AP® Calculus BC

Flashcards

361 cards with the terms and ideas you need to know cold. The ones you find hard come back sooner.

Multiple choice practice

292 exam-style questions, timed like the real exam, with every answer explained.

Limits and Continuity

16 topics · 91 videos

• A limit describes where a function is heading, not where it is
• Graphs, tables and formulas can all show the same limit
• Rewriting a 0/0 expression often reveals the limit
• Continuous means the limit and the function value match
• Infinite limits and limits at infinity explain asymptotes

Differentiation: Definition and Fundamental Properties

10 topics · 58 videos

• A derivative is the limit of average rates of change
• The derivative at a point is the slope of the tangent line
• Differentiable means continuous, but not the other way around
• Derivative rules let you skip the limit definition
• Know the derivatives of sin x, cos x, tan x, eˣ and ln x by heart

Differentiation: Composite, Implicit, and Inverse Functions

6 topics · 40 videos

• Chain rule: derivative of the outside times derivative of the inside
• Implicit differentiation treats y as a function of x
• An inverse function's slope is the reciprocal of the original's slope
• Picking the right mix of rules is half the work
• The second derivative is the rate of change of the first

Contextual Applications of Differentiation

7 topics · 43 videos

• A derivative is a rate, so always give its units
• Velocity is the derivative of position; acceleration is the derivative of velocity
• Related rates: link the quantities, then differentiate with respect to time
• Near a point, the tangent line is a good stand-in for the curve
• L'Hospital's Rule handles 0/0 and ∞/∞ limits

Analytical Applications of Differentiation

12 topics · 69 videos

• The sign of f′ tells you where f is increasing or decreasing
• The sign of f″ tells you which way f is bending (concavity)
• Extreme values can only happen at critical points or endpoints
• Theorems like the MVT and EVT guarantee a value exists, but only when their conditions are met
• Optimization means building a function and finding its max or min

Integration and Accumulation of Change

14 topics · 82 videos

• Area under a rate graph gives the net change
• Riemann sums estimate a definite integral, and their limit is the exact value
• The Fundamental Theorem of Calculus links derivatives and integrals
• Antidifferentiation undoes differentiation, plus a constant C
• Choosing the right integration technique is a skill in itself

Differential Equations

9 topics · 56 videos

• A differential equation describes a rate of change, and its solutions are functions
• Slope fields let you see solutions without solving
• Separation of variables solves many differential equations exactly
• An initial condition picks one particular solution out of a whole family
• "Rate proportional to amount" leads to exponential growth or decay

Applications of Integration

13 topics · 76 videos

• An integral adds up thin slices
• Average value is the integral divided by the length of the interval
• Integrate velocity for displacement and speed for total distance
• Area between curves is top minus bottom, or right minus left
• Volume is the integral of the area of a cross section

Parametric Equations, Polar Coordinates, and Vector-Valued Functions

9 topics · 51 videos

• For parametric curves, dy/dx = (dy/dt) ÷ (dx/dt)
• Work with a vector one component at a time
• Speed is the length of the velocity vector
• Polar area adds up thin pie slices: ½∫r² dθ

Infinite Sequences and Series

15 topics · 92 videos

• A series converges when its partial sums approach a limit
• Pick the convergence test that fits the series
• Terms going to 0 is needed for convergence, but it isn't enough
• Taylor polynomials approximate a function near a point
• Error bounds tell you how far off an approximation can be