AP Calc BC Fall Final

What does lim_{x→c} f(x)=L mean?

As x gets close to c (from both sides), f(x) gets close to L. L must be a finite number.

Three common reasons a limit does not exist (DNE).

1) Left-hand limit ≠ right-hand limit. | 2) Unbounded behavior (→ ±∞). | 3) Oscillation (keeps bouncing).

When analyzing a piecewise limit at x=c, what must you show?

Compute left-hand and right-hand limits separately and check if they are equal. If equal, that common value is the limit.

First step for an analytic limit problem?

Direct substitution (plug in the value).

If direct substitution gives 0/0, what are common algebra moves?

Factor and cancel, multiply by conjugate (if radicals), get LCD, or simplify until cancellation appears.

Classic trig limit: lim_{x→0} sin(ax)/x = ?

a

Classic trig limit: lim_{x→0} sin(ax)/(ax) = ?

1

Trig limit: lim_{x→0} tan(x)/x = ? (idea)

1, because tan(x)/x = (sin(x)/x)·(1/cos(x)) → 1·1 = 1.

Approximation near 0: tan(kx) ≈ ?

kx (for x→0).

Limit: lim_{x→0} tan(ux)/tan(kx) = ?

u/k

Why does lim_{x→0} cos(x)/x DNE?

cos(x)→1 but x→0, so the ratio blows up (unbounded).

Continuity 3-step test at x=a.

1) f(a) exists. | 2) lim{x→a} f(x) exists (LHL = RHL). | 3) lim{x→a} f(x) = f(a).

Two types of discontinuities mentioned.

Removable (hole; can cancel) and nonremovable (asymptote/cusp/etc.; cannot be fixed by redefining a point).

Limit trick: lim_{x→a^-} |x-a|/(x-a) = ? and from right?

From left: −1. From right: +1. (Base form.)

Intermediate Value Theorem (IVT) statement (root form).

If f is continuous on [a,b] and f(a)<0<f(b) (or vice versa), then there exists c in (a,b) with f(c)=0.

Squeeze Theorem setup.

If f(x) ≤ g(x) ≤ h(x) near c and lim f = lim h = L, then lim g = L.

What is an indeterminate form? Give examples.

A form that doesn’t determine a limit by substitution alone: 0/0, ∞/∞, 0·∞, 1^∞, ∞−∞, 0^0, ∞^0.

For limits at infinity of rational functions, quickest method?

Compare degrees / horizontal asymptote idea: | deg(top)

Brute force for rational limits at infinity: what do you divide by?

Divide numerator and denominator by the highest power of x in the denominator.

Composite limit lim f(g(x)): what’s the key idea?

The inside g(x) output becomes the input (x-value) for f. Track whether it approaches from the left or right when needed.

Definition of derivative (difference quotient limit).

f′(x)=lim_{h→0} [f(x+h)−f(x)]/h

What does 'differentiable at a point' mean?

A derivative (slope) exists at that point.

Quick checklist to prove differentiability at x=a (as written in notes).

1) f is continuous at a. | 2) f′ is continuous at a (derivative exists/behaves).

Basic derivative rule: d/dx[c] = ?

0 (c constant).

Power rule.

d/dx[x^n] = n x^(n−1)

Constant multiple rule.

d/dx[c·u] = c·u′

Sum rule.

d/dx[u+v] = u′ + v′

Derivative of sin(x).

cos(x)

Derivative of cos(x).

−sin(x)

Derivative of tan(x).

sec^2(x)

Derivative of sec(x).

sec(x)tan(x)

Derivative of csc(x).

−csc(x)cot(x)

Derivative of cot(x).

−csc^2(x)

Product rule.

(fg)′ = f g′ + g f′

Quotient rule.

(f/g)′ = (g f′ − f g′)/g^2

Chain rule.

d/dx[f(g(x))] = f′(g(x))·g′(x)

Higher-order derivatives: what happens if the polynomial degree is less than the derivative order?

Eventually becomes 0 once you differentiate more times than the degree.

Cyclical derivatives example: if f(x)=sin x, what is f^(103)(x)?

−cos x (since derivatives repeat every 4).

Implicit differentiation key fact for y terms.

d/dx[y] = dy/dx; more generally d/dx[#·y] = #·dy/dx.

When finding a second derivative via implicit differentiation, what must you often do at the end?

Plug back using the original equation if needed to eliminate y′ / simplify in terms of x and y.

Tangent line concept: derivative equals what?

Slope of the tangent line at that point.

Sign rule in related rates: if a quantity is increasing, its rate is…

Positive (derivative > 0). If decreasing, negative.

Key reminder: derivative of an unchanging measurement is…

0 (e.g., ladder length fixed).

Standard related rates steps (5).

1) Draw picture. | 2) List known/unknown rates & quantities. | 3) Write relationship equation. | 4) Differentiate w.r.t. time. | 5) Substitute values and solve.

Pythagorean relationship used often (ladder/boat).

a^2 + b^2 = c^2

Area of rectangle (for related rates).

A = l·w

Area of triangle.

A = (1/2) b h

Circle area and circumference.

A = πr^2; C = 2πr

Cube volume and surface area (edge s).

V = s^3; S = 6s^2

Sphere volume and surface area (radius r).

V = (4/3)πr^3; A = 4πr^2

Cylinder volume.

V = πr^2 h

Cone volume and similar triangles relation.

V = (1/3)πr^2 h; r/h = R/H (constant ratio).

Net volume change model (inflow/outflow).

dV/dt = (rate in) − (rate out)

Angle model used in some related rates.

tan(θ) = y/x

Inverse proportionality model mentioned.

L = k/M (L inversely proportional to M).

Distance between two moving points model.

z^2 = x^2 + y^2

Calculator/rounding reminders from notes (keep it practical).

Use correct degree/radian mode; keep units; round to ~3 decimals at end (per notes).

Critical number vs. critical point.

Critical number: x-value where f′(x)=0 or undefined. | Critical point: (x, f(x)) for a critical number.

Where can absolute extrema occur on a closed interval?

Only at endpoints or critical numbers (then compare f-values).

What does f′(x) > 0 tell you?

f is increasing on that interval.

What does f′(x) < 0 tell you?

f is decreasing on that interval.

First Derivative Test: f′ changes +→− at c means…

Relative maximum at x=c.

First Derivative Test: f′ changes −→+ at c means…

Relative minimum at x=c.

Concavity test using second derivative.

f″(x)>0 → concave up; f″(x)<0 → concave down.

What does f″(x)=0 indicate?

PPOI: possible point of inflection (must check for concavity change).

Point of inflection must satisfy what?

Concavity changes (up↔down) at that x, and the point is on the function (use (x,y)).

Second Derivative Test for extrema at critical number c (when f′(c)=0).

If f″(c)>0 → relative min; if f″(c)<0 → relative max; if f″(c)=0 → test fails.

Rolle’s Theorem statement (core conditions + conclusion).

If f is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then ∃c in (a,b) with f′(c)=0.

Mean Value Theorem statement (core formula).

If f is continuous on [a,b] and differentiable on (a,b), then ∃c in (a,b): f′(c) = [f(b)−f(a)]/(b−a).

MVT caution when solving for c.

Reject any c not in the open interval (a,b). Endpoints don’t count.

Interpreting derivative graphs: what can area under f′ tell you about f?

Signed area under f′ over an interval gives net change in f (increase if positive, decrease if negative).

Derivative sketches: From f to f′, what maps to what?

Slope of f → y-values of f′. Zero slope on f → x-intercepts of f′. Sharp corners/cusps → f′ undefined (holes/asymptotes).

Derivative sketches: From f′ to f, what’s a key limitation?

You can’t know vertical shift (+C) without an initial point/value for f.

Curve dissection workflow (high-level).

Analyze f for intercepts; analyze f′ for inc/dec & relative extrema; analyze f″ for concavity & inflection points; then sketch with interval notes.

Optimization: first step idea.

Use the constraint (perimeter/area/volume/cost) to write everything in one variable.

Optimization critical step.

Optimize F(x): solve F′(x)=0 (and check endpoints/second derivative).

Rectangle formulas (A, P).

A = L·W; P = 2L + 2W.

Isosceles triangle: height in terms of side s and base b.

h = sqrt(s^2 − (b/2)^2)

Circle area and circumference (optimization use).

A = πr^2; C = 2πr.

Cone: volume and slant height.

V = (1/3)πr^2h; ℓ = sqrt(r^2 + h^2).

Cylinder: closed surface area and volume.

V = πr^2h; A_closed = 2πr^2 + 2πrh.

Square-base box: volume and surface area parts.

V = x^2h; top+bottom area = 2x^2; side area = 4xh.

Profit/revenue model.

R(q) = p(q)·q (revenue); often maximize profit = revenue − cost.

Cost modeling idea (general).

Total cost = Σ (rate)·(length or area) for each material/region.

Left Riemann sum: which y-values do you use?

Use function values at left endpoints of each subinterval.

Right Riemann sum: which y-values do you use?

Use function values at right endpoints.

Midpoint Riemann sum: which y-values do you use?

Use function values at midpoints of each subinterval.

Trapezoidal Rule idea (weights).

Endpoints counted once; interior y-values counted twice (then multiply by Δx/2).

Don’t assume Δx is always constant — what’s the warning?

Some problems use non-uniform subinterval widths, so you must use the given Δx for each piece.

Basic power integral (n≠−1).

∫ x^n dx = x^(n+1)/(n+1) + C

∫ cos x dx = ?

sin x + C

∫ sin x dx = ?

−cos x + C

∫ sec^2 x dx = ?

tan x + C

∫ sec x tan x dx = ?

sec x + C

∫ csc^2 x dx = ?

−cot x + C

∫ csc x cot x dx = ?

−csc x + C

Fundamental Theorem of Calculus (FTC) evaluation rule.

If F is an antiderivative of f, then ∫_a^b f(x)dx = F(b) − F(a).

Integral property: constant multiple.

∫a^b c f(x) dx = c ∫a^b f(x) dx (c constant).

Integral property: add/subtract functions.

∫a^b [f(x) ± g(x)] dx = ∫a^b f(x) dx ± ∫_a^b g(x) dx

Integral property: reversing bounds.

∫b^a f(x) dx = −∫a^b f(x) dx