AP Calculus AB/BC

Average speed over a stretch of time is easy — distance over time.

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A limit answers “where is the function headed as the input closes in on a point?” — not necessarily where it lands at the point itself.

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Put your finger on the curve and slide toward $x=c$ from the left, then from the right.

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Feed the function inputs that close in on $c$ from both sides and watch the outputs settle toward a number.

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When each piece of an expression has a limit, the limit passes through sums, products, and quotients — so you can often just substitute.

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When substitution gives $0/0$, the expression is hiding a removable factor.

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Try the cheapest tool first (substitution).

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If a wiggly function is trapped between two functions that meet at the same height, it is forced to that height too.

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A real limit is the same whether you read it from a graph, a table, or algebra — the representations must agree.

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A break in a graph comes in three flavors: a hole, a jump, or a blow-up.

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A function is continuous at a point if you can draw through it without lifting your pen — no hole, jump, or blow-up there.

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Continuous on an interval means unbroken across the whole stretch — with endpoints only needing the one-sided approach that stays inside.

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If a graph has a single hole, you can plug it — define the missing point to be exactly the limit.

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Near certain inputs a function shoots up or down without bound — the graph races along a vertical line.

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Zoom out: what height does the function level off to as $x$ runs far right or far left?

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An unbroken curve that starts below a height and ends above it must cross that height somewhere in between.

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The average rate of change over $[a,a+h]$ is the slope of the SECANT line through the two points.

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The DERIVATIVE is exactly that limiting instantaneous rate, packaged as one object.

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With only a graph or a table (no formula), estimate the tangent slope from nearby points.

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A function that has a tangent slope everywhere cannot jump or break — so differentiable forces continuous.

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For a power of $x$, differentiating just brings the exponent down front and drops it by one — a shortcut proved from the definition.

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Differentiation is linear: it passes through sums and pulls out constant factors, and a constant has zero slope.

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A few fundamental derivatives to memorize — the slope of $\sin$ is $\cos$, and $e^x$ is its own slope.

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A product changes because EACH factor changes — hold one fixed while the other varies, then swap.

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Differentiate a fraction: top-rate times bottom minus top times bottom-rate, all over the bottom squared — order and sign matter.

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Build the remaining trig derivatives from $\sin$ and $\cos$ using the quotient rule.

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A composite function is one function inside another.

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When an equation is not solved for $y$, treat $y$ as a hidden function of $x$.

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An inverse function is the graph reflected across $y=x$, so its tangent slopes are RECIPROCALS of the original’s at the matching (reflected) point.

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The inverse-trig derivatives are clean algebraic expressions (no trig left) — memorize them, and apply the chain rule to any inner function.

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Before differentiating, read the STRUCTURE: is it a product, a quotient, a composite, or an implicit equation?

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Differentiate again to get the rate of the rate.

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In a real situation, $f'(a)$ is the INSTANTANEOUS RATE of change of the quantity at the input $a$ — and it carries UNITS: (units of $f$) per (unit of the input).

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For motion on a line, velocity is the derivative of position and acceleration is the derivative of velocity.

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Any derivative is a rate of change of one quantity with respect to another — cost per item, people per year, liters per second.

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When several changing quantities are tied together by an equation, their RATES are tied together too.

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Model the relationship, differentiate with respect to time, THEN substitute the values true at that instant, and solve for the unknown rate — reporting units.

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Zoomed in near a point, a smooth curve looks like its tangent line.

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When a limit gives the indeterminate form $\tfrac{0}{0}$ or $\tfrac{\infty}{\infty}$, compare how fast the numerator and denominator change — replace each by its derivative.

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If a function is smooth over an interval, then at some interior point its instantaneous rate (tangent slope) exactly equals its average rate (secant slope) over the interval.

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A continuous function on a closed interval must reach a highest and lowest value; those extremes happen at critical points or at the endpoints.

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Where the slope is positive the function climbs; where it is negative the function falls.

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At a critical point, watch how the slope changes sign: from up to down is a peak; from down to up is a valley.

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To find the overall highest and lowest values on a closed interval, just test all the candidates: critical points and both endpoints.

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The second derivative tells you how the curve bends: $f''>0$ bends upward (like a cup), $f''<0$ bends downward.

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At a point where the slope is zero, the bending tells you the type: concave up means a valley, concave down means a peak.

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The three graphs are linked: where $f'=0$, $f$ has a horizontal tangent; where $f'>0$, $f$ rises; where $f''>0$, $f$ is concave up.

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Given any one of the three graphs, you can deduce features of the others — position, slope, and bending are all connected.

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To optimize a real quantity, first write it as a function of a single variable using the constraint that ties the variables together.

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Once the objective is a single-variable function, use calculus: find critical points, confirm which is the max or min, and report the quantity that was asked.

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For a curve given implicitly, the tangent slope comes from implicit differentiation.

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If you know the RATE at which something changes, adding up (accumulating) that rate over time gives the TOTAL change.

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To estimate the area under a curve, slice the region into rectangles and add their areas.

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As the rectangles get infinitely thin, the Riemann sum becomes exact — that limit is the definite integral.

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An accumulation function $g(x)=\int_a^x f(t)\,dt$ builds up area as $x$ moves right.

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Because $g'=f$, the shape of an accumulation function $g$ is read from $f$: $g$ rises where $f>0$, has extrema where $f$ changes sign, and is concave up where $f$ is increasing.

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Definite integrals behave like sums: they split over adjoining intervals, pull out constants, add over sums, and flip sign when you reverse the limits.

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To get the exact area, find an antiderivative and subtract its values at the two limits — no rectangles needed.

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Antidifferentiation reverses the derivative.

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Substitution reverses the chain rule: spot an inner function and its derivative, rename the inner function $u$, and the integral simplifies.

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When a rational integrand is “top-heavy” or leads to an inverse-trig form, reshape it first: long-divide improper fractions, or complete the square in a denominator.

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For a product of two different kinds of function, trade the integral for an easier one using the reverse of the product rule.

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A rational function with a factorable denominator can be split into simpler fractions, each easy to integrate as a logarithm.

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When a limit of integration is infinite (or the integrand blows up), define the integral as a LIMIT and see whether it converges to a finite value.

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Read the form of the integrand first, then pick the tool: basic rule, u-substitution, or (BC) parts / partial fractions / a reshaping step.

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A differential equation is a sentence about a RATE.

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A proposed solution is just a candidate function.

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A slope field is a picture of a differential equation: at a grid of points, draw a tiny segment whose slope is what $\dfrac{dy}{dx}$ evaluates to there.

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Given a slope field, you can describe solutions without a formula: where they rise or fall, where equilibria sit, and roughly what a particular solution through a given point looks like.

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When you cannot solve a DE exactly, march along it in small steps.

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If a DE can be rearranged so all the $y$’s (with $dy$) sit on one side and all the $x$’s (with $dx$) on the other, you can integrate each side separately to recover $y$.

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The general solution has a free constant — a whole family of curves.

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Whenever a quantity changes at a rate proportional to itself, it grows or decays exponentially.

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Real populations cannot grow forever.

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The average value of a function over an interval is the constant height that would enclose the same area — you total the function (integrate) and spread it evenly across the width.

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Integration walks the motion chain backwards: integrate acceleration to recover velocity, integrate velocity to recover position.

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Any “rate in / rate out” story is an accumulation: integrate the net rate to get the net change, and add it to a starting amount to get the total present at a later time.

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Between two curves, each thin vertical strip has height (top − bottom).

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When curves are easier to read as $x$ in terms of $y$ (they open sideways), slice with horizontal strips instead: each has width (right − left) and you integrate over $y$.

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If two curves cross several times, which one is on top switches.

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Build a solid from a base region by standing a known shape on each thin slice.

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Same slice-and-add idea, but the standing shape is a triangle or semicircle.

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Revolve a region against its bounding axis and each slice becomes a solid disc.

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Revolving about a line that is not the region’s own axis just shifts the radius: measure the distance from the curve to that line.

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If the region does not touch the axis, each slice is a washer — a disc with a hole.

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Combine the two adjustments: it is a washer (hole) AND the axis is shifted.

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To measure the true length of a curve, add up tiny hypotenuses: each small piece has horizontal run $dx$ and vertical rise $f'(x)\,dx$, so its length is $\sqrt{1+[f'(x)]^2}\,dx$.

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When a curve is traced by $x(t)$ and $y(t)$, the slope is still rise over run — but both rise and run are measured against the parameter $t$.

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To get $\dfrac{d^2y}{dx^2}$ you differentiate the SLOPE $\dfrac{dy}{dx}$ again with respect to $x$ — but since everything is in terms of $t$, differentiate with respect to $t$ and divide by $\dfrac{dx}{dt}$ once more.

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The length of a parametric path adds up tiny hypotenuses: each has horizontal piece $x'(t)\,dt$ and vertical piece $y'(t)\,dt$, so its length is $\sqrt{x'^2+y'^2}\,dt$.

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A vector-valued function $\langle x(t),y(t)\rangle$ traces a moving point.

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Integrating a velocity vector recovers position, one component at a time.

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Planar motion bundles two 1-D motions.

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A polar curve $r=f(\theta)$ is really a parametric curve: $x=r\cos\theta,\ y=r\sin\theta$.

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Sweeping a ray from the origin, each thin slice is nearly a circular sector of radius $r(\theta)$ and tiny angle $d\theta$.

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For the region between an outer and inner polar curve, each sector is a washer: subtract the inner sector area from the outer over the angles where the region lives.

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An infinite series is the limit of its partial sums.

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A geometric series multiplies by the same ratio each term.

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If the terms themselves do not shrink to zero, there is no way the running total can settle — the series must diverge.

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If the terms come from a positive, decreasing function, the series and the matching improper integral are like a staircase and a ramp — they rise together, so they converge or diverge together.

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A p-series $\sum 1/n^p$ converges exactly when the terms shrink fast enough — that threshold is $p>1$.

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Compare a messy positive series to a known one (usually a p-series or geometric).

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When terms alternate sign and steadily shrink to zero, the partial sums zig-zag inward toward a limit — like closing in from both sides.

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Look at how each term compares to the one before.

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A series can converge only because of sign cancellation (conditional) or robustly even if you make every term positive (absolute).

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Because an alternating series’ partial sums straddle the true sum, the error after stopping is no bigger than the first term you left out.

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A Taylor polynomial matches a function’s value and first several derivatives at a center, so it hugs the curve near that point — the more terms, the closer the fit.

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The error of a Taylor approximation is controlled by the size of the next derivative over the interval — it behaves like the first term you dropped.

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A power series converges only within a certain distance of its center — the radius.

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Letting a Taylor polynomial run to infinitely many terms gives a power series that equals the function on its interval of convergence.

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New series come cheaply from known ones: substitute, multiply, differentiate, or integrate a known series term by term to represent a related function.

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