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The second component of the AP Exam includes 6 Free Response Questions (FRQs).

You're given 30 minutes to complete the first two-Part A. These both require a calculator. Then you'll have 1 hour for the remaining four-Part B. These, however, are tech-free. Not to worry, though, in the sections ahead we'll detail what you can expect in each part and how to maximize your score.

Speaking of scoring, the AP weights each problem equally, 9 points apiece (more later). And, taken together, this section accounts for 50% of your total exam score.

The AP course emphasizes 3 big ideas: change, limits, and analysis of functions. While the specific content varies from year-to-year, in aggregate the exam will test your ability to understand and interpret functions represented in a variety of forms: graphically, from a table, in written form, and as an equation.

Each FRQ contains 3 or 4 related sub-tasks, and you can expect a fairly even split between process-oriented and conceptual questions.

Process-Oriented: find the derivative; estimate the value of a definite integral; or identify the location or value of a relative maximum

Conceptual: interpret the graph of the first derivative, f'; verify the conditions of the Mean Value Theorem (MVT); explain why the rate of change of a system is positive or negative with justification

Note: these lists are not exhaustive, but the FAQs section below discusses other examples

In addition, two questions (at minimum) will feature an authentic scenario. This means they'll give you more context than an abstract equation or graph.

So what do you need to do differently in the FRQ section? In the words of every math teacher you've ever had: show your work.

This emphasis on justification is more than mere suggestion. Approximately half of the points possible depend on your ability to support your answer. Of course, what constitutes good justification depends on the question. See the comparisons below.

While this is accurate, it isn't clear that this is the derivative nor how we arrived here. A better answer would include evidence of our process.

Here, we clearly show we're taking the derivative-using the power rule and the chain rule. If we make an arithmetic error, the reader knows what we were attempting. In these situations you may earn yourself some of the available points. However, if you make a calculation error in the first solution, you'll earn no points at all.

Again, technically the result is accurate. Still, we could do much better to communicate our line of thinking.

Step one indicates that we've taken the derivative at x=3. Next we show the location of the point of intersection (3, p(3)). Finally, we label our result.

Before we move on, just a few more quick examples.

*NOTE: simply providing a +/- chart (a common practice for these types of problems) does not supply sufficient justification. A simple sentence indicating a change, however, will suffice.

As mentioned earlier, each FRQ counts equally. Within each sub-question, however the weightings can vary slightly. That said, normally each part counts 2 or 3 points toward the 9 point total. Thus, the maximum possible tally for the entire Free Response Section is 54 points. For reference, the MC section contributes 54 points, as well. Your raw score will sum these two.

If you're aiming for a 5? The cutoff changes every year, but a general guide is that raw scores above 70 should qualify.

And what about a 4? Here, again, it varies year-to-year. Usually, raw scores between 60-69 will translate to a 4.

AP readers-those people you're trying to please with your answers-award points based on accuracy and on completeness. To earn maximum marks you'll need both.

Above, we discussed some particulars of how to justify your answers. It's important because they contribute significantly to your total score.

That said, you will earn points for other things. You'll earn points for setting up derivatives and integrals properly; using complete and proper notation; logically moving step-by-step through your mathematical process; and providing the correct answer with the corresponding units, where appropriate. So, what does this look like?

Rates: At t = 3, the rate at which people are leaving is 142 every hour.

Average: On [3, 7] the average rate of change of v is 6 meters/sec.

Accumulation: The liquid has cooled 8 degrees Fahrenheit over the interval from t = 0 to t = 15 mins.

Exactly what do they mean when they say calculator required?

Well, you can expect to be asked to graph gnarly functions. They often won't give you a useful window, so you'll also need to find that. In these scenarios, try the ZoomFit option. It isn't always perfect, but it can be a lifesaver and (at minimum) point you in the right direction. This is usually just a first step.

Often the questions ask you to find zeros (keep at least 3 decimal places) or locate relative (or absolute) minimums and maximums.

Remember, if they ask for location, they're looking for the x-coordinate.

And if they ask for the absolute maximum? Then they want the y-value.

What else is your calculator good for? Calculating! The AP also may ask you to compute the value of a first or second derivative at a specific point. Similarly, they could ask for the numerical value of any type of definite integral. Need to practice? We've got you covered.

OK. But how am I supposed to justify? I'm glad you asked. Let's look at an example.

A local citrus vendor stocks her stand with 100 pounds of clementines at 9 a.m, at which time the market opens. She leaves her partner in control of the stand and returns to the farm to pick up more fruit. Customers buy her fruit at a rate of

where p(t) is in pounds and t is the number of hours since 9 a.m. Three hours later, the vendor returns with a resupply and moves them from the truck to the stand at a rate modeled by

And there you have it! In Part A, you'll lean heavily on your calculator. And as long as you provide good, clear mathematics notation, appropriate reasoning, and correct units, you'll be prepared to collect maximum points!

Many AP Calculus teachers attempt to predict the types of FRQs that will appear on the upcoming exam. As you'll see below, Modeling Rates and Particle Motion have been popular in the past few years.
2019: Modeling Rates and Particle Motion

1-integrate a difference of rates (definite); average value calculation (not average rate of change); find maximum; interpret rate of change (of a rate)

2-from a table, justify intermediate value theorem (IVT); approximate using trapezoidal rule; compute distance traveled given velocity; calculate a relative distance between two moving bodies

2018: Modeling Rates and Particle Motion

1-definite integral given a rate and bounds; integral evaluation with constant function adjustment; finding zeros; minimize a difference of functions

2-find and calculate derivative; evaluate a definite integral with initial condition; interpret displacement vs. total distance traveled

2017: Modeling Volume/Related Rates and Modeling Rates

1-Riemann sums approximation from a table; determine over/underestimate with justification; modeling volume from an equation given cross-sectional heights; applying the fundamental theorem of calculus (FTC)

2-compute a definite integral given a rate and bounds; calculate and interpret the derivative of a rate; find and explain the difference of two rates; compute a definite integral with an initial condition

Above all, whichever calculator you select, make sure you're familiar with it. It's useful to refresh (and practice!) the calculator functions you'll need on exam day.

In short, you'll need a graphing calculator for the exam. And while you've likely used one all year that falls within the guidelines, the AP is quite particular about which graphing calculator model you may use.

You are permitted to bring a backup calculator, though if you're using fresh batteries and pack a few extras (just in case) you should have plenty of insurance.

TI-73; T1-80; TI-81; TI-82; TI-83 (Plus and Plus Silver)

TI-84 (Plus/CE, Plus Silver, Plus C Silver, Plus T, Plus CE-T)

TI-85; TI-86; TI-89 and Titanium

TI-NSpire (CX, CX II, CX II-T)

TI-NSpire (CX II CAS, CX II-T CAS, CX II-C CAS)

FX-6000 Series; FX-6200 Series; FX-6300 Series

FX-6500 Series; FX-7000 Series; FX-7300 Series

FX-7400 Series; FX-7500 Series; FX-7700 Series

FX-7800 Series; FX-8000 Series; FX-8500 Series

FX-8700 Series; FX-8800 Series; FX-9700 Series

FX-9750 Series; FX-9860 Series; FX-CG-5000 (No Stylus)

FX 1.0 Series; FX-CG-10; FX-CG-20 Series; FX-CG-50

CFX-9800 Series; CFX-9850 Series; CFX-9950 Series

CFX-9970 Series; Algebra FX 2.0 Series

Graph25 Series; Graph35 Series; Graph75 Series

Graph95 Series; Graph100 Series

HP-9G; HP-28 Series; HP-38G; HP-39 Series; HP-40 Series HP-48 Series; HP-49 Series; HP-50 Series; HP Prime

4 Questions; 60 minutes; Graphing Calculator Prohibited
Alas, you'll need to put away your trusty TI (or other approved model) in the final section of the exam. Looking for consolation? Well, on a number of the problems your calculator wouldn't offer much help or time-savings anyway. How could that be?

Here the AP likes to test your ability to interpret graphs, tables, and slope fields.

What can you discern about g''(t)at t=a if you're given a picture of g(t)?

Use a table of f(t)values to guarantee the existence of a particular f'(t) value.

Could you produce a specific solution to a differential equation from a slope field?

These examples should give you some idea, but for clarity let's work an example.

The function g is differentiable on the closed interval [-6, 3]and satisfies g(-2) = 15. The graph of g' consists of a semicircle, a line segment, and a quarter-circle, as shown above, where g'(2) = 1.

Congratulations! Once you complete Part B, you've completed your exam. On test day remember to use good, clear mathematics notation, and always justify your answer!

As mentioned earlier, it's a popular pastime of some AP Calculus teachers to predict the content that will appear in the upcoming FRQs. You'll notice that a problem requiring you to analyze a graph and apply FTC has appeared with some frequency.
2019

Graphical Analysis of f and application of FTC

Modeling using Separable Differential Equations

Calculating Area and Volumes of Revolution

Function Analysis-L'Hospital and Squeeze

2018

Graphical Analysis of f' and application of FTC

Modeling from a table of data and analytically

Function Analysis-L'Hospital

Separable Differential Equation including Slope Fields

2017

Graphical Analysis of f' and application of FTC

Modeling using Separable Differential Equations

Particle Motion

Function Analysis from a table of data and graphically

We're not done quite yet. Here's a final reference for things we may not have covered above.
Q: I made a mistake, should I erase it?
A: If it's minor, sure. With more significant mistakes, however, it's quicker and clearer to cross out your work. You can always point the reader forward with an arrow.
Q: What if I found two solutions and I'm not sure which is correct?
A: Ideally, you'll have a hunch one way or the other and you'll clearly identify that answer. If you're flummoxed, though, or you run out of time, here's what's likely to happen. Readers will usually score both solutions, take the arithmetic mean, and then round down. You won't earn your top score, but it's possible you may earn more points. As we've seen though, with good justification, you can overcome an errant final answer.
Q: Do I have to round my answers to 3 decimal places?
A: Unless the question specifies specific rounding (usually this is to the nearest integer), you won't be penalized if you leave more than 3 correct places.
Q: Can use the nDeriv notation my calculator uses?
A: No. Stick to proper mathematics notation: f'(x) or ddx

A: Answers always require justification to receive full marks. Just justify it.
Q: What happens if I make an error in part (b) and need to use my answer in part (c)?
A: Here, the reader is on your side. You shouldn't be penalized for carrying an incorrect value from a prior part of the question. Your reader will score your answer differently. Usually, she will re-compute the sub-question with the erroneous input. If your process and justification are correct, you'll earn the corresponding points.

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AP Calc - AP Calculus Free Response Question (FRQ) Overview | Fiveable

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Study with friends using the #1 virtual studying platform for students. Create a study room, invite your buddies, & keep each other accountable.

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All about limits: the basic idea, finding limits from graphs, different ways to calculate limits, limit laws, one-sided limits, the squeeze theorem, limits a...

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Unit 2 Derivative with the lim as h approaches 0

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Unit 2 Derivative with the lim as h approaches 0

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In this unit, you will learn more about derivatives, how they are defined, and the basic derivative rules (and exceptions). As one of the core foundations of analyzing functions and change in calculus itself, it is of utmost importance to gain an in-depth understanding of this unit before moving forward to more advanced concepts.📑 This unit makes up 10-12% of the Calculus AB exam and 4-7% of the Calculus BC exam.

This list will be a good place to start in terms of self-assessing 📖 what you need to study or learn. As you are looking through this list write down what topics you don't remember or still need to learn. You can then go to the full study guide page for each topic to get some more information!

The rate of change at a particular point.

This notation is derived from the limit definition you learned in Unit 1. Rate of change at a particular point.

Think about the speed of something at a particular moment. For example, if you were running a race, what is your speed at exactly 11 minutes?

There are 3 situations where a derivative doesn't exist. These places are where there is a discontinuity, a corner/cusp, or a vertical tangent.

For example, if you were taking the derivative of x^2, it would become 2x

You will need to memorize these! 🧠

Use this when taking the derivative of two functions that are multiplied together:

Use this when taking the derivative of two functions which are divided:

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AP Calc Unit 2 Notes: Overview | Fiveable

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In this unit, you will learn more about derivatives, how they are defined, and the basic derivative rules (and exceptions). As one of the core foundations of analyzing functions and change in calculus itself, it is of utmost importance to gain an in-depth understanding of this unit before moving forward to more advanced concepts.📑 This unit makes up 10-12% of the Calculus AB exam and 4-7% of the Calculus BC exam.

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Set students up for success in Calculus and beyond! Explore the entire Calculus curriculum: polynomials, derivatives, and more. Try it free!

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Unit 2/3 Study Guide - AP Calculus AB/BC

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Math is fun (see the index for Derivatives)

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Math is fun (see the index for Derivatives)

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There are rules we can follow to find many derivatives.

For example:

The slope of a constant value (like 3) is always 0

The slope of a line like 2x is 2, or 3x is 3 etc

and so on.

Here are useful rules to help you work out the derivatives of many functions (with examples below). Note: the little mark ' means derivative of, and f and g are functions.

"The derivative of" is also written

Examples

Example: what is the derivative of sin(x) ?

From the table above it is listed as being cos(x)

It can be written as:

Or:

sin(x)' = cos(x)

Power Rule

The question is asking "what is the derivative of x ³ ?"

We can use the Power Rule, where n=3:

(In other words the derivative of x ³ is 3x ²)

So it is simply this:

"multiply by power
then reduce power by 1"

It can also be used in cases like this:

1/x is also x⁻¹

We can use the Power Rule, where n = −1:

So we just did this:

which simplifies to −1/x²

Multiplication by constant

the derivative of cf = cf'

the derivative of 5f = 5f'

We know (from the Power Rule):

So:

Sum Rule

Example: What is the derivative of x²+x3 ?

The Sum Rule says:

the derivative of f + g = f' + g'

So we can work out each derivative separately and then add them.

Using the Power Rule:

And so:

Difference Rule

What we differentiate with respect to doesn't have to be x, it could be anything. In this case v:

The Difference Rule says

the derivative of f − g = f' − g'

So we can work out each derivative separately and then subtract them.

Using the Power Rule:

And so:

Sum, Difference, Constant Multiplication And Power Rules

Using the Power Rule:

And so:

Product Rule

Example: What is the derivative of cos(x)sin(x) ?

The Product Rule says:

the derivative of fg = f g' + f' g

In our case:

We know (from the table above):

So:

the derivative of cos(x)sin(x) = cos(x)cos(x) − sin(x)sin(x)

=

Quotient Rule

To help you remember:

The derivative of "High over Low" is:
"Low dHigh minus High dLow, over the line and square the Low"
Example: What is the derivative of cos(x)/x ?

In our case:

We know (from the table above):

So:

Reciprocal Rule

The Reciprocal Rule says:
With f(x)= x, we know that f'(x) = 1
So:

Which is the same result we got above using the Power Rule.

Chain Rule

The Chain Rule says:

the derivative of f(g(x)) = f'(g(x))g'(x)

The individual derivatives are:

So:

Let's do the previous example again using that formula:

Let u = x ², so y = sin(u):

Differentiate each:

Substitute back u = x ² and simplify:

Same result as before (thank goodness!)

Another couple of examples of the Chain Rule:

The Chain Rule says:

the derivative of f(g(x)) = f'(g(x))g'(x)

The individual derivatives are:

So:
= sin(x)cos²(x)
Note: is also or many other forms.

The Chain Rule says:

the derivative of f(g(x)) = f'(g(x))g'(x)

The individual derivatives are:

So:

6800, 6801, 6802, 6803, 6804, 6805, 6806, 6807, 6808, 6809, 6810, 6811, 6812

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About this unit

The derivative of a function describes the function's instantaneous rate of change at a certain point - it gives us the slope of the line tangent to the function's graph at that point. See how we define the derivative using limits, and learn to find derivatives quickly with the very useful power, product, and quotient rules.

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Differentiation: definition and basic derivative rules | Khan Academy

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The derivative of a function describes the function's instantaneous rate of change at a certain point - it gives us the slope of the line tangent to the function's graph at that point. See how we define the derivative using limits, and learn to find derivatives quickly with the very useful power, product, and quotient rules.

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AP Calculus AB and BC Unit 2 Review [Differentiation: Definition and Basic Derivative Rules]

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Differentiation Learn with flashcards, games and more — for free.