One of the most popular study sites used by AP students is Quizlet and for good
reason! Quizlet combines the classic flashcard studying method with unique, fun
games to learn vocabulary. However, the number of resources provided by Quizlet
can make it challenging to find the best decks for each AP Calculus AB and BC
For that reason, here are the most comprehensive Quizlet decks for effective
studying! Vocabulary is critical for understanding different concepts, theorems, and rules. However, Quizlet cannot substitute for practice. As a math class, AP Calc requires intensive practice as to your skills by doing practice FRQs.
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➡️ Unit 1: Limits and Continuity
Unit 1 can be described as both the most important unit and one of the least important units. Limits and continuity form the basis for every other topic in AP Calculus, throwing students straight into the deep end with infinitesimals and the concept of limits. You’ll also learn what it means for a function to be continuous and how limits form the foundation of calculus. Unlike other units, this unit has quite a few true vocab terms and theorems like the IVT where without knowing these you may struggle. Quizlet to the rescue!
- Limit – The value that a function approaches as x gets infinitely close to a constant (notated as lim x→a f(x))
- Continuous – The graph is a smooth unbroken curve (The limit exists!)
- Removable Discontinuity – A hole in the graph, typically removed via factoring and canceling out
✂️ Unit 2: Differentiation: Definition and Fundamental Properties
Unit 2 begins the first of the two major parts of this class: derivatives. Derivatives help calculus students define and describe how a function changes at an instant. While that may seem paradoxical, instantaneous rates of change help us show the behavior of thousands of scenarios and knowing how to calculate basic derivatives is crucial.
- Derivative (Limit Definition) – df/dx = lim h→0 [f(x+h) – f(x)]/h
- Tangent Line – A line that touches a function only once
- Basic Derivative Rules – Product rule, quotient rule, sum and difference rule
⛓️ Unit 3: Differentiation: Composite, Implicit, and Inverse Functions
Once you’ve learned the basics of derivatives, you can start getting into some of the more complicated rules, such as differentiating functions in the form of f(g(x)) with the chain rule or differentiating implicit functions like x^2 + y^2 = 4 using implicit differentiation. These rules, especially the chain rule and implicit differentiation, will be super important to know to go into the next unit where you apply derivatives.
- Derivative of a Composite Function – Chain rule – d/dx f(g(x)) = f’(g(x)) * g’(x)
- Derivative of Implicit Functions – Differentiate with respect to x and use the chain rule. Ex. In the equation x^2 + y^2 = 4, 2x + 2y dy/dx = 0.
- Derivative of Inverse Functions – d/dx (f^-1(x)) = 1/f’(f^-1(x))
💥 Unit 4: Contextual Applications of Differentiation
Now that you know how to compute derivatives, we can start getting into why derivatives matter and how they can be applied to real-life problems. One of the most important problem types you’ll encounter is related rates, in which, by using implicit differentiation with respect to time, you can relate rates to each other. You can also optimize certain functions by finding minimums and maximums which you’ll get more in-depth into in the next unit.
- Related Rates – A type of problem that requires you to use implicit differentiation to find a relationship between two rates
- Position, Velocity, and Acceleration – If position = s(t), velocity is v(t) = s’(t) and acceleration is a(t) = v’(t) = s’’(t).
- L’Hopitals Rule – If we have an indeterminate form 0/0 or ∞/∞ all we need to do is differentiate the numerator and differentiate the denominator and then take the limit
📈 Unit 5: Analytical Applications of Differentiation
The final derivative unit! After spending 3 units in derivatives, you’ll get to learn how to analyze functions using derivatives. Skills such as concavity and extrema along with critical points are so powerful that with just a few data points, you can sketch pretty accurate graphs of functions! These skills combined with your already existing application skills will turn you into a derivative master!
- Absolute and Relative Extrema – The minimum and maximum points on a graph, absolute extrema require you to test endpoints and find the highest or lowest point on an interval, relative extrema require you to find where f’(x) switches sign
- Concavity – The rate at which the derivative is changing, notated by f’’(x). If f’’(x) < 0, it is concave down and vice versa.
- Mean Value Theorem – If f(x) is continuous on the interval [a, b] and differentiable on the interval (a, b), there is a point c ∈ [a, b] where f’(c) = f(b) – f(a)/b – a (there is a point where the derivative equals the average rate)
∫ Unit 6: Integration and Accumulation of Change
Now that you’re a derivative wizard (or a derivawizard for short), you can get into the second big topic in AP Calculus: integrals. Where derivatives described instantaneous change, integration describes how change accumulates over a function. Integrals also can be used to calculate the area under a curve. Integration, as you will come to learn, is an inverse for differentiation through the two parted fundamental theorem of calculus.
- Antiderivative: The opposite of a derivative. Ex. the antiderivative of 2x is x^2 + C. NEVER FORGET THE +C!!! Because constants become zero in a derivative, the +C accounts for it.
- Riemann Sum – A way of approximating an integral by using infinitesimally small rectangles
- Accumulation of Change – Integrals represent an accumulation of change over a given region and can be used to calculate aggregates and totals this way
- (BC Only) Integration by Parts – A way to integrate products with the formula: the integral of udv = uv – the integral of vdu
- (BC Only) Integration by Partial Fractions – A way to integrate complex fractions by splitting the fraction into smaller partial fractions and then integrating each term separately
- (BC Only) Improper Integrals – An integral containing a point in which the value of the function is undefined or contains x = positive or negative infinity
🦠 Unit 7: Differential Equations
Let’s suppose you knew the derivative of a function. Would you be able to figure out information about the specifics of that function? Well with differential equations you can! By using integration to solve separable differential equations, you can figure out exactly what a function looks like if given an initial condition.
- Separable Diff. Eq – A differential equation in the form dy/dx = f(x)g(y)
- Exponential Growth – A situation in which a function is proportional to itself, represented by dy/dx = ky where k is a constant (The solution is in the form Ce^kx)
- (BC Only) Euler’s Method – A numerical method to approximate a particular solution by taking tiny steps from an initial condition
- (BC Only) Logistic Model – A differential equation in the form dy/dx=ky(L-y) where k is a constant and L is the carrying capacity. The solution is in the form L/(1+Ce^(-Lkt))
💿 Unit 8: Applications of Integration
For Calc AB students, this is where your journey ends. After a triumphant battle against derivatives and integrals, you come to your final foe. When applying integration, the majority of this unit focuses on two things: area between curves and volumes. By using integration, one can find the exact volume of function-defined solids, and by carefully using formulas, you can calculate areas between curves and average values.
- Solids of Revolution – A solid formed by revolving a function (or area between two functions) around either the x or y-axes
- Average Value Formula – On the interval [a, b], the average value of a function is 1/b-a * the integral from a to b of f(x) dx.
- Disk Method – Where you have one function bounded by an axis. V = pi * the integral from a to b of R(x)^2 dx.
- Washer Method – Used when your volume has a hole in it, or if you have a major and minor radius. V = pi * integral from a to b of (R(x)^2 – r(x)^2) dx.
- (BC Only) Arc length – Use to find the arc length of a function. S = integral from a to b of sqrt(1+(dy/dx)^2) dx
❄️️ (BC Only) Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions
For BC students, there’s still some more challenges to overcome! You’ve dealt with the x-y coordinate system, but now it’s time to transition to other methods of showing graphs. In this unit, you’ll learn about parametric equations, polar coordinates, and vector-valued functions as the unit title says. By using these, we can represent graphs and do calculus on graphs that would be almost impossible to solve using cartesian x-y coordinates.
- Polar Equation – A function in terms of the angle counterclockwise from the x-axis and the distance from the origin
- Parametric Equations – A function in which the x- and y-coordinates are in terms of a parameter t
- Vector-Valued Functions – Like parametric equations, but each point on the graph is represented by a vector from the origin
➕ (BC Only) Unit 10: Infinite Sequences and Series
This is the end of the AP Calculus journey. Combining everything you’ve learned, you now apply this to infinite sequences and series. There are 2 main topics that we shall cover, convergence of series, which is finding out which infinite series can be evaluated, and power series approximations, which is finding polynomial representations of functions.
- Convergence – When the value of an infinite series can be found
- Power series representation – An infinite series where the terms are of form a_nx^n where n is a non-negative integer
- Taylor Series – The power series representation of a function not centered at x=0
- MacLaurin Series – The power series representation of a function centered at x=0
- Alternating Series/Lagrange Error Bound – Although two separate concepts used in two very different situations, these are almost the same. The difference between the actual function and a partial sum approximation. No more than the value of the next term in the series