What AP Calculus Students Should Know for their Unit 2 Review

Need extra practice for your AP Calculus Unit 2 review? Outlined are the topics and derivative practice problems aligned with College Board’s curriculum to study for an AP Calculus AB or AP Calculus BC Unit 2 test.

As College Board outlines, the topics to review for an AP Calculus Unit 2 test are:

· Defining Average and Instantaneous Rates of Change at a Point

This may have been introduced in Unit 1. Recall that average rate of change is the slope of the secant line (how we find slope in Algebra 1) and instantaneous rate of change is the slope of the tangent line. Formulas for both are below. Also, "instantaneous rate of change," "slope," "slope of the tangent line," and "derivative" all mean the same thing!

Worked out solutions to these problems will be at the end of the post. The second problem is from the 2022 AP Calculus AB/BC exam. Every year on the AP Calculus exam, there is a free response question with a table, and one part of the question always asks about estimating a derivative using average rate of change. Carefully check the scoring guidelines at the end. You want to make sure to have the difference quotient in order to earn full credit.

· Defining the Derivative of a Function and Using Derivative Notation

Finding the derivative is the same as finding the instantaneous rate of change, so the example will be similar to #1, but extra practice is helpful! Notice this problem says to use the definition of derivative, not the power rule. But you can use the power rule to check your answer!

· Estimating Derivatives of a Function at a Point

This section of the AP Calculus Unit 2 review introduces taking a derivative on a calculator, writing equations of tangent lines, and estimating the value of a derivative given a table. I’ll assume your teacher covered how to evaluate a derivative on a calculator if a calculator is allowed on this test. Then for estimating the value of a derivative given a table, that’s the same as average rate of change like problem #2 above. So we’ll do an equation of a tangent line question for this section:

If you haven't covered Chain Rule yet, use your calculator to find the derivative/slope of the tangent line.

· Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist

Differentiability means the derivative exists. The derivative will exist as long as the function is continuous and “smooth.” By smooth, I mean the derivative (slope) from the left and derivative from the right match: there isn’t a corner (where one-sided derivatives differ), a cusp (where the slopes of the secant lines approach +/- infinity), or a vertical tangent. In simpler terms, look at a graph of the function and pretend it’s an outline of a roller coaster ride. Does it look like a smooth ride? If so, the function is differentiable. The derivative exists. If there are sharp corners or a jump in the graph, for example, that wouldn’t be a smooth ride and is not differentiable at those places.

· Applying the Power Rule

When I taught AP Calculus, I taught the power rule the same day as the constant rule, sum and difference rules, and constant multiple rule so see examples in the next section. Be aware that you may have to rewrite some terms (hint: radicals and fractions) in order to apply the power rule!

· Derivative Rules: Constant, Sum, Difference, and Constant Multiple

· Derivatives of cosx, sinx, e^x, and lnx

These derivative rules will be in future sections. For extra Unit 2 derivative practice, check out Flipped Math.

· The Product Rule

9. Write the equation of the tangent line for f(x) = xsinx at x=π

Some textbooks use the product rule as the derivative of f*g is fg’ + f’g. I flip those terms to f’g+fg’ to make it similar to the numerator of the quotient rule (less to memorize!)

· The Quotient Rule

· Finding the Derivatives of Tangent, Cotangent, Secant, and Cosecant

12. Find the derivative of y=secxtanx

AP Calculus Unit 2 Review

For your AP Calculus Unit 2 review, make sure you are practicing derivative problems from multiple perspectives: given functions, tables, graphs, and word problems. Most of these problems (and probably your homework problems) are given functions. The main goal for this unit is to memorize these derivative rules, so most questions will likely be with functions. If you need extra practice, check out the 7 best resources to study for AP Calculus tests. These resources have questions that match the rigor of your tests and have a variety of graph and table questions.

If you need further explanation on how to approach some of these difficult Unit 2 review questions, especially when all of the topics are mixed together, consider individual Calculus tutoring with me. I answer any questions students have, then provide practice solving my past test questions and previous AP exam questions (like #2). Getting more practice with problems given graphs, tables, and word problems will help you be prepared for in class tests and the AP exam.

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Solutions to the above problems:

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