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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
(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.

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

Implicit Differentiation

When y is tangled up with x, differentiate both sides and solve for dy/dx.
  • Treat y as a function of x: d/dx (y²) = 2y · dy/dx
  • Differentiate every term, collect dy/dx terms on one side, then factor and divide
  • For x² + y² = 25: 2x + 2y y' = 0, so y' = −x/y
  • Second derivatives: differentiate y' implicitly again and substitute y' back in
  • Useful when solving for y explicitly is hard or impossible

Inverse Functions and Inverse Trig

Derivatives of inverses come from the reciprocal of the original slope.
  • If g = f⁻¹, then g'(x) = 1 / f'(g(x))
  • If f(2) = 5 and f'(2) = 4, then (f⁻¹)'(5) = 1/4
  • d/dx arcsin x = 1/√(1 − x²); d/dx arctan x = 1/(1 + x²)
  • d/dx aˣ = aˣ ln a; d/dx log_a x = 1/(x ln a)
  • Combine with the chain rule: d/dx arctan(3x) = 3/(1 + 9x²)
Don't forget the inner derivative: d/dx sin(3x) = 3cos(3x), not cos(3x).
In implicit differentiation, every y term picks up a dy/dx.
For (f⁻¹)'(b), evaluate f' at a = f⁻¹(b), not at b.

Derivatives as Rates in Context

A derivative has units of 'output per input' and describes how fast a quantity changes.
  • If V(t) is volume in liters and t is minutes, V'(t) is in liters per minute
  • Interpret V'(5) = −2: at t = 5 minutes, the volume is decreasing at 2 liters per minute
  • Motion: velocity v(t) = s'(t), acceleration a(t) = v'(t) = s''(t)
  • Speed is |v(t)|; speed increases when v and a have the same sign
  • A particle is at rest when v = 0 and changes direction where v changes sign

Related Rates

When quantities are linked by an equation, their rates are linked by its derivative.
  • Draw a picture, name variables, and write an equation relating them
  • Differentiate both sides with respect to time t (chain rule on every variable)
  • Substitute known values only after differentiating
  • Common equations: Pythagorean theorem, similar triangles, area and volume formulas
  • Answer with units and a sign that matches increasing (+) or decreasing (−)

Linearization and L'Hôpital's Rule

Tangent lines approximate values; derivatives also resolve indeterminate limits.
  • L(x) = f(a) + f'(a)(x − a) approximates f near a
  • The approximation is an overestimate if f is concave down there, an underestimate if concave up
  • L'Hôpital's Rule: if a limit gives 0/0 or ∞/∞, lim f/g = lim f'/g' (when the latter exists)
  • Check the indeterminate form before applying it; it doesn't apply to forms like 1/0
  • Example: lim_{x→0} (eˣ − 1)/x = lim eˣ/1 = 1
Substituting numbers before differentiating in related rates freezes quantities that are changing.
Speed isn't velocity: speed is |v|.
L'Hôpital differentiates top and bottom separately, not as a quotient-rule derivative.

Mean Value and Extreme Value Theorems

Two existence theorems guarantee special points for continuous, differentiable functions.
  • MVT: if f is continuous on [a, b] and differentiable on (a, b), some c has f'(c) = [f(b) − f(a)]/(b − a)
  • Meaning: at some instant, the instantaneous rate equals the average rate
  • EVT: a continuous function on a closed interval has an absolute max and an absolute min
  • Candidates for absolute extrema: critical points inside the interval and the endpoints
  • Critical point: where f'(x) = 0 or f'(x) doesn't exist (and f is defined)

Increasing, Decreasing and Relative Extrema

The sign of f' tells you which way f is heading.
  • f' > 0 → f increasing; f' < 0 → f decreasing
  • First Derivative Test: f' changes + to − → relative max; − to + → relative min
  • Make a sign chart for f' using the critical points as dividers
  • Justify answers with the sign change of f', not just 'f' = 0'
  • A critical point without a sign change (like x³ at 0) isn't an extremum

Concavity and Optimization

The second derivative describes the bending of the graph and helps find maxima and minima.
  • f'' > 0 → concave up (f' increasing); f'' < 0 → concave down
  • Inflection point: where concavity changes (f'' changes sign)
  • Second Derivative Test: f'(c) = 0 and f''(c) < 0 → relative max; f''(c) > 0 → relative min
  • Optimization: write the quantity as a function of one variable, find critical points, check endpoints
  • Always confirm the answer is a max or min (sign chart, second derivative or endpoint comparison)
f'(c) = 0 alone doesn't guarantee an extremum. Check for a sign change.
f''(c) = 0 alone doesn't make an inflection point. Concavity must change.
Don't forget the endpoints when finding absolute extrema on a closed interval.

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
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.

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

Separation of Variables

When the variables can be separated, integrate each side on its own.
  • Rewrite dy/dx = g(x)h(y) as (1/h(y)) dy = g(x) dx
  • Integrate both sides, adding one constant C
  • Use the initial condition to find C, then solve for y if possible
  • Example: dy/dx = xy with y(0) = 3 gives y = 3e^{x²/2}
  • Check the domain: some solutions only exist on an interval containing the initial x

Exponential Models

When a quantity's rate of change is proportional to its size, it grows or decays exponentially.
  • dy/dt = ky has solution y = y₀e^{kt}
  • k > 0: growth; k < 0: decay
  • Half-life and doubling time come from solving e^{kt} = 1/2 or 2
  • Newton's Law of Cooling: dT/dt = k(T − T_room)
  • Logistic growth (BC): dP/dt = kP(1 − P/L); solutions approach the carrying capacity L, growing fastest at P = L/2
Add the constant of integration before solving for y, not after exponentiating incorrectly.
A slope field shows slopes, not the solution values themselves.
Don't separate dy/dx = x + y: it isn't separable.

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).
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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: Definition & Basic Rules

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.

Unit 3: Differentiation: Composite, Implicit & Inverse

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.

Unit 4: Contextual Applications of Differentiation

Velocity and acceleration
v(t) = s'(t); a(t) = v'(t) = s''(t).
Speeding up
When v and a have the same sign; slowing down when they differ.
Related rates steps
Picture → variables → equation → differentiate in t → substitute → answer with units.
Local linearization
L(x) = f(a) + f'(a)(x − a).
L'Hôpital's Rule
For 0/0 or ∞/∞: lim f/g = lim f'/g'.
Units of a derivative
Units of the output per unit of the input.

Unit 5: Analytical Applications of Differentiation

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 6: Integration & Accumulation of Change

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).

Unit 7: Differential Equations

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 8: 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.
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: Definition & Basic Rules

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.
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).

Unit 3: Differentiation: Composite, Implicit & Inverse

The Chain Rule
Differentiate compositions from the outside in.
Implicit Differentiation
When y is tangled up with x, differentiate both sides and solve for dy/dx.
Inverse Functions and Inverse Trig
Derivatives of inverses come from the reciprocal of the original slope.
Key fact
Chain rule: (f∘g)' = f'(g(x))·g'(x).
Key fact
d/dx ln g(x) = g'(x)/g(x).
Key fact
(f⁻¹)'(b) = 1/f'(a) where f(a) = b.
Key fact
d/dx arctan x = 1/(1 + x²).

Unit 4: Contextual Applications of Differentiation

Derivatives as Rates in Context
A derivative has units of 'output per input' and describes how fast a quantity changes.
Related Rates
When quantities are linked by an equation, their rates are linked by its derivative.
Linearization and L'Hôpital's Rule
Tangent lines approximate values; derivatives also resolve indeterminate limits.
Key fact
Units of f'(t): units of f per unit of t.
Key fact
v = s', a = v' = s''.
Key fact
Speeding up when v and a have the same sign.
Key fact
L'Hôpital: only for 0/0 or ∞/∞.

Unit 5: Analytical Applications of Differentiation

Mean Value and Extreme Value Theorems
Two existence theorems guarantee special points for continuous, differentiable functions.
Increasing, Decreasing and Relative Extrema
The sign of f' tells you which way f is heading.
Concavity and Optimization
The second derivative describes the bending of the graph and helps find maxima and minima.
Key fact
MVT: f'(c) = average rate of change on [a, b] for some c in (a, b).
Key fact
f' changes + → − : relative max; − → + : relative min.
Key fact
Inflection point: f'' changes sign.
Key fact
Absolute extrema on [a, b]: compare critical points and endpoints.

Unit 6: Integration & Accumulation of Change

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.
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.

Unit 7: Differential Equations

Differential Equations and Slope Fields
A differential equation relates a function to its derivatives; a slope field sketches its solutions.
Separation of Variables
When the variables can be separated, integrate each side on its own.
Exponential Models
When a quantity's rate of change is proportional to its size, it grows or decays exponentially.
Key fact
dy/dt = ky → y = y₀e^{kt}.
Key fact
Separate: (1/h(y)) dy = g(x) dx, then integrate both sides.
Key fact
Use the initial condition to find C.
Key fact
Logistic: growth fastest at half the carrying capacity.

Unit 8: 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.
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: Definition & Basic Rules

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.

Unit 3: Differentiation: Composite, Implicit & Inverse

Watch out
Don't forget the inner derivative: d/dx sin(3x) = 3cos(3x), not cos(3x).
Watch out
In implicit differentiation, every y term picks up a dy/dx.
Watch out
For (f⁻¹)'(b), evaluate f' at a = f⁻¹(b), not at b.

Unit 4: Contextual Applications of Differentiation

Watch out
Substituting numbers before differentiating in related rates freezes quantities that are changing.
Watch out
Speed isn't velocity: speed is |v|.
Watch out
L'Hôpital differentiates top and bottom separately, not as a quotient-rule derivative.

Unit 5: Analytical Applications of Differentiation

Watch out
f'(c) = 0 alone doesn't guarantee an extremum. Check for a sign change.
Watch out
f''(c) = 0 alone doesn't make an inflection point. Concavity must change.
Watch out
Don't forget the endpoints when finding absolute extrema on a closed interval.

Unit 6: Integration & Accumulation of Change

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.

Unit 7: Differential Equations

Watch out
Add the constant of integration before solving for y, not after exponentiating incorrectly.
Watch out
A slope field shows slopes, not the solution values themselves.
Watch out
Don't separate dy/dx = x + y: it isn't separable.

Unit 8: 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).