AP Calculus BC

01

Limits

L'Hôpital's Rule — 0/0 & ∞/∞ Forms

EASY

When a limit gives the indeterminate forms 0/0 or ∞/∞, take the derivative of numerator and denominator separately, then evaluate again.

📌 Worked Example

Find: lim(x→0) (sin 3x) / (5x)

Step 1 — Check form: sin(0)/0 = 0/0 ✓ → apply L'Hôpital

Step 2 — Differentiate top & bottom: 3cos(3x) / 5

Step 3 — Plug in x = 0: 3cos(0)/5 = 3/5

⚡ Common trap: Do NOT use the quotient rule here. L'Hôpital means differentiate numerator and denominator separately, not as a fraction product rule.

Memory Keys

INDETERMINATE = 0/0 or ∞/∞ TOP' / BOTTOM' (not quotient rule!) CHECK FORM FIRST REPEAT if still indeterminate

🎯 Your Turn — Practice

a) Find lim(x→0) (tan 2x) / (sin 7x)

b) Find lim(x→∞) (3x² + 1) / (x² − 5)

c) Find lim(x→0) (1 − cos x) / x²

02

Differentiation

Chain Rule — Trig & Exponential Compositions

EASY

The chain rule handles functions inside functions. Always identify the outer and inner function first, then multiply their derivatives.

[f(g(x))]' = f '(g(x)) · g'(x)

📌 Worked Example

Differentiate: y = sin(e^(3x))

Outer → sin( ), Inner → e^(3x)

dy/dx = cos(e^(3x)) · e^(3x) · 3 = 3e^(3x) cos(e^(3x))

⚡ Multi-layer chains: For y = sin²(3x), there are THREE layers: (·)², sin(·), 3x. Work outside → in.

Memory Keys

OUTSIDE → INSIDE COPY INNER (don't simplify yet) MULTIPLY by inner' PEEL the onion

🎯 Your Turn

a) Find dy/dx for y = cos⁴(x³)

b) Find dy/dx for y = e^(sin 2x)

c) Find dy/dx for y = ln(tan x)

03

Differentiation

Implicit Differentiation — Finding dy/dx

MEDIUM

When y is not isolated, differentiate both sides with respect to x. Every time you differentiate a y-term, multiply by dy/dx (chain rule on y).

📌 Worked Example

Find dy/dx: x³ + y³ = 6xy

Step 1 differentiate both sides: 3x² + 3y²·(dy/dx) = 6y + 6x·(dy/dx)

Step 2 collect dy/dx terms: 3y²(dy/dx) − 6x(dy/dx) = 6y − 3x²

Step 3 factor & solve: dy/dx = (6y − 3x²)/(3y² − 6x) = (2y−x²)/(y²−2x)

⚡ Forget dy/dx? Every y derivative NEEDS it. Missing even one ruins the answer.

Memory Keys

d/dx of y = dy/dx COLLECT dy/dx on one side FACTOR OUT dy/dx DIVIDE to isolate

🎯 Your Turn

a) Find dy/dx for x²y + y³ = 5

b) Find dy/dx for sin(xy) = x

c) Find the slope of x² + y² = 25 at the point (3, 4).

04

Integration — FTC

Fundamental Theorem of Calculus Part 1

MEDIUM

FTC Part 1 says: d/dx [∫ₐˣ f(t) dt] = f(x). But when the upper limit is a function g(x), chain rule applies!

d/dx [∫ₐ^(g(x)) f(t) dt] = f(g(x)) · g'(x)

📌 Worked Example

Find: d/dx [∫₀^(x²) sin(t) dt]

Step 1 Upper limit = x², so g(x) = x², g'(x) = 2x

Step 2 Apply formula: sin(x²) · 2x = 2x sin(x²)

⚡ Don't forget: Multiply by the derivative of the upper limit. Students almost always forget this step!

Memory Keys

PLUG IN upper limit × derivative of upper limit KILL the integral sign f(g(x)) · g'(x)

🎯 Your Turn

a) d/dx [∫₁^(x³) cos(t²) dt]

b) d/dx [∫₀^(√x) e^(t²) dt]

c) d/dx [∫_x^(x²) ln(t) dt] (Hint: split at a constant!)

05

Integration Techniques

Integration by Parts — LIATE Strategy

MEDIUM

Formula: ∫u dv = uv − ∫v du

Choose u using LIATE: Logarithmic → Inverse trig → Algebraic → Trig → Exponential. The first matching type is u.

📌 Worked Example

Compute: ∫ x·eˣ dx

LIATE: x is Algebraic (A), eˣ is Exponential (E) → pick u = x, dv = eˣ dx

du = dx, v = eˣ

= x·eˣ − ∫eˣ dx = xeˣ − eˣ + C

⚡ Cyclic trick: For ∫eˣsin(x)dx, do IBP twice, then add the original integral to both sides and solve.

Memory Keys

LIATE picks u uv MINUS ∫v du DIFFERENTIATE u INTEGRATE dv CYCLIC if eˣ·trig

🎯 Your Turn

a) ∫ x cos(x) dx

b) ∫ ln(x) dx

c) ∫ eˣ sin(x) dx

06

Integration Techniques

Partial Fraction Decomposition

TRICKY

Break a rational function with a factorable denominator into simpler fractions before integrating.

📌 Worked Example

Compute: ∫ 5/(x²−x−6) dx

Step 1 Factor denom: (x−3)(x+2)

Step 2 Set up: 5/[(x−3)(x+2)] = A/(x−3) + B/(x+2)

Step 3 Multiply through: 5 = A(x+2)+B(x−3). Plug x=3: A=1; x=−2: B=−1

Step 4 Integrate: ln|x−3| − ln|x+2| + C

⚡ Repeated factors: If the denominator has (x−2)², you need both A/(x−2) AND B/(x−2)² as separate terms.

Memory Keys

FACTOR denominator first ONE letter per factor COVER-UP trick for A,B RESULT → ln integrals

🎯 Your Turn

a) ∫ 3/(x²−4) dx

b) ∫ (x+1)/(x²+x−6) dx

07

Improper Integrals

Convergence & Divergence — Infinite Limits

MEDIUM

Replace ∞ with a limit variable t, evaluate the definite integral, then take the limit as t → ∞. If the limit exists = converges; if not = diverges.

📌 Worked Example

Does ∫₁^∞ (1/x²) dx converge?

= lim(t→∞) ∫₁ᵗ x⁻² dx = lim(t→∞) [−x⁻¹]₁ᵗ = lim(t→∞) (−1/t + 1) = 1 ✓ Converges

⚡ p-test shortcut: ∫₁^∞ (1/xᵖ) dx converges if p > 1, diverges if p ≤ 1. Memorize this!

Memory Keys

REPLACE ∞ with t EVALUATE → take limit p > 1 CONVERGES p ≤ 1 DIVERGES

🎯 Your Turn

a) Does ∫₁^∞ (1/x) dx converge or diverge?

b) Evaluate ∫₀^∞ e^(−3x) dx

c) Does ∫₁^∞ (1/x^(2/3)) dx converge?

08

Series

Geometric Series — Sum & Convergence

EASY

A geometric series Σ a·rⁿ (n=0 to ∞):

Sum = a / (1 − r), converges ONLY when |r| < 1

📌 Worked Example

Find the sum: Σ (2/3)ⁿ from n=0 to ∞

a = 1, r = 2/3. |r| = 2/3 < 1 → converges

Sum = 1/(1 − 2/3) = 1/(1/3) = 3

⚡ Index trap: If the series starts at n=1 instead of n=0, the first term a changes! Always identify the actual first term.

Memory Keys

|r| < 1 → converges SUM = a/(1−r) a = FIRST TERM CHECK INDEX (n=0 vs n=1)

🎯 Your Turn

a) Sum of Σ (1/4)ⁿ from n=0 to ∞

b) Does Σ (−3/2)ⁿ converge? If yes, find sum.

c) Write 0.777... as a fraction using geometric series.

09

Series — Convergence Tests

Ratio Test — Most Versatile Test

MEDIUM

Compute L = lim(n→∞) |a(n+1) / aₙ|

If L < 1 → converges absolutely | L > 1 → diverges | L = 1 → inconclusive

📌 Worked Example

Test: Σ n!/3ⁿ

|a(n+1)/aₙ| = [(n+1)!/3^(n+1)] · [3ⁿ/n!] = (n+1)/3 → ∞

L = ∞ > 1 → Diverges

⚡ L = 1 tells you nothing! This is the ratio test's blind spot. Try a different test (comparison, p-series, etc.).

Memory Keys

NEXT TERM / CURRENT TERM L < 1 → CONVERGES L > 1 → DIVERGES L = 1 → TRY ANOTHER TEST GREAT for n! or rⁿ

🎯 Your Turn

a) Test Σ n²/2ⁿ for convergence.

b) Test Σ (n!)²/(2n)! using Ratio Test.

10

Series — Taylor & Maclaurin

Taylor Series — Building the Polynomial

TRICKY

A Taylor series centered at a: f(x) = Σ f⁽ⁿ⁾(a)/n! · (x−a)ⁿ

Maclaurin = Taylor centered at a = 0. These four are the most tested — memorize them:

eˣ

Σ xⁿ/n!

sin x

Σ (−1)ⁿ x^(2n+1)/(2n+1)!

cos x

Σ (−1)ⁿ x^(2n)/(2n)!

1/(1−x)

Σ xⁿ, |x|<1

📌 Worked Example — Modify a known series

Find the Maclaurin series for f(x) = e^(−x²)

Start from eˣ = Σ xⁿ/n!, substitute x → −x²:

e^(−x²) = Σ (−x²)ⁿ/n! = Σ (−1)ⁿ x^(2n)/n!

Memory Keys

MEMORIZE Big 4 series SUBSTITUTE don't derive ALTERNATING (−1)ⁿ for sin/cos CENTER at a: replace x with (x−a)

🎯 Your Turn

a) Find the Maclaurin series for f(x) = sin(x²)

b) Find the first 3 non-zero terms of the Maclaurin series for cos(3x)

c) Find the Maclaurin series for 1/(1+x²)

11

Series

Lagrange Error Bound

TRICKY

The error of the nth-degree Taylor polynomial satisfies:

|Error| ≤ M · |x−a|^(n+1) / (n+1)!

where M is the maximum value of |f^(n+1)| on the interval between a and x.

📌 Worked Example

Use a 3rd-degree Maclaurin polynomial for sin(x) to estimate sin(0.1). Find the error bound.

f⁽⁴⁾(x) = sin(x) → |M| ≤ 1 on [0, 0.1]

|Error| ≤ 1 · (0.1)⁴ / 4! = 0.0001/24 ≈ 4.17 × 10⁻⁶

⚡ Off-by-one: For the nth-degree polynomial, you need the (n+1)th derivative for M, not the nth.

Memory Keys

NEXT derivative for M |x−a|^(n+1) / (n+1)! MAX of |f^(n+1)| = M BOUND not exact error

🎯 Your Turn

a) Use the 2nd-degree Maclaurin polynomial for eˣ to approximate e^(0.3). Give the Lagrange error bound.

12

Series — Power Series

Interval & Radius of Convergence

TRICKY

Use the Ratio Test to find where a power series converges. Set L < 1 and solve for x. Then check the endpoints separately!

📌 Worked Example

Find the IOC for Σ (x−2)ⁿ / (n·3ⁿ)

Ratio test: |(x−2)/3| · n/(n+1) → |x−2|/3 < 1 → |x−2| < 3

Radius = 3. Center = 2. Potential IOC: (−1, 5)

Check endpoints: x=−1 → Σ(−1)ⁿ/n → converges (alternating). x=5 → Σ1/n → diverges.

IOC = [−1, 5)

⚡ Always check endpoints! They can converge or diverge independently. Open vs closed bracket matters for full credit.

Memory Keys

RATIO TEST → solve |·| < 1 RADIUS R = result CHECK each endpoint [ or ( depends on convergence

🎯 Your Turn

a) Find the radius and interval of convergence for Σ xⁿ/n!

b) Find the IOC for Σ n(x+1)ⁿ

13

Parametric Equations

First & Second Derivative — dy/dx and d²y/dx²

MEDIUM

dy/dx = (dy/dt) / (dx/dt)

d²y/dx² = [d/dt(dy/dx)] / (dx/dt)

Note: do not just take (d²y/dt²)/(d²x/dt²) — that is WRONG!

📌 Worked Example

x = t², y = t³. Find dy/dx and d²y/dx².

dx/dt = 2t, dy/dt = 3t²

dy/dx = 3t²/2t = 3t/2

d/dt(3t/2) = 3/2 → d²y/dx² = (3/2)/(2t) = 3/(4t)

⚡ Second derivative trap: d²y/dx² ≠ (d²y/dt²)/(d²x/dt²). You must differentiate dy/dx with respect to t, then divide by dx/dt.

Memory Keys

dy/dx = (dy/dt)÷(dx/dt) 2nd: DIFFERENTIATE dy/dx w.r.t. t THEN divide by dx/dt again

🎯 Your Turn

a) x = cos t, y = sin t. Find dy/dx. Where is the tangent horizontal?

b) x = t − sin t, y = 1 − cos t. Find d²y/dx².

14

Parametric — Arc Length

Arc Length Formula for Parametric Curves

MEDIUM

L = ∫ₐᵇ √[(dx/dt)² + (dy/dt)²] dt

📌 Worked Example

Find the arc length of x = 3t, y = 4t for t ∈ [0, 1].

dx/dt = 3, dy/dt = 4

L = ∫₀¹ √(9 + 16) dt = ∫₀¹ 5 dt = 5

⚡ Don't mix up: Parametric arc length uses (dx/dt)² + (dy/dt)², while regular y = f(x) uses √(1 + (dy/dx)²). They look different!

Memory Keys

SQUARE both derivatives ADD under root INTEGRATE over t PYTHAGOREAN feel

🎯 Your Turn

a) Find the arc length of x = t², y = t³/3 from t = 0 to t = 1.

b) Set up (don't evaluate) the arc length for x = cos t, y = sin t from t = 0 to π. What does this represent?

15

Polar Coordinates

Polar Area — Setting Up the Integral

TRICKY

A = (1/2) ∫ₐᵇ [r(θ)]² dθ

For the area between two polar curves: A = (1/2)∫[r_outer² − r_inner²] dθ

📌 Worked Example

Find the area enclosed by r = 2cos(θ)

This is a circle. It goes from θ = −π/2 to π/2 (one full loop).

A = (1/2) ∫_{−π/2}^{π/2} (2cosθ)² dθ = (1/2)∫4cos²θ dθ = 2∫½(1+cos2θ)dθ

= [θ + sin(2θ)/2]₋π/₂^π/₂ = π

⚡ Limits of integration: Always sketch the curve and find the correct θ range. Tracing the full curve is the most common polar mistake.

Memory Keys

HALF times r² integrated SKETCH to find θ range OUTER² MINUS INNER² between curves cos²θ → use half-angle

🎯 Your Turn

a) Find the area of one petal of r = sin(2θ)

b) Find the area inside r = 3 and outside r = 2 + cosθ.

16

Differential Equations

Separable Differential Equations

EASY

A separable DE has the form dy/dx = f(x)·g(y). Separate variables, integrate both sides, then solve for y.

📌 Worked Example

Solve: dy/dx = xy, y(0) = 2

Separate: dy/y = x dx

Integrate: ln|y| = x²/2 + C

Solve: y = Ae^(x²/2). Use y(0)=2: A=2 → y = 2e^(x²/2)

⚡ Don't forget +C on ONE side only! And always apply the initial condition after solving for y, not before.

Memory Keys

SEPARATE x and y INTEGRATE both sides +C on one side SOLVE for y (explicit form) APPLY initial condition last

🎯 Your Turn

a) Solve dy/dx = 3y, y(0) = 5

b) Solve dy/dx = (x+1)/(y+1), y(0) = 2

c) Solve dy/dx = y·cos(x), y(0) = 1

17

Differential Equations

Logistic Growth — Reading the Graph & Equation

MEDIUM

Model: dP/dt = kP(1 − P/M) where M = carrying capacity. Growth slows as P → M.

Fastest growth occurs when P = M/2 (inflection point of logistic curve).

⚡ AP Exam love: They ask "at what population is growth fastest?" Answer: ALWAYS M/2, not M.

Memory Keys

S-CURVE shape FASTEST growth at P = M/2 P → M as t → ∞ kP(1−P/M) = logistic

🎯 Your Turn

a) dP/dt = 0.2P(1 − P/1000). At what P is growth fastest? What is dP/dt at that P?

b) A population has carrying capacity 500. If P = 500, what is dP/dt? Explain.

18

Differential Equations

Euler's Method — Numerical Approximation

EASY

Euler's Method approximates solutions step by step:

y(new) = y(old) + h · f(x, y)

where h = step size, f(x,y) = dy/dx

📌 Worked Example

dy/dx = x + y, y(0) = 1. Use h = 0.1. Approximate y(0.2).

Step 1: At (0,1): slope = 0+1 = 1. y(0.1) = 1 + 0.1(1) = 1.1

Step 2: At (0.1, 1.1): slope = 0.1+1.1 = 1.2. y(0.2) = 1.1 + 0.1(1.2) = 1.22

⚡ Smaller h = more accurate. Euler's method is always an approximation. Overestimates if concave down, underestimates if concave up.

Memory Keys

y_new = y_old + h·slope UPDATE x too: x_new = x+h CONCAVE UP → underestimate CONCAVE DOWN → overestimate

🎯 Your Turn

a) dy/dx = 2x, y(0) = 3. Use Euler's method with h = 0.5 to approximate y(1).

b) For the same DE, is Euler's estimate too high or too low? Justify.

19

Applications of Derivatives

Related Rates — Implicit Differentiation w.r.t. Time

TRICKY

Differentiate an equation with respect to time t (not x). Every variable that changes with time needs a rate (·dt).

📌 Worked Example — Ladder Problem

A 10 ft ladder leans against a wall. The base slides away at 2 ft/s. How fast does the top slide down when the base is 6 ft from the wall?

Relation: x² + y² = 100

Differentiate: 2x(dx/dt) + 2y(dy/dt) = 0

When x=6: y=8. Plug in: 2(6)(2) + 2(8)(dy/dt) = 0 → dy/dt = −3/2 ft/s

⚡ Don't substitute early! Plug in known values ONLY after differentiating. If you substitute before, you lose the rate terms.

Memory Keys

DRAW and label WRITE the geometric relation DIFFERENTIATE w.r.t. t SUBSTITUTE values AFTER differentiation

🎯 Your Turn

a) A spherical balloon is inflated at 10 cm³/s. How fast is the radius increasing when r = 5 cm? [V = (4/3)πr³]

b) Two cars leave an intersection: one going north at 60 mph, one going east at 80 mph. How fast is the distance between them increasing after 1 hour?

20

Applications of Integrals

Volumes of Revolution — Washer & Shell Method

TRICKY

Disk/Washer Method (rotating about horizontal axis, integrating with respect to x):

V = π ∫[R(x)² − r(x)²] dx

Shell Method (integrating with respect to x, rotating about y-axis):

V = 2π ∫ x · f(x) dx

📌 Worked Example — Washer

Region bounded by y = x² and y = x, rotated about the x-axis.

Intersections: x = 0 and x = 1. On [0,1]: x > x² so outer = x, inner = x²

V = π ∫₀¹ [x² − x⁴] dx = π[x³/3 − x⁵/5]₀¹ = π(1/3 − 1/5) = 2π/15

⚡ Which method to use? If integrating parallel to axis of rotation → Shell. If integrating perpendicular → Disk/Washer. Pick the one with simpler algebra!

Memory Keys

WASHER: π(R²−r²) SHELL: 2πx·f(x) FIND intersections first OUTER minus INNER radius PARALLEL → Shell / PERP → Disk

🎯 Your Turn

a) Find the volume when y = √x, y = 0, x = 4 is rotated about the x-axis.

b) Use the shell method to find the volume when y = x², x = 0, y = 4 is rotated about the y-axis.

c) Region bounded by y = x and y = x² rotated about y = −1. Set up the integral (washer method).

Essential Formulas

20 Problems at a Glance