2022 AP Calculus BC Exam Guide

Your guide to the 2022 AP Calculus BC exam

Format of the 2022 AP Calculus BC exam

• Section 1: Multiple Choice - 50% of your total score
• 45 questions in 1 hr 45 mins
• Part A: 30 questions in 60 minutes (calculator not permitted).
• Part B: 15 questions in 45 minutes (graphing calculator required).

• Section 2: Free Response - 50% of your total score
• 6 questions in 1 hr 30 mins
• Part A: 2 questions in 30 minutes (graphing calculator required).
• Part B: 4 questions in 60 minutes (calculator not permitted).

When is the 2022 AP Calculus BC exam and how do I take it?

How should I prepare for the exam?

• First, download the AP Calculus Cram Chart PDF - a single sheet that covers everything you need to know at a high level. Take note of your strengths and weaknesses! 
• Review every unit and question type, and focus on the areas that need the most improvement and practice. We’ve put together this plan to help you study between now and May. This will cover all of the units and essay types to prepare you for your exam
• Practice problems are your best friends! Both the FRQs and MCQs had several questions that were similar to previous FRQs and the open-sourced multiple-choice questions on College Board (essentially the same questions with different numbers).
•  Join Hours 🤝to talk to real students just like you studying for this exam! We have TAs in each subject channel to support you this Spring. 
• Finally, check out our cram replays so that you can review for the AP Calc exam with a rockstar teacher!  

Pre-work: set up your study environment

Create a study space.

Organize your study materials.

Plan designated times for studying.

Decide on an accountability plan.

Get support from your peers. 

AP Calculus BC 2022 Study Plan

UNIT 1: Limits and Continuity

Big takeaways:

Content to Focus On:

• Methods of Finding Limits
• From a table - Be able to estimate values not given in a table by using values that were given in a table.
• From a graph - Be able to use a graph to interpret a limit at an x value.
• Algebraically
• Limit Properties
• Rationalization
• Some limits look unsolvable, (0/0 or ∞/∞), but using algebraic manipulation, you can solve them! In unit 4 we learn about another helpful tool for this, L’Hospital’s rule!
• Infinite Limits
• Squeeze Theorem
• The squeeze theorem helps us to find limits of functions that we do not know, by using the limits of a function that is greater than or equal to and a function that is less than or equal to. If we can find the limits of those two functions, and they are equal, then our function should have that limit too!
• Continuity and Discontinuity
• Students should be able to prove if a function is continuous as a point, and know the different types of discontinuity! Removable, jump, infinite.
• First existence theorem: Intermediate Value Theorem
  • This theorem helps us to prove points that exist given certain information. Don’t forget to always state or prove the conditions! In this case, it would be that the function is continuous on [a, b].

Resources:

• Defining Limit and using Limit Notation: Introduction to limits

• Graphical Limits: FInding limits given a graph.

• Algebraic Limits: How do you find a limit?

• Continuity Part I & II: Defining a continuous function

• Limits at Infinity: Using limits at infinity to demonstrate function behavior

• Working with the Intermediate Value Theorem: There are two theorems that are related to Unit 1: The Intermediate Value Theorem and The Extreme Value Theorem. Learn about the Intermediate Value Theorem here.

UNIT 2: Differentiation: Definition and Basic Derivative Rules

Big takeaways:

Content to Focus On:

• Average Rate of Change vs Instantaneous Rate of Change
• If we apply a limit to an average rate of change, we are looking at an instantaneous rate of change!
• Definition of the Derivative
• A long time ago, before we learned the power rule, etc, for the derivative, we learned how to solve the instantaneous rate of change using the limit definition of a derivative.
• Estimating The Derivative
• Connecting differentiability and continuity. Remember, differentiability implies continuity, but not the other way around! Make sure you know how to tell if a function is differentiable. No cusps or corners, polynomials are always differentiable!
• Finding Simple Derivatives
• Power Rule
• Derivative Properties
• Trigonometric Derivatives
• Exponential and Logarithmic Derivatives
• **While this year it may not be necessary to memorize certain derivatives like natural log or tangent, using time out of your exam time to look them up can be costly! Make sure you are still on your game when it comes to the derivatives you need to know.
• Product and Quotient Rules

Resources to use:

• The Limit Definition of the Derivative: The derivative from first principles

• Introduction to Finding Derivatives: The power rule and trigonometric derivatives, a must-watch!

• The Product and Quotient Rules: The product and power rules, a powerful tool in your derivative-finding toolkit!

• Practicing Derivative Rules: Applying what you’ve learned in finding derivatives so far!

UNIT 3: Differentiation: Composite, Implicit & Inverse Functions

Big takeaways:

Content to Focus On:

• Chain Rule
• The chain rule allows us to differentiate composite functions. But make sure you see every function! Sometimes the chain rule can be more than one chain! For example: (sin(x2))2. We have 3 functions to consider here!
• Implicit Differentiation
• Implicit differentiation gives us the tools to derive any variable with respect to any variable. The main concept is that if the variable you are deriving is not the same as the variable you are taking the derivative of, the chain rule applies and you must also multiply by the derivative of that variable. For example, the derivative of y2with respect to x is 2ydydx. 
• Derivatives of Inverse Functions (Including Trigonometric Functions)
• Remember how to find an inverse? Switch x and y and solve for y? Well then take the derivative of that!, or follow the formula we have! The important part to remember on problems like this is the x and y values. If you are supposed to be using the x value of an inverse function, that means it was the y-value of your original function!

Resources to use:

• The Chain Rule: The chain rule, used to evaluate the derivative of composite functions, very important!

• Implicit Derivatives: Derivatives of implicitly defined functions/curves

• Practicing Derivative Rules II: Applying what you have learned through the previous 2 units!

• Using Tables to Find Derivatives: Finding derivatives when the equation may not be present

UNIT 4: Contextual Applications of Differentiation

Big takeaways:

Content to Focus On:

• Rates of Change
• Position, Velocity, Acceleration
• Velocity is the derivative of position, and acceleration is the derivative of velocity. Velocity has a direction. If the velocity and acceleration match signs, the particle is speeding up. If their signs are opposite, the particle is slowing down. Speed is the absolute value of velocity and has no direction. On the AP test, you can be asked to map out the path of a particle, which direction it is going, and if it is speeding up or slowing down.
• Related Rates
• Related Rates have to do with nothing other than relating rates! There are a set of steps to follow when solving a related rate: 
• 1. Assign letters to quantities and label their derivatives with respect to time. If A=area, then dA/dt would be the rate of change of the area.
• 2. Identify the rates that are known and the rate that needs to be found.
• 3. Find an equation that relates the variables whose rates of change you need in step 2. Draw a picture to help you find this equation!
• 4. Differentiate the equation with respect to time. Remember it is necessary to think about this as implicit differentiation.
• 5.  Substitute in all known values and variables, and solve for the unknown rate of change.
• One special type of related rate is that of a cone. You should practice a related rate problem with a cone, because it is necessary to write r, the radius, in terms of h before differentiating.

• Linearization and Tangent Line Approximations
• In these types of problems, we use a tangent line approximation for one value that we know and use it to approximate another very close value. Let day, x0 is the one we know about, and x1 is the one we need. Then we would use: f(x1)-f(x0)=f'(x0)(x1–x0) to find f(x1).
• L’Hopital’s Rule
• When using L’Hospital’s rule, we have a few things we need to remember! To apply L’Hospital’s rule, your limit needs to be in one of two forms: 0/0 or ∞/∞. To show this in a free-response question, you need to show the limits in the numerator and the denominator goes to either 0 or infinity separate from each other, then you can use L’Hospital’s rule, which says: xaf(x)g(x)=xaf'(x)g'(x). 

Resources to use:

• Related Rates: How to solve related rates problems

UNIT 5: Analytical Applications of Differentiation

Big takeaways:

Content to Focus On:

• Two of the three Existence Theorems
• It is important when using theorems to make sure their conditions are met before you use them!
• Mean Value Theorem
• In order to use the Mean Value Theorem, the functions must be continuous on a closed interval and differentiable on that same interval, but open. Once you can say for sure that these things are true, the mean value theorem tells us that there must be a point in that interval where the average rate of change equals the instantaneous rate of change, or the derivative, at a point within the interval. f'(c)=f(b)-f(a)b-a.
• Extreme Value Theorem
• For this existence theorem, the condition is that the function continuous on a finite closed interval. If it is, that means there must be an absolute maximum and an absolute minimum.
• Increasing/Decreasing Functions and The First Derivative
• If the derivative of a function is positive, then the function is increasing! If the derivative is negative, then the function is decreasing!
• Local and Global Extrema
• First and Second Derivative Tests for Local Extrema
• Candidates Test for Global Extrema
• Concavity, Inflection Points, and The Second Derivatives
• If the second derivative is positive, the function will be concave up. If the second derivative is negative, the function will be concave down. Whenever the second derivative changes sign, there will be an inflection point, a change in concavity, on the original function.
• Curve/Derivative Sketching
• All of the information you learned from how the first and second derivatives relate to the original function can help you to sketch graphs based on the information you have about their derivatives!
• Optimization Problems
• These are also known as applied maximum and minimum problems. These are problems about finding an absolute maximum and an absolute minimum in an applied situation. When trying to find an absolute max or min, you must find all critical points (f'(x)=0 or undefined). Then plug those and the endpoints into the original function to find out which has the highest or lowest value, depending on what you are looking for.

Resources to use:

• Interpreting Derivatives Through Graphs: Graphical interpretation of derivatives

• Existence Theorems: Mean Value Theorem, Extreme Value Theorem, Intermediate Value Theorem

• Increasing and Decreasing Functions: Using the first derivative to show where function increases and decreases

• Concavity: Using the second derivative to find concavity

• f, f’, and f”: Relating a function and its 

• Optimization Problems: How to solve optimization problems

UNIT 6: Integration and Accumulation of Change

Big takeaways:

Content to Focus On:

• Approximating an integral as an area
• When given a graph, we can find the area under the curve and between the x-axis to find the integral. However, if an area is underneath the x-axis, that would be considered negative.
• Riemann Sums
• A way of evaluating an area under a curve by making rectangles to find the area and adding them up. This is useful with a table of values or with a curve that we do not know the exact area of its shape. There are four kinds: Left, Right, Midpoint, and Trapezoidal sum. Depending on the function, these may be over or underestimates of the actual area. 
• Students should also be able to write a Reimann sum in summation notation and integral notation.
• The Fundamental Theorem of Calculus
• This theorem tells us two very important things: 
• abf(x)dx=F(b)-F(a)
• ddx(axf(t)dt)=f(x), where a is a constant.
• We use this theorem very often when working with integrals. 
• Definite vs. Indefinite Integrals
• Integrals must always include a dx (if not x, whichever variable you are using) at the end. 
• If an integral is indefinite, it has no bounds. This means it cannot be evaluated using the fundamental theorem of calculus. Instead, we add on a +C, to make it know that there could have been a constant at the end of the function and that there are lots of possibilities for what that constant would be. 
• Definite integrals have bounds, and we use the fundamental theorem of calculus often with them. The bounds let us know the endpoints of the integral, or in terms of a graph, what two points we are finding the area under the curve between. 
• Integral Rules
• Make sure you remember how to integrate and its connection to deriving!
• You can check an integral by taking its derivative to see if you get back to where you started. 
• U-Substitution
• When you have a composition of functions in an integral, it is necessary to use u-substitution to make sure you are integrating each part. It is usually the inside function that is chosen. Sometimes it can be hard to tell. You will know once you try! If you find yourself going in circles and needing to substitute more, I would try using a different piece of the function as u. 

• Integrating using integration by parts
• Integration by parts you can use when you are multiplying two functions, it looks like this:
• udv=uv-vdu.
• When trying to figure out which function to prioritize as u, make sure to go to this helpful order:
• LIPET (Logs, Inverse Trig, Polynomials, Exponentials, Trig). Whichever one appears first, use that for u, then use the other for dv. Find du and v, and plug-in from there!
• Integrating using partial fraction decomposition
• This is helpful when dealing with a rational function in your integral. The denominator should be a higher power than the numerator to use this. It helps turn terms into sums of ratios of linear nonrepeating decimals that are integrable. 
• Evaluating Improper Integrals
• An improper integral has one or both limits at infinity. In order to integrate these, we denote that infinite bound as a variable, a, and take the limit as a approaches infinity of the integral of our function. The fundamental theorem of calculus really helps us here!

Resources to use:

• The Fundamental Theorem of Calculus: Explains the Fundamental Theorem of Calculus and its uses as the most important theorem in calculus

• Some Integration Techniques: How to do integration techniques found in Calculus AB

UNIT 7: Differential Equations

Big takeaways:

Content to Focus On:

• Differential Equations
• Modeling
• If you are given information, can you write a differential equation from it? 
• Verifying Solutions
• Being able to plug in given information and deduce if the solution is true. 
• Separation of variables
• This often appears in the FRQ section. Students are given a differential equation and need to put all of the terms for one variable on one side and for the other variable on the other, then they integrate both sides and solve for y, usually y, but really whichever the dependent variable it. 
• NOTE: If you skip the step of separating variables on AP exams in the past, they have offered you no credit for the rest of that section of the problem. 
• Using initial conditions
• Once you have done your separation of variables, don’t forget to have a +C! This is where the initial condition comes in. You plug in the terms given from x and y in your initial condition, then you solve for C, rewriting the function at the end with C plugged in!
• Slope fields
• Slope fields are coordinate planes of 1 by 1 sections of slopes at each x and y value. If given a differential equation and asked to find a slope field, you plug in an x and y pair into the differential equation, the value it outputs is the slope at the point, and is the steepness you used to draw a line at that particular pair. 

BC Only

• Euler’s Method
• This is another method for approximating a solution to a differential equation or a point on a solution curve. 
• Logistic Models with Differential Equations
• This is a joint proportional differential equation. dy/dt=ky(a-y). You can interpret the initial condition and the differential equation without solving the differential equation. 
• The limiting value, also known as the carrying capacity, can be determined using a logistic growth model. 
• The logistic growth model can also be used to determine the value of the dependent variable in the logistic differential equation at the point when it is changing the fastest. 

Resources to use:

• Separable Differential Equations: How to solve separable differential equations

UNIT 8: Applications of Integration

Big takeaways:

Content to Focus On:

• Average Value of a function
• The average value of a function is denoted by 1b-aabf(x)dx. We can only find the average value of a function that we have a definite integral for. 
• Position, Velocity, Acceleration
• Position is the integral of velocity, and velocity is the integral of acceleration! If you take the absolute value of velocity, you are able to find speed. 
• Distance, the integral of the absolute value of velocity
• Displacement, the integral of velocity
• To remember this, I think about a track! After one lap around, your distance is 400 m, but your displacement is 0. Make sure if you are asked for distance, you remember to use absolute value!
• Finding the area between curves
• If given two curves on a coordinate plane and endpoints, students should be able to integrate those functions, either with respect to the x or y-axis, in order to determine the area between them. When talking about the x-axis, all functions should be in terms of x, the bounds should be in terms of x, and it will be the integral of the upper functions minus the lower. When talking about with respect to the y-axis, all functions should be in terms of y, the bounds should be in terms of y, and it will be an integral of an outer function minus the inner.
• Finding the area between curves that intersect at more than two points. 
• Sometimes, your function will have different parts where you may need to split the area and add your answers together, because the upper functions changes or another function changes. 
• When in doubt, break it up into pieces, you know how to work with, and add those together. 
• The area under the x-axis is already accounted for as negative when evaluating using an integral, so don’t worry about that!
• Volumes of cross-sections
• When using cross-sections, you integrate the area of whatever shape is given to you (this could be a square, rectangle, triangle, or semicircle). 
• You will integrate the area on a given interval, but it is important that you interpret the function for the variable in the shape that it represents. 
• If it says perpendicular to the x-axis, all should be in terms of x.
• Perpendicular to the y-axis, all should be in terms of y. 
• Volumes by disks and washers
• In this case, we are integrating an area again, but this time the shape will always be a circle, so we will always be using the area formula for a circle. The variable that changes here is r, the radius, which you can interpret using the function. 
• If the rotation is around just the x-axis, all should be in terms of x (bounds and functions) and the radius will be your function. 
• If the rotation is around just the y-axis, all should be in terms of y (bounds and functions) and the radius will be your function.
• If it is rotated around any other vertical or horizontal line, you may need to add or subtract value from your function to find the radius. 
• A washer is when two functions are used, and you must subtract out the volume the inner function would have added into your total volume to account for the missing piece. 

• Arc Length
• BC Students are also required to know how to find the arc length of a smooth, planar curve and distance traveled. 
• A definite integral can be used to calculate this. s(x)=ax1+[f'(t)]2dt. 

Resources to use:

• Interpreting the Meaning of the Derivative and the Integral: Showing derivatives and integrals applied in different contexts

• Position, Velocity, and Acceleration: Exploring the relationship between position, velocity, and acceleration

UNIT 9: Parametric Equations, Polar Coordinates & Vector-Valued Functions

Big takeaways:

Content to Focus On:

• 9.1: Defining and Differentiating Parametric Equations
• Parametric functions are a set of related functions where x and y are independent of each other but connected using the variable t.
• x(t)=t^2-1, y(t)=3t
• To find the slope of the tangent line, we need to find dy/dx
• 9.4 Defining and Differentiating Vector-Valued Functions
• Parametric functions are associated with time and are generally used to calculate motion and velocity.
• Each point on a vector-valued function is determined by a position vector, a vector that starts at the origin.
• Vector-valued functions tell us how much the particle is moving in the direction of x and how much it is moving in the direction of y.
• 9.7: Defining Polar Coordinates and Differentiating in Polar Form 
• Polar functions are functions that are graphed around a pole in a circular system and with the points (r, theta) instead of (x, y).
• The slope of the tangent lines for polar functions can be found using the formula dy/dx = (rcos(theta)+(dx/d(theta))sin(theta))/(-rsin(theta)+cos(theta)(dx/d(theta))
• 9.9: Finding the Area of the Region Bounded by Two Polar Curves
• To find the area between two curves, you subtract “outer minus inner”.
• Arc length for polar functions can be found using the formula

Resources to Use:

UNIT 10: Infinite Sequences and Series

Big takeaways:

Content to Focus On:

• 10.2: Working with geometric series
• A geometric series has a constant ratio between successive terms. 
• n=0arn=a1-r, and r>0
• If r<1, then the series converges but diverges otherwise. 
• 10.5 Harmonic Series and p-series
• This section includes the harmonic series, the alternating harmonic series, and the p-series. 
• The harmonic series is a divergent infinite series. k=11k. This form should be familiar to you. 
• The alternating harmonic series converges conditionally. k=1(-1)k-1k because this series converges, but the absolute value of its series, the harmonic series, does not. 
• P series is any series in the form n=101np. If p is greater than 1, it converges. Otherwise, it diverges!
• 10.7: Alternating series test for convergence
• An alternating series is one that changes sign, so it is in the form an=(-1)n+1bn where bn>0 for all n and bn is decreasing, and nbn =0, then n=1an converges.
• This means you need to find nbn. If it is zero and all of the other conditions are met, you have a convergent series. 
• 10.8: Ratio Test for Convergence
• The ratio test compares a term from the series to a different value in the series by taking its limit.
• If our series is and then to use the ratio test, we want to find L, where L=nan+1an. If L < 1, the series is absolutely convergent. If L > 1, the series is divergent, if L=1, we have more work to do, because the series could still be divergent, conditionally convergent, or absolutely convergent. 

• 10.11: Finding Taylor Polynomial Approximation of Functions
• A Taylor polynomial centered at x=a can be found using this form:

• These Taylor polynomials can be used to approximate function values near x=a. 
• The more terms you find in an nth degree Taylor polynomial, the closer you are to approximating the actual function.

Resources to Use: