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AP Calc

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Unit 2

2.12 MC Answers and Review

4 min read•december 29, 2021

AP Calculus AB/BC ♾️

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Answers and Review for Multiple Choice Practice on Differentiation: Definition & Fundamental Properties

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⛔STOP!⛔ Before you look at the answers make sure you gave this practice quiz a try so you can assess your understanding of the concepts covered in unit 2. Click here for the practice questions: AP Calculus Unit 2 Multiple Choice Questions. Facts about the test: The AP Calculus exam has two sections of multiple choice: 30 questions in 60 minutes with no calculator, and 15 questions in 45 minutes with a calculator. *The following questions were not written by CollegeBoard and although they cover information outlined in the AP Calculus Course and Exam Description the formatting on the exam may be different.

Explanation: D is correct because to find the average rate of change, you will take the interval and plug into the function. This will allow you to determine the y-coordinate, then you can substitute the information into the average rate of change formula.

📄 Study Unit 2.1: Defining Average and Instantaneous Rates of Change at a Point

Explanation: C is correct because the instantaneous rate of change represents the slope of the tangent or the derivative. It can also be represented by the limit of the average rate of change.

📄 Study Unit 2.1: Defining Average and Instantaneous Rates of Change at a Point

Explanation: C is correct because the derivative at a point can be found by finding the slope at that point. Take the derivative of f(x) using the power rule and substitute into the derivative for the given value.

📄 Study Unit 2.5: Applying the Power Rule

Explanation: B is correct because the derivative represents the slope of the tangent line and the point is given as (5, -2). Take this information and plug into point-slope form.

📄 Study Unit 2.3: Estimating Derivatives of a Function at a Point

Explanation: A is correct because the slope of the tangents can be obtained by plugging the two given points into the slope formula. This is also the formula of a derivative.

📄 Study Unit 2.1: Defining Average and Instantaneous Rates of Change at a Point

Explanation: C is correct because the derivative at maximums are horizontal tangent lines. Horizontal lines have a slope of 0.

🎥 Watch: Introduction to Finding Derivatives

Explanation: B is correct because first you will need to rewrite the fraction with a negative exponent. Using the power rule for derivatives, multiply the constant by the exponent and subtract 1 from the exponent. Do not forget to simplify. 🎥 Watch: Introduction to Finding Derivatives

Explanation: A is correct because first you will need to convert the radical to an exponent. Using the power rule for derivatives, multiply the constant by the exponent and subtract 1 from the exponent. Do not forget to simplify. 📄 Study Unit 2.5: Applying the Power Rule

10.

Explanation: D is correct because to determine h'(x), you must utilize the product rule. The product rule consist of f'(x) g(x) + f(x) g'(x).

📄 Study Unit 2.8: The Product Rule

11.

Explanation: C is correct because to determine h'(x), you must utilize the quotient rule. The quotient is f(x) g'(x) - g(x) f'(x) all divided by f(x) squared.

📄 Study Unit 2.9: The Quotient Rule

12.

Explanation: B is correct because the derivative utilizes the power rule and the rules for exponential and trigonometric functions. Be sure to keep e intact and change the sign with the derivative of cos x.

📄 Study Unit 2.7: Derivatives of cos x, sinx, e^x, and ln x

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Explanation: A is correct because taking the derivatives of trigonometric functions requires you to understand/memorize the derivatives of trigonometric functions. In some cases (like this one), you may be able to simplify before taking the derivative to make it a less complicated derivative.

📄 Study Unit 2.10: Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions

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Explanation: D is correct because when a function has a sharp corner or turn, is not continuous, and/or has a vertical tangent line it is not differentiable. These are the special case for nondifferentiability.

📄 Study Unit 2.4: Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist

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Explanation: B is correct because the function f(x) passes the tests for continuity. Also, the function does not create a sharp turn nor a vertical asymptote at a. Therefore, the function is continuous and differentiable.

📄 Study Unit 2.4: Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist