this inverse derivatives 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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