(x−h)²+(y−k)² = r²

“Satellite dishes are parabolas because a parabola focuses every incoming signal to one point.”

Equation of a circle with centre (h,k), radius r. Use it when any circle.

(x−h)²+(y−k)² = r²

y² = 4ax

“Satellite dishes are parabolas because a parabola focuses every incoming signal to one point.”

Standard parabola opening rightward. Use it when vertex at the origin.

y² = 4ax

ellipse vs hyperbola

“Satellite dishes are parabolas because a parabola focuses every incoming signal to one point.”

An ellipse is a closed curve with e<1 (sum of focal distances constant); a hyperbola is an open two-branch curve with e>1 (difference of focal distances constant).

parabola vs hyperbola

“Satellite dishes are parabolas because a parabola focuses every incoming signal to one point.”

A parabola has eccentricity exactly 1 and one focus; a hyperbola has e>1 and two branches with two foci.

Myth: Eccentricity mix-up

“Satellite dishes are parabolas because a parabola focuses every incoming signal to one point.”

Remember: circle e=0, ellipse 0<e<1, parabola e=1, hyperbola e>1.

Limit — common mistake

“A limit is how mathematics talks about 'getting infinitely close' without ever arriving.”

A frequent error is assuming the limit always equals the function's value at that point. In reality, the value a function approaches as the input nears a point, even if it isn't defined there.

Memory trick: a limit can exist where f(a) doesn't — it's about approaching, not arriving.

Continuity — common mistake

“A limit is how mathematics talks about 'getting infinitely close' without ever arriving.”

A frequent error is thinking every function is continuous everywhere. In reality, a function is continuous at a point if its limit there equals its actual value (no breaks or jumps).

Memory trick: continuous means you can draw it without lifting your pen.

Standard limit

“A limit is how mathematics talks about 'getting infinitely close' without ever arriving.”

Lim(x→0) sin x / x = 1 is a key result used everywhere.

Memory trick: 0/0 is indeterminate — use standard limits or L'Hôpital's rule.

Standard limit — common mistake

“A limit is how mathematics talks about 'getting infinitely close' without ever arriving.”

A frequent error is evaluating it as 0/0 and stopping. In reality, lim(x→0) sin x / x = 1 is a key result used everywhere.

Memory trick: 0/0 is indeterminate — use standard limits or L'Hôpital's rule.

lim(x→0) sin x / x = 1

“A limit is how mathematics talks about 'getting infinitely close' without ever arriving.”

A fundamental trigonometric limit. Use it when x measured in radians.

lim(x→0) sin x / x = 1

limit vs value of a function

“A limit is how mathematics talks about 'getting infinitely close' without ever arriving.”

The limit is what f(x) approaches near a point; the value is what f actually equals there — they can differ at a discontinuity.

continuous function vs discontinuous function

“A limit is how mathematics talks about 'getting infinitely close' without ever arriving.”

A continuous function has no breaks (limit = value everywhere); a discontinuous one has jumps, holes, or asymptotes.

Myth: Limit equals f(a) always

“A limit is how mathematics talks about 'getting infinitely close' without ever arriving.”

That's only true when f is continuous at a; limits can exist where the function is undefined.

Derivative — common mistake

“Your car's speedometer is a live derivative — the rate your distance is changing.”

A frequent error is confusing the derivative (a rate) with the function's value. In reality, the instantaneous rate of change of a function — the slope of the tangent at a point.

Memory trick: derivative = how fast the output changes as the input changes.

Chain rule — common mistake

“Your car's speedometer is a live derivative — the rate your distance is changing.”

A frequent error is forgetting to multiply by the inner derivative. In reality, to differentiate a composite function, differentiate the outer function and multiply by the derivative of the inner.

Memory trick: for sin(2x), it's cos(2x)×2 — don't drop the ×2.

Product and quotient rules

“Your car's speedometer is a live derivative — the rate your distance is changing.”

The derivative of a product is u'v+uv'; of a quotient is (u'v−uv')/v².

Memory trick: the derivative of a product is NOT the product of derivatives.

Product and quotient rules — common mistake

“Your car's speedometer is a live derivative — the rate your distance is changing.”

A frequent error is differentiating a product as just u'v'. In reality, the derivative of a product is u'v+uv'; of a quotient is (u'v−uv')/v².

Memory trick: the derivative of a product is NOT the product of derivatives.

d/dx(xⁿ) = n·x^(n−1)

“Your car's speedometer is a live derivative — the rate your distance is changing.”

The power rule. Use it when any real exponent n.

d/dx(xⁿ) = n·x^(n−1)

dy/dx = dy/du · du/dx

“Your car's speedometer is a live derivative — the rate your distance is changing.”

The chain rule for composite functions. Use it when y is a function of u, u of x.

dy/dx = dy/du · du/dx

derivative vs differential

“Your car's speedometer is a live derivative — the rate your distance is changing.”

The derivative dy/dx is a rate (slope); the differential dy = f'(x)dx is the small change it predicts for a small dx.

product rule vs quotient rule

“Your car's speedometer is a live derivative — the rate your distance is changing.”

The product rule differentiates u·v as u'v+uv'; the quotient rule differentiates u/v as (u'v−uv')/v².

Myth: Forgetting the chain rule

“Your car's speedometer is a live derivative — the rate your distance is changing.”

For composite functions you must multiply by the derivative of the inner function.

Antiderivative

“Integration adds up infinitely many slivers — it's how we find the area of any curved shape.”

Integration reverses differentiation; an indefinite integral always carries a constant of integration +C.

Memory trick: no +C means an incomplete indefinite integral.

Antiderivative — common mistake

“Integration adds up infinitely many slivers — it's how we find the area of any curved shape.”

A frequent error is forgetting to add +C to an indefinite integral. In reality, integration reverses differentiation; an indefinite integral always carries a constant of integration +C.

Memory trick: no +C means an incomplete indefinite integral.