Each card is a single, complete idea: what it is, the formulas you need, and one worked example. Built to be re-read.
A limit is the value a function approaches as x gets infinitely close to a certain value. It describes the function's behavior near a point, not necessarily the value at that point.
Approach the point from the left (x→c−) and from the right (x→c+). The general (two-sided) limit exists only when both one-sided limits exist and are equal.
The limit is about the approach; f(c) is about the actual output. They can disagree at holes and at redefined points of piecewise functions.
Read limits from a graph by tracing each side toward the target x-value, or from a table by watching the numerical trend from both directions (round to three decimal places).
Try direct substitution first. If it gives 00, manipulate the expression — factor and cancel, combine fractions, or multiply by the conjugate — then substitute again.
Limits distribute over sums, differences, products, quotients (nonzero denominator), and constant multiples, which lets you break a messy limit into pieces.
Two identities handle most trig limits as the input approaches zero. Rewrite the expression so the angle inside matches the denominator.
For rational functions, compare degrees of numerator and denominator to find horizontal asymptotes. Divide by the highest power of x when in doubt.
If a function is trapped between two others that converge to the same value at a point, the trapped function must converge to that value too.
f is continuous at x=c only if all three hold: f(c) is defined, limx→cf(x) exists, and the limit equals f(c). Check them in order and stop at the first failure.
Removable holes come from factors that cancel; vertical asymptotes come from factors left in the denominator; jumps come from piecewise pieces whose one-sided limits disagree.
Polynomial, rational, exponential, logarithmic, and trig functions are continuous on their domains. For piecewise functions, force the pieces to agree at each boundary.
If f is continuous on [a,b], then f takes on every value K between f(a) and f(b) at least once. Always state continuity and that K lies between f(a) and f(b).
The average rate of change over [a,b] is the slope of the secant line joining the two endpoints of the interval. It answers 'how fast on average,' not 'how fast right now.'
The instantaneous rate of change is the slope of the tangent line at a single point. You get it exactly by taking a limit of secant slopes as the gap h shrinks to zero.
Instead of computing a slope one point at a time, differentiate once to get a derivative function that returns the instantaneous rate of change at any x. All notations mean the same thing.
A tangent line uses point-slope form with the derivative as the slope. Find the point, find f' at that point, then substitute.
A function is differentiable where it has a single well-defined, smooth slope. It fails at discontinuities, corners/cusps, and vertical tangents. Differentiable ⟹ continuous, but not the reverse.
These shortcuts replace the limit definition. Constants die, constant factors ride along, and sums differentiate term by term.
Bring the exponent down as a coefficient and drop it by one. It works for negative and fractional exponents too — rewrite radicals and fractions as powers first.
For a product of two functions, differentiate one factor at a time and add. The derivative of a product is never just the product of the derivatives.
For a quotient, it is 'low d-high minus high d-low, over low squared.' Order matters here, unlike the product rule.
Memorize the six trig derivatives. The co-functions (cos, cot, csc) all pick up a negative sign.
ex is its own derivative; lnx gives x1. Other bases carry an extra factor of ln(a).
For composite (nested) functions, differentiate the outside while leaving the inside alone, then multiply by the derivative of the inside. Nearly every derivative from here on needs it.
When a function sits inside a function inside a function, apply the chain rule repeatedly — work from the outermost layer inward and multiply every layer's derivative.
Use it when y is not isolated. Differentiate every term with respect to x, attaching dxdy each time you differentiate a y term, then solve algebraically for dxdy.
Differentiate repeatedly to get the second, third, or nth derivative. Done implicitly, you must substitute the first derivative back in to simplify.
Critical points occur where f′(x)=0 or is undefined. Use sign changes of f′ to classify them.
f′′>0⟹ concave up; f′′<0⟹ concave down. Inflection at sign changes of f′′.
A definite integral measures the total change in a quantity over an interval. Geometrically, it is the signed area between the curve and the x-axis: regions above the axis are positive, and regions below are negative.
When a region is not made of simple shapes, approximate its area with rectangles or trapezoids. Left, right, and midpoint sums use one endpoint per sub-interval; the trapezoidal rule averages left and right. More sub-intervals give better accuracy.
Definite integrals behave like signed areas: equal bounds give zero, swapping bounds flips the sign, constants factor out, and integrals split over adjacent intervals.
An indefinite integral is the family of all antiderivatives. Always include +C because the derivative of any constant is zero. The reverse power rule raises the exponent by one and divides by the new exponent.
Integration reverses differentiation. Memorize the basic antiderivatives for exponentials, logarithms, and the six trigonometric functions. Watch the sign on sine and cosine.
Differentiation and integration are inverse operations. If an accumulation function has a variable upper limit, its derivative is the integrand evaluated at that limit. If the upper limit is a function of x, apply the chain rule by multiplying by the derivative of the bound.
To evaluate a definite integral, find any antiderivative F, then subtract its values at the bounds. This is the workhorse of integration on the AP exam.
u-Substitution reverses the chain rule. Choose u to be the inner function, compute du, and rewrite the integral so that the remaining dx terms match. Don't forget to change the bounds for definite integrals or substitute back for indefinite integrals.
When an integrand doesn't fit a basic rule, rewrite it algebraically. Split fractions, expand products, simplify radicals into powers, or use polynomial long division to make a 'top-heavy' rational function manageable. Completing the square can reveal inverse-trig forms (BC).
Integrate top minus bottom for vertical slices, or right minus left for horizontal slices.
Find the area of one geometric slice, then integrate that area across the base region.
Rotate a region about an axis: use discs when the region touches the axis, washers when a hole is created.
Move all y terms to one side and all x terms to the other, then integrate both sides.
The derivative is the limit of the average rate of change. Differentiability implies continuity, but not the reverse.
An inverse swaps x and y coordinates. Its derivative is the reciprocal of the original function's derivative, evaluated at the matching point — perfect for table or single-point questions.
Memorize only arcsin, arctan, and arcsec — the co-function versions (arccos, arccot, arccsc) are identical with a negative sign. Pair them with the chain rule when the inside is more than x.
Differentiate a relationship between quantities with respect to time, then substitute the known rates.
Velocity is the derivative of position and acceleration is the derivative of velocity. Speed is |v(t)|.
In an applied problem, f'(a) is the instantaneous rate of change of the output with respect to the input at x = a. A full interpretation names the value, the units, the time/input, and the direction of change.
The units of a derivative are output units divided by input units. The second derivative divides again by the input units, which is why applied second derivatives carry squared units.
Speed is |v(t)|. Compare the signs of velocity and acceleration: matching signs mean the particle is speeding up, opposite signs mean it is slowing down. Positive velocity means motion right/up, negative means left/down.
Displacement is the net change in position over an interval, not the total distance traveled. Average velocity is that net change divided by the elapsed time — an average rate of change, not the average of velocities.
Most related-rates setups come from a standard geometry formula. Treat every variable as a function of t, so differentiating A=lw needs the product rule and a2+b2=c2 generates 2a⋅dtda terms.
Use the tangent line at a convenient point to estimate f at a nearby input. Concave up graphs sit above their tangent line, so the estimate is too low; concave down graphs give an overestimate.
Only after direct substitution gives 00 or ∞∞ may you differentiate the numerator and denominator separately (not with the quotient rule) and re-evaluate the limit. Repeat if the form is still indeterminate.
If f is continuous on [a, b] and differentiable on (a, b), some interior point has instantaneous rate equal to the average rate.
Local (relative) extrema are any peaks or valleys; absolute (global) extrema are the single highest and lowest points on the whole interval. Concave up looks like a U (rate of change increasing), concave down like an upside-down U (rate of change decreasing). A point of inflection is where concavity actually changes.
If f is continuous on a closed interval [a, b], then f must attain both an absolute maximum and an absolute minimum somewhere on that interval. This is the guarantee that makes the Candidates Test valid.
All three are existence theorems: verify the hypotheses in writing, then state what must exist. IVT gives a y-value, MVT gives a slope, EVT gives extreme points.
At a critical point where f′(c)=0, the sign of f′′(c) classifies it by concavity. If f′′(c)=0 the test is inconclusive — fall back on the first derivative test.
On a closed interval, absolute extrema can only occur at critical points or endpoints. Evaluate the original function f (not f') at every candidate and compare the outputs.
Read behavior of f off the graph of f'. Height of f' gives the sign (increasing/decreasing), and the slope of f' gives concavity. Relative extrema of f sit where f' crosses the x-axis; inflection points of f sit where f' turns around.
Write a primary equation for the quantity to maximize or minimize, use the constraint to reduce it to one variable, then find and test critical points. Always answer the question that was asked and justify why your answer is the extreme value.
A differential equation relates a function to one or more of its derivatives. To verify that a proposed function is a solution, differentiate it and plug both the function and its derivative(s) back into the equation. If both sides simplify to the same expression, the function satisfies the equation.
A slope field draws short line segments at grid points to show the slope prescribed by a differential equation. Use the field to sketch solution curves that follow the flow, and match equations to fields by looking for global sign patterns (for example, dxdy=x2+y2 never produces negative slopes).
When a differential equation can be written with all y terms on one side and all x (or t) terms on the other, integrate both sides. Add the constant of integration on the independent-variable side, then use an initial condition to solve for the particular solution.
The rate-of-change model dtdy=ky describes quantities that grow or decay in proportion to their current size. Its solution is y=Cekt, where C is the initial value and k is determined by additional data points.
For a function continuous on [a, b], the average value is the height of a rectangle with the same area as the region under the curve.
Integration moves backwards through motion: velocity is an antiderivative of acceleration, and position is an antiderivative of velocity.
The definite integral of a rate function gives the net change in the quantity over that interval — the backbone of calculator-active free response.