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What a Limit Means

A limit describes the value f(x) approaches as x gets close to a number, not necessarily the value at that number.
  • lim_{x→a} f(x) = L means f(x) can be made as close to L as we like by taking x close enough to a (x ≠ a)
  • The limit exists only if the left-hand and right-hand limits both exist and are equal
  • A limit can exist even when f(a) is undefined, e.g. a hole in the graph
  • Infinite limits (vertical asymptotes) and limits at infinity (horizontal asymptotes) describe unbounded or end behavior
  • Estimate limits from graphs and tables, then confirm algebraically

Evaluating Limits Algebraically

Most limits come from direct substitution or a little algebra to remove a 0/0 form.
  • If f is continuous at a, lim_{x→a} f(x) = f(a): just substitute
  • 0/0 is indeterminate: factor and cancel, rationalize, or combine fractions, then substitute
  • Special limits: lim_{x→0} sin x / x = 1 and lim_{x→0} (1 − cos x)/x = 0
  • Limits at infinity of rational functions: compare the degrees of numerator and denominator
  • The Squeeze Theorem: if g ≤ f ≤ h and g and h share a limit L, then f → L too

Continuity and the IVT

A function is continuous at a point when there is no break, jump or hole there.
  • f is continuous at a if f(a) is defined, lim_{x→a} f(x) exists, and the two are equal
  • Discontinuities: removable (hole), jump (one-sided limits differ), infinite (vertical asymptote)
  • Polynomials, rational functions (on their domains), exponentials, logs and trig functions are continuous where defined
  • Intermediate Value Theorem: if f is continuous on [a, b], it takes every value between f(a) and f(b)
  • The IVT guarantees a root exists when a continuous function changes sign on an interval
0/0 doesn't mean the limit is 0 or undefined. It means you need more algebra.
f(a) and lim_{x→a} f(x) can differ; the limit ignores the value at a.
The IVT says a value exists, not where it is or how many times it's hit.

The Derivative as a Limit

The derivative measures instantaneous rate of change: the slope of the tangent line.
  • f'(x) = lim_{h→0} [f(x + h) − f(x)] / h
  • Average rate of change on [a, b] is [f(b) − f(a)]/(b − a), the slope of a secant line
  • The derivative at a point is the slope of the tangent line there
  • Differentiable implies continuous, but continuous doesn't imply differentiable (corners, cusps, vertical tangents)
  • Recognize a limit that IS a derivative: lim_{h→0} [(2 + h)³ − 8]/h = derivative of x³ at x = 2 = 12

Basic Rules

A handful of rules handle most derivatives without using the limit definition.
  • Power rule: d/dx xⁿ = n·xⁿ⁻¹; constants differentiate to 0
  • Sum and constant-multiple rules let you differentiate term by term
  • d/dx sin x = cos x, d/dx cos x = −sin x, d/dx eˣ = eˣ, d/dx ln x = 1/x
  • Product rule: (fg)' = f'g + fg'
  • Quotient rule: (f/g)' = (f'g − fg')/g²

Tangent Lines

The derivative turns into an equation for the line that best fits the curve at a point.
  • Tangent line at x = a: y − f(a) = f'(a)(x − a)
  • A horizontal tangent occurs where f'(x) = 0
  • The normal line is perpendicular to the tangent: slope −1/f'(a)
  • Local linearization: f(x) ≈ f(a) + f'(a)(x − a) for x near a
  • More trig derivatives: d/dx tan x = sec² x, d/dx sec x = sec x tan x

The Chain Rule

Differentiate compositions from the outside in.
  • d/dx f(g(x)) = f'(g(x)) · g'(x)
  • d/dx [g(x)]ⁿ = n[g(x)]ⁿ⁻¹ g'(x)
  • d/dx e^{g(x)} = e^{g(x)} g'(x); d/dx ln g(x) = g'(x)/g(x)
  • d/dx sin(g(x)) = cos(g(x)) g'(x)
  • Nested compositions: apply the chain rule once per layer
(fg)' is not f'g'. Use the product rule.
Quotient rule order matters: the numerator is f'g − fg', not fg' − f'g.
A function can be continuous at a point where it isn't differentiable, like |x| at 0.
Don't forget the inner derivative: d/dx sin(3x) = 3cos(3x), not cos(3x).

Accumulation and Riemann Sums

A definite integral adds up a rate over an interval to give total change.
  • ∫_a^b f(x) dx is the signed area between f and the x-axis; area below the axis counts as negative
  • Left, right and midpoint Riemann sums approximate it with rectangles; the trapezoidal rule uses trapezoids
  • For an increasing f, a left sum underestimates and a right sum overestimates
  • The integral of a rate gives net change: ∫_a^b r(t) dt = amount gained from a to b
  • Properties: ∫_a^a f = 0, ∫_b^a f = −∫_a^b f, and integrals over adjacent intervals add

The Fundamental Theorem of Calculus

Differentiation and integration undo each other.
  • FTC part 2: ∫_a^b f(x) dx = F(b) − F(a), where F' = f
  • FTC part 1: d/dx ∫_a^x f(t) dt = f(x)
  • With a variable upper limit: d/dx ∫_a^{g(x)} f(t) dt = f(g(x)) g'(x)
  • Accumulation function: g(x) = g(a) + ∫_a^x g'(t) dt
  • The FTC turns area and total-change questions into antiderivative problems

Antiderivatives and u-Substitution

Finding antiderivatives reverses the derivative rules.
  • ∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C (n ≠ −1); ∫ 1/x dx = ln|x| + C
  • ∫ eˣ dx = eˣ + C; ∫ cos x dx = sin x + C; ∫ sin x dx = −cos x + C
  • u-substitution reverses the chain rule: let u be the inner function and replace du = u'(x) dx
  • For definite integrals, change the limits to u-values or substitute back before evaluating
  • Always include + C for indefinite integrals

Differential Equations and Slope Fields

A differential equation relates a function to its derivatives; a slope field sketches its solutions.
  • dy/dx = f(x, y) gives the slope of a solution curve at every point
  • A slope field draws short segments with those slopes at grid points
  • Solution curves follow the segments like currents in a stream
  • Verify a solution by substituting it (and its derivative) into the equation
  • An initial condition like y(0) = 2 picks out one particular solution
The definite integral is net (signed) area. Total area needs absolute values.
Don't forget + C on indefinite integrals.
With u-substitution in a definite integral, change the limits too.
Add the constant of integration before solving for y, not after exponentiating incorrectly.

Average Value and Motion

Integrals turn rates into totals and averages.
  • Average value of f on [a, b]: (1/(b − a)) ∫_a^b f(x) dx
  • Displacement on [a, b] = ∫_a^b v(t) dt
  • Total distance traveled = ∫_a^b |v(t)| dt
  • Position: s(b) = s(a) + ∫_a^b v(t) dt
  • Units: integrating a rate multiplies its units by the input's units (m/s × s = m)

Area Between Curves

Subtract the lower curve from the upper curve and integrate.
  • Area = ∫_a^b [top(x) − bottom(x)] dx
  • Find intersection points to get the limits of integration
  • If the curves cross, split the integral where top and bottom switch
  • Integrating with respect to y: ∫ [right(y) − left(y)] dy
  • Area is always positive, so check which function is on top

Volumes

Slice a solid into thin pieces whose volumes you can add up.
  • Disc method: V = π ∫_a^b [R(x)]² dx
  • Washer method: V = π ∫_a^b ([R(x)]² − [r(x)]²) dx
  • Radii are distances from the axis of rotation, so adjust when rotating about lines like y = −1
  • Known cross sections: V = ∫_a^b A(x) dx, with A(x) the area of the cross section (square, semicircle, …)
  • Square cross sections with side s have A = s²; semicircles with diameter s have A = (π/8)s²
Displacement and total distance differ when velocity changes sign.
In the washer method it's R² − r², not (R − r)².
The average value of f is not the average of f(a) and f(b).

Parametric Equations

Parametric curves give x and y separately as functions of a parameter t.
  • dy/dx = (dy/dt)/(dx/dt), provided dx/dt ≠ 0
  • d²y/dx² = [d/dt (dy/dx)] / (dx/dt)
  • Horizontal tangent where dy/dt = 0 (dx/dt ≠ 0); vertical tangent where dx/dt = 0 (dy/dt ≠ 0)
  • Arc length: L = ∫_a^b √((dx/dt)² + (dy/dt)²) dt
  • Eliminate the parameter to identify the curve, e.g. x = cos t, y = sin t is the unit circle

Vector-Valued Motion

Position, velocity and acceleration can be written as vectors in the plane.
  • Position r(t) = ⟨x(t), y(t)⟩; velocity v(t) = ⟨x'(t), y'(t)⟩; acceleration a(t) = ⟨x''(t), y''(t)⟩
  • Speed = |v(t)| = √(x'(t)² + y'(t)²)
  • Displacement = ∫ v(t) dt componentwise; distance traveled = ∫ speed dt
  • Position at time b: r(b) = r(a) + ∫_a^b v(t) dt
  • A particle moves left when x'(t) < 0 and up when y'(t) > 0

Polar Curves

Polar coordinates describe points by distance r and angle θ.
  • x = r cos θ, y = r sin θ, r² = x² + y²
  • Area enclosed: A = (1/2) ∫_α^β r² dθ
  • Area between curves: (1/2) ∫ (r_outer² − r_inner²) dθ
  • Slope: dy/dx = (dy/dθ)/(dx/dθ) using x = r cos θ, y = r sin θ
  • Rose curves r = cos(nθ) have n petals if n is odd and 2n if n is even
Polar area uses (1/2)r², not r.
The second derivative for parametric curves divides by dx/dt again.
Speed is a scalar; velocity is a vector.

Sequences and Series

A series adds up the terms of a sequence; it converges if its partial sums approach a finite limit.
  • A sequence aₙ converges if lim_{n→∞} aₙ exists
  • A series Σaₙ converges if the partial sums Sₙ = a₁ + … + aₙ approach a limit
  • nth-term test: if lim aₙ ≠ 0, the series diverges (if it is 0, the test says nothing)
  • Geometric series Σ arⁿ converges to a/(1 − r) when |r| < 1 and diverges when |r| ≥ 1
  • Telescoping series collapse when consecutive terms cancel

Convergence Tests

Different series shapes call for different tests.
  • p-series Σ1/nᵖ converges if p > 1 and diverges if p ≤ 1 (the harmonic series Σ1/n diverges)
  • Integral test: for positive, decreasing f, Σf(n) and ∫_1^∞ f(x) dx both converge or both diverge
  • Comparison and limit comparison tests compare to a known series
  • Ratio test: L = lim |aₙ₊₁/aₙ|; L < 1 converges absolutely, L > 1 diverges, L = 1 inconclusive
  • Alternating series test: if |aₙ| decreases to 0, Σ(−1)ⁿaₙ converges

Absolute and Conditional Convergence

Some series converge only because their signs alternate.
  • Absolutely convergent: Σ|aₙ| converges (then Σaₙ converges too)
  • Conditionally convergent: Σaₙ converges but Σ|aₙ| diverges, e.g. the alternating harmonic series
  • Alternating series error bound: |S − Sₙ| ≤ |aₙ₊₁|
  • Choose tests by form: factorials or powers → ratio test; rational functions → p-series comparison
  • Always name the test and check its conditions when justifying
aₙ → 0 does not mean Σaₙ converges (harmonic series).
The ratio test is inconclusive when L = 1.
Conditional convergence needs both: Σaₙ converges and Σ|aₙ| diverges.

Taylor and Maclaurin Polynomials

Polynomials built from derivatives at one point approximate a function near that point.
  • Taylor polynomial about x = a: Pₙ(x) = Σ_{k=0}^n f^{(k)}(a)(x − a)ᵏ / k!
  • Maclaurin means a = 0
  • The coefficient of (x − a)ᵏ is f^{(k)}(a)/k!, so derivatives can be read off coefficients
  • Higher degree usually gives a better approximation near a
  • P₁ is the tangent line; P₂ also matches concavity

Common Maclaurin Series

A few series are worth memorizing and adapting.
  • eˣ = 1 + x + x²/2! + x³/3! + … (all x)
  • sin x = x − x³/3! + x⁵/5! − … (all x)
  • cos x = 1 − x²/2! + x⁴/4! − … (all x)
  • 1/(1 − x) = 1 + x + x² + x³ + … (|x| < 1)
  • ln(1 + x) = x − x²/2 + x³/3 − … (−1 < x ≤ 1)

Building New Series

Substitute, multiply, differentiate or integrate known series instead of computing derivatives.
  • Substitute: e^{−x²} = 1 − x² + x⁴/2! − …
  • Multiply by powers of x: x·sin x = x² − x⁴/3! + …
  • Differentiate or integrate term by term (the radius of convergence stays the same)
  • 1/(1 + x²) = 1 − x² + x⁴ − …, and integrating gives arctan x = x − x³/3 + x⁵/5 − …
  • Use series to evaluate limits, e.g. lim_{x→0} (sin x − x)/x³ = −1/6
Don't forget the factorial in each Taylor coefficient.
sin x has only odd powers; cos x has only even powers.
A series only equals its function inside the interval of convergence.

Power Series and Intervals of Convergence

A power series converges on an interval centered at its center.
  • Σcₙ(x − a)ⁿ converges for |x − a| < R, where R is the radius of convergence
  • Use the ratio test to find R, then test each endpoint separately
  • Endpoints can converge absolutely, conditionally, or diverge
  • Differentiating or integrating term by term keeps R (endpoints may change)
  • At the center x = a, every power series converges

Lagrange Error Bound

The Lagrange bound limits how far a Taylor polynomial can be from the function.
  • |f(x) − Pₙ(x)| ≤ M|x − a|ⁿ⁺¹/(n + 1)!
  • M is a bound for |f^{(n+1)}| between a and x
  • For sin x and cos x, M = 1 always works
  • Choose n large enough that the bound is below the required accuracy
  • The bound is a guarantee, not the actual error

Alternating Series Error and Choosing Approximations

For alternating series with decreasing terms, the next term bounds the error.
  • |S − Sₙ| ≤ |aₙ₊₁| when the terms alternate, decrease in size and approach 0
  • The sign of the first omitted term tells whether the partial sum over- or underestimates
  • Example: sin(0.1) ≈ 0.1 − 0.1³/6 with error < 0.1⁵/120
  • Use the alternating bound when the series alternates; otherwise use Lagrange
  • Approximating integrals: integrate the Taylor polynomial, then bound the error
The ratio test never decides the endpoints. Test them separately.
In Lagrange, M bounds the (n + 1)th derivative, not the nth.
The alternating bound needs decreasing terms.
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Unit 1: Limits & Continuity

Limit
lim_{x→a} f(x) = L: f(x) gets arbitrarily close to L as x approaches a (x ≠ a).
One-sided limits
Approach from the left (x→a⁻) or right (x→a⁺); the limit exists only if both agree.
Continuity at a
f(a) is defined, lim_{x→a} f(x) exists, and lim_{x→a} f(x) = f(a).
Removable discontinuity
A hole: the limit exists but doesn't equal f(a) (or f(a) is undefined).
Intermediate Value Theorem
Continuous f on [a, b] takes every value between f(a) and f(b).
Squeeze Theorem
If g ≤ f ≤ h near a and g, h → L, then f → L.

Unit 2: Differentiation & Applications

Derivative (definition)
f'(x) = lim_{h→0} [f(x + h) − f(x)]/h; instantaneous rate of change.
Power rule
d/dx xⁿ = n xⁿ⁻¹.
Product rule
(fg)' = f'g + fg'.
Quotient rule
(f/g)' = (f'g − fg')/g².
Tangent line
y − f(a) = f'(a)(x − a).
Differentiability
Implies continuity; fails at corners, cusps, vertical tangents and discontinuities.
Chain rule
d/dx f(g(x)) = f'(g(x))·g'(x).
Implicit differentiation
Differentiate both sides with respect to x, treating y as y(x); solve for dy/dx.
Inverse function derivative
(f⁻¹)'(b) = 1/f'(a), where f(a) = b.
d/dx ln u
u'/u.
d/dx arctan u
u'/(1 + u²).
d/dx aˣ
aˣ ln a.
Mean Value Theorem
Continuous on [a, b] and differentiable on (a, b) → f'(c) = [f(b) − f(a)]/(b − a) for some c.
Extreme Value Theorem
Continuous on [a, b] → f has an absolute max and min there.
Critical point
Where f'(x) = 0 or f' is undefined (f defined).
First Derivative Test
f' + to − → relative max; − to + → relative min.
Concavity
f'' > 0 concave up; f'' < 0 concave down; sign change = inflection point.
Candidates Test
Absolute extrema on [a, b]: evaluate f at critical points and endpoints.

Unit 3: Integration & Differential Equations

Definite integral
∫_a^b f(x) dx: signed area; the net accumulation of f over [a, b].
FTC part 2
∫_a^b f(x) dx = F(b) − F(a), where F' = f.
FTC part 1
d/dx ∫_a^x f(t) dt = f(x).
u-substitution
Let u = inner function, du = u' dx; reverses the chain rule.
Riemann sum
Sum of f(xᵢ)Δx using left, right or midpoint heights.
Net change theorem
∫_a^b F'(t) dt = F(b) − F(a).
Differential equation
An equation involving a function and its derivatives.
Slope field
Short segments showing dy/dx at grid points; solutions follow them.
Separation of variables
Rewrite as h(y) dy = g(x) dx, integrate both sides, apply the initial condition.
Exponential model
dy/dt = ky → y = y₀e^{kt}.
Newton's Law of Cooling
dT/dt = k(T − T_room).
Logistic model
dP/dt = kP(1 − P/L); approaches carrying capacity L, fastest growth at L/2.

Unit 4: Applications of Integration

Average value
(1/(b − a)) ∫_a^b f(x) dx.
Displacement vs. distance
∫ v dt vs. ∫ |v| dt.
Area between curves
∫ (top − bottom) dx or ∫ (right − left) dy.
Disc method
V = π ∫ R(x)² dx.
Washer method
V = π ∫ (R² − r²) dx.
Known cross sections
V = ∫ A(x) dx, A = cross-section area.

Unit 5: Parametric, Polar & Vector Functions

Parametric derivative
dy/dx = (dy/dt)/(dx/dt).
Parametric second derivative
d²y/dx² = [d/dt(dy/dx)]/(dx/dt).
Speed (vector motion)
|v(t)| = √(x'(t)² + y'(t)²).
Polar ↔ rectangular
x = r cos θ, y = r sin θ, r² = x² + y².
Polar area
A = (1/2)∫_α^β r² dθ.
Arc length (parametric)
∫_a^b √((dx/dt)² + (dy/dt)²) dt.

Unit 6: Infinite Sequences & Series

Geometric series
Σarⁿ converges to a/(1 − r) when |r| < 1.
p-series
Σ1/nᵖ converges iff p > 1.
nth-term test
If lim aₙ ≠ 0, Σaₙ diverges. Can't prove convergence.
Ratio test
L = lim|aₙ₊₁/aₙ|: < 1 converges, > 1 diverges, = 1 inconclusive.
Alternating series test
|aₙ| decreasing to 0 → Σ(−1)ⁿaₙ converges; error ≤ next term.
Conditional convergence
Σaₙ converges but Σ|aₙ| diverges.

Unit 7: Taylor & Maclaurin Series

Taylor polynomial
Pₙ(x) = Σ f^{(k)}(a)(x − a)ᵏ/k!.
Maclaurin eˣ
1 + x + x²/2! + x³/3! + …
Maclaurin sin x
x − x³/3! + x⁵/5! − …
Maclaurin cos x
1 − x²/2! + x⁴/4! − …
Maclaurin 1/(1 − x)
1 + x + x² + … for |x| < 1.
Reading derivatives
f^{(k)}(a) = k! × (coefficient of (x − a)ᵏ).

Unit 8: Polynomial Approximations & Series Convergence

Radius of convergence
R such that the series converges for |x − a| < R; found with the ratio test.
Interval of convergence
(a − R, a + R) plus whichever endpoints converge.
Lagrange error bound
|f(x) − Pₙ(x)| ≤ M|x − a|ⁿ⁺¹/(n + 1)!.
Alternating series error
|S − Sₙ| ≤ |aₙ₊₁|.
Term-by-term calculus
Differentiating or integrating keeps the radius.
Center
A power series always converges at x = a.
Press 1–4 to answer · Enter for next

Unit 1: Limits & Continuity

What a Limit Means
A limit describes the value f(x) approaches as x gets close to a number, not necessarily the value at that number.
Evaluating Limits Algebraically
Most limits come from direct substitution or a little algebra to remove a 0/0 form.
Continuity and the IVT
A function is continuous at a point when there is no break, jump or hole there.
Key fact
A limit exists only when the left- and right-hand limits agree.
Key fact
lim_{x→0} sin x / x = 1.
Key fact
Continuity at a: f(a) defined, the limit exists, and they're equal.
Key fact
IVT needs continuity on a closed interval.

Unit 2: Differentiation & Applications

The Derivative as a Limit
The derivative measures instantaneous rate of change: the slope of the tangent line.
Basic Rules
A handful of rules handle most derivatives without using the limit definition.
Tangent Lines
The derivative turns into an equation for the line that best fits the curve at a point.
The Chain Rule
Differentiate compositions from the outside in.
Key fact
f'(x) = lim_{h→0} [f(x + h) − f(x)]/h.
Key fact
Power rule: d/dx xⁿ = n xⁿ⁻¹.
Key fact
Product: f'g + fg'. Quotient: (f'g − fg')/g².
Key fact
Tangent line: y − f(a) = f'(a)(x − a).
Key fact
Chain rule: (f∘g)' = f'(g(x))·g'(x).

Unit 3: Integration & Differential Equations

Accumulation and Riemann Sums
A definite integral adds up a rate over an interval to give total change.
The Fundamental Theorem of Calculus
Differentiation and integration undo each other.
Antiderivatives and u-Substitution
Finding antiderivatives reverses the derivative rules.
Differential Equations and Slope Fields
A differential equation relates a function to its derivatives; a slope field sketches its solutions.
Key fact
∫_a^b f(x) dx = F(b) − F(a).
Key fact
d/dx ∫_a^x f(t) dt = f(x).
Key fact
∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C, n ≠ −1.
Key fact
The integral of a rate is net change.
Key fact
dy/dt = ky → y = y₀e^{kt}.

Unit 4: Applications of Integration

Average Value and Motion
Integrals turn rates into totals and averages.
Area Between Curves
Subtract the lower curve from the upper curve and integrate.
Volumes
Slice a solid into thin pieces whose volumes you can add up.
Key fact
Average value = (1/(b − a)) ∫_a^b f(x) dx.
Key fact
Total distance = ∫ |v(t)| dt; displacement = ∫ v(t) dt.
Key fact
Area = ∫ (top − bottom) dx.
Key fact
Washer: V = π ∫ (R² − r²) dx.

Unit 5: Parametric, Polar & Vector Functions

Parametric Equations
Parametric curves give x and y separately as functions of a parameter t.
Vector-Valued Motion
Position, velocity and acceleration can be written as vectors in the plane.
Polar Curves
Polar coordinates describe points by distance r and angle θ.
Key fact
Parametric slope: dy/dx = (dy/dt)/(dx/dt).
Key fact
Speed = √(x'² + y'²).
Key fact
Polar area = (1/2)∫ r² dθ.
Key fact
Arc length = ∫ √(x'² + y'²) dt.

Unit 6: Infinite Sequences & Series

Sequences and Series
A series adds up the terms of a sequence; it converges if its partial sums approach a finite limit.
Convergence Tests
Different series shapes call for different tests.
Absolute and Conditional Convergence
Some series converge only because their signs alternate.
Key fact
Geometric: Σarⁿ = a/(1 − r) if |r| < 1.
Key fact
p-series converges iff p > 1.
Key fact
nth-term test can only prove divergence.
Key fact
Alternating error ≤ first omitted term.

Unit 7: Taylor & Maclaurin Series

Taylor and Maclaurin Polynomials
Polynomials built from derivatives at one point approximate a function near that point.
Common Maclaurin Series
A few series are worth memorizing and adapting.
Building New Series
Substitute, multiply, differentiate or integrate known series instead of computing derivatives.
Key fact
Coefficient of (x − a)ᵏ = f^{(k)}(a)/k!.
Key fact
eˣ = Σxⁿ/n!.
Key fact
sin x = Σ(−1)ⁿx^{2n+1}/(2n + 1)!.
Key fact
1/(1 − x) = Σxⁿ for |x| < 1.

Unit 8: Polynomial Approximations & Series Convergence

Power Series and Intervals of Convergence
A power series converges on an interval centered at its center.
Lagrange Error Bound
The Lagrange bound limits how far a Taylor polynomial can be from the function.
Alternating Series Error and Choosing Approximations
For alternating series with decreasing terms, the next term bounds the error.
Key fact
Radius from the ratio test; always check endpoints separately.
Key fact
Lagrange: |error| ≤ M|x − a|ⁿ⁺¹/(n + 1)!.
Key fact
Alternating error ≤ first omitted term.
Key fact
Every power series converges at its center.
Common mistakes for each unit — read the mistake, then make sure you know why it's wrong.

Unit 1: Limits & Continuity

Watch out
0/0 doesn't mean the limit is 0 or undefined. It means you need more algebra.
Watch out
f(a) and lim_{x→a} f(x) can differ; the limit ignores the value at a.
Watch out
The IVT says a value exists, not where it is or how many times it's hit.

Unit 2: Differentiation & Applications

Watch out
(fg)' is not f'g'. Use the product rule.
Watch out
Quotient rule order matters: the numerator is f'g − fg', not fg' − f'g.
Watch out
A function can be continuous at a point where it isn't differentiable, like |x| at 0.
Watch out
Don't forget the inner derivative: d/dx sin(3x) = 3cos(3x), not cos(3x).

Unit 3: Integration & Differential Equations

Watch out
The definite integral is net (signed) area. Total area needs absolute values.
Watch out
Don't forget + C on indefinite integrals.
Watch out
With u-substitution in a definite integral, change the limits too.
Watch out
Add the constant of integration before solving for y, not after exponentiating incorrectly.

Unit 4: Applications of Integration

Watch out
Displacement and total distance differ when velocity changes sign.
Watch out
In the washer method it's R² − r², not (R − r)².
Watch out
The average value of f is not the average of f(a) and f(b).

Unit 5: Parametric, Polar & Vector Functions

Watch out
Polar area uses (1/2)r², not r.
Watch out
The second derivative for parametric curves divides by dx/dt again.
Watch out
Speed is a scalar; velocity is a vector.

Unit 6: Infinite Sequences & Series

Watch out
aₙ → 0 does not mean Σaₙ converges (harmonic series).
Watch out
The ratio test is inconclusive when L = 1.
Watch out
Conditional convergence needs both: Σaₙ converges and Σ|aₙ| diverges.

Unit 7: Taylor & Maclaurin Series

Watch out
Don't forget the factorial in each Taylor coefficient.
Watch out
sin x has only odd powers; cos x has only even powers.
Watch out
A series only equals its function inside the interval of convergence.

Unit 8: Polynomial Approximations & Series Convergence

Watch out
The ratio test never decides the endpoints. Test them separately.
Watch out
In Lagrange, M bounds the (n + 1)th derivative, not the nth.
Watch out
The alternating bound needs decreasing terms.