Fundamentals
The core facts every aspirant should own — each a titled nugget with a real-world story, the concept in plain words, and a memory trick. Works even when the internet doesn't.
1,200 fundamentals
“A round pothole on the road is described by (x−h)²+(y−k)²=r², where (h,k) is the centre you’d mark on a map.”
For any circle, (x−h)²+(y−k)²=r² encodes its centre (h,k) and radius r. It works for circles anywhere on the plane. A common error is writing (x+h)² or (y+k)², which shifts the centre opposite to intended.
(x−h)²+(y−k)² = r²
Memory trick: Center (h,k), radius r: (x−h)²+(y−k)²=r².
“A satellite dish reflects signals toward its focus, just like y² = 4ax curves rightward toward (a,0).”
The curve y² = 4ax is a parabola with vertex at (0,0), focus at (a,0) and directrix x = –a. Any point is equidistant from focus and directrix. In exams, watch the sign of a: a>0 opens right, a<0 opens left; also ensure axis is the x-axis.
y² = 4ax
Memory trick: Focus (a,0), directrix x=−a → opens right if a>0
“A planet’s closed orbit around the Sun is an ellipse, while a comet that escapes follows an open hyperbola.”
An ellipse is a closed curve with eccentricity e < 1; the sum of distances from any point to the two foci is constant (2a). A hyperbola is an open two-branch curve with e > 1; the absolute difference of distances to the foci is constant (2a). Remember that e = 1 marks a parabola, not a hyperbola.
Memory trick: EllipSe = Sum constant; Hyperbola = Difference constant.
“In a Cassegrain telescope the primary mirror is a parabola and the secondary hyperbola redirects light from one focus to the other.”
A parabola has eccentricity exactly 1, one focus and a directrix; its points satisfy equal distance to focus and directrix. A hyperbola has e > 1, two foci, and the absolute difference of distances to them is constant. Beware mixing up e > 1 with the parabola’s e = 1 when identifying the curve.
Memory trick: Parabola = 1 focus, e=1; Hyperbola = 2 foci, e>1.
“A satellite’s elliptical orbit keeps the ratio of its distance to Earth (focus) and to the straight line (directrix) fixed – that ratio is eccentricity.”
For any point P on a conic, e = PF ÷ PD, where PF is the distance to the focus and PD the perpendicular distance to the directrix. This ratio is constant for the whole curve. The mistake is treating e as the focus-directrix distance or a variable; it is a fixed ratio, giving e=0 (circle), 0<e<1 (ellipse), e=1 (parabola), e>1 (hyperbola).
Memory trick: E = PF/PD – ‘PF over PD’ sounds like a photo (P-F-O-P-D).
“A door knob may be missing, but you can still feel the direction the door wants to turn – the limit exists without the function value.”
A limit can exist even if the function is not defined at that point; it cares only about nearby values. The common mistake is to equate limit with f(a). Always verify existence of the approaching value first.
Memory trick: Limit ≠ f(a) if f(a) missing.
“When you slide a phone screen across a table, the motion feels smooth only if the path has no sudden jumps.”
A function f is continuous at x = a when limₓ→a f(x) exists and equals f(a). Hence there is no hole, jump or break at that point. The exam trap is assuming any piecewise or |x| graph is discontinuous without checking the limit; |x| is continuous at 0 because both sides approach 0.
Memory trick: Continuity: limit = value → L=V
“Zooming into a tiny slice of a pizza, the chord length and the arc become almost the same – that’s why sin x/x →1.”
The standard limit lim (x→0) sin x / x = 1 resolves the 0/0 indeterminate form; it underlies many JEE limit problems. Remember to work in radians; otherwise the limit equals π/180, not 1.
Memory trick: sin x/x →1 (rad).
“When you tilt a tiny angle, the arc length and chord become almost the same, so sin θ/θ ≈1.”
The limit limₓ→0 (sin x)/x equals 1. It follows from the squeeze theorem on the unit circle or from the series sin x = x – x³/6… . The usual exam trap is to stop at the 0/0 form and write 0 or claim the limit is undefined.
Memory trick: Tiny angle rule: sin θ ≈ θ, so sin θ/θ → 1
“A carpenter measuring a tiny angle on a circular protractor finds the arc length equals the chord length, illustrating sin θ≈θ.”
This limit holds only when x is measured in radians; in degrees sin θ/θ → π/180. The radian definition makes the arc length equal to the angle, giving the result. A common mistake is plugging degrees and getting a wrong factor.
lim(x→0) sin x / x = 1
Memory trick: Radians only: sin θ/θ→1, not in degrees
“A TV channel may show a picture that approaches a scene as you tune, yet the exact frequency may still be blank—limit ≠ value.”
The limit describes the value that f(x) approaches as x nears a point, independent of the actual function value at that point. At a removable discontinuity the limit exists but f(a) may be undefined or different. Remember to check f(a) separately; continuity is not automatic.
Memory trick: Limit ≠ f(a) unless continuous
“A river flowing without a dam represents a continuous function; a dam causing water to jump is a discontinuity.”
A continuous function has no breaks: for every a, limₓ→a f(x) = f(a). Discontinuous functions show jumps, holes (removable), or vertical asymptotes. Even if the limit exists (hole), the function is still discontinuous. Identify the type to apply the right theorem.
Memory trick: Continuous = no jumps, holes, or asymptotes
“A GPS route may point to a road that ends abruptly; the path’s direction approaches a point (limit) even though the road isn’t there (f(a) missing).”
Limit equals f(a) only if the function is continuous at a. If f is undefined or jumps, the limit can still exist, but f(a) differs or does not exist. Exams often give a limit value and ask for f(a); verify continuity first.
Memory trick: Limit = f(a) ⇔ continuity at a
“Seeing a speedometer reading (function value) and assuming it’s the acceleration (derivative) is the classic mix-up.”
A derivative measures the instantaneous rate of change, not the function’s value. It is the limit of Δy/Δx as Δx→0, giving the tangent slope. Trap: plugging the point into the original function instead of the derivative, or confusing units.
Memory trick: Derivative = Δy/Δx limit.
“Turning a blender’s speed knob (inner) changes the blade’s rotation speed (outer) – the total effect is the product of both.”
The chain rule states d/dx f(g(x)) = f'(g(x))·g'(x). Differentiate the outer function, then multiply by the inner derivative. Forgetting g'(x) (e.g., writing d/dx sin 2x = cos 2x) loses a factor and costs marks.
Memory trick: Chain: outer × inner – never drop the inner derivative.
“When two cyclists ride side by side, the combined acceleration equals each rider’s acceleration added, not multiplied.”
For u(x)v(x), (uv)' = u'v + uv'. For u/v, (u/v)' = (u'v – uv')/v². The plus sign in the product rule and the squared denominator in the quotient rule are frequent sources of error.
Memory trick: Product: +; Quotient: – on top, v² bottom.
“A chef tasting two sauces thinks the flavor change is just the product of each sauce’s change, but the real change adds both contributions.”
A common slip is writing (uv)' = u'v' or (u/v)' = u'v'/v². The correct formulas add the missing terms: (uv)' = u'v + uv' and (u/v)' = (u'v – uv')/v². Missing a term gives a wrong answer.
Memory trick: Never multiply derivatives – add the missing term.
“Whisking batter: volume ∝ x³, the rate grows as 3·x².”
Derivative of xⁿ is n·x^(n-1) for any real constant n, provided x lies in the function’s domain. Works for fractions like √x (n=½) but only where x>0. Mistake: using the rule when n varies with x.
d/dx(xⁿ) = n·x^(n−1)
Memory trick: Only constant n – if n varies, stop.
“If a river’s speed varies with depth and depth varies with distance downstream, the overall speed change along the bank equals both rates multiplied.”
When y = f(u) and u = g(x), dy/dx = (dy/du)·(du/dx). The derivative passes through each intermediate variable, multiplying the rates. Mixing the order or omitting a factor leads to an incorrect chain.
dy/dx = dy/du · du/dx
Memory trick: Chain link: dy/du × du/dx.
“A car’s speedometer reads km/h (derivative); the tiny distance traveled in a second dt is dy = speed·dt, the differential.”
dy/dx is the instantaneous rate (slope). The differential dy = f'(x)·dx gives the predicted small change for a tiny increment dx. Students often treat dy as the same as the derivative; remember dy includes the factor dx.
Memory trick: dy = f'(x)·dx – derivative times tiny step.
“When a rectangle’s length and width both expand with time, its area changes as dA/dt = L·dW/dt + W·dL/dt.”
For y = u·v, dy/dx = u'v + uv' (product rule). For y = u/v, dy/dx = (u'v – uv')/v² (quotient rule). The denominator is always v²; missing the square or flipping the sign of the second term is a frequent exam error.
Memory trick: Quotient: “minus, then square” → (u'v – uv')/v²
“Turning the volume knob (outer function) changes the loudness, but the effect also depends on how fast the knob itself is turned (inner function).”
For a composite function y = f(g(x)), dy/dx = f'(g(x))·g'(x). Forgetting g'(x) drops a crucial factor, especially when g(x) is a power, trig or exponential inside another function.
Memory trick: Chain rule: outer × inner – link the chain!
“Knowing a road’s slope tells you direction, but you need the starting altitude to locate the exact point.”
Indefinite integration reverses differentiation, producing a family F(x)+C. The constant C is lost when differentiating, so it must be added back. Dropping +C gives an incomplete solution, especially in initial-value problems.
Memory trick: ∫ → always write +C.
“A recipe without a pinch of salt tastes flat; the missing ingredient changes the final dish.”
When finding antiderivatives, forgetting the +C omits one degree of freedom, leading to a wrong general solution. In differential equations, the missing constant makes the answer fail the given condition.
Memory trick: No +C = incomplete answer.