AP Calculus Bootcamp: Fast-Paced Daily Unit-by-Unit Review
SAT Score Range
•
11 sessions
•
JM
SM
About
Meeting every day for 10 days, we’ll cover every major unit—from limits and continuity to differentiation, implicit and inverse functions, related rates, optimization, integrals, the Fundamental Theorem of Calculus, differential equations, parametric/polar calculus, and infinite series.Each session blends clear concept explanations, step-by-step examples, guided practice, and quick concept checks to reinforce understanding.Perfect for students who want a structured, supportive, high-efficiency learning plan to get ready for quizzes, unit tests, midterms, and the AP exam—without feeling overwhelmed.Get ready to truly understand AP Calculus!
Tutored by
✋ ATTENDANCE POLICY
Please do not miss more than two sessions in a row, and message the tutor if you cannot attend.
SESSION 1
20
Dec
SESSION 1
Limits and continuity
Limits and continuity
Sat 3:00 PM - 5:00 PM UTCDec 20, 3:00 PM - 5:00 PM UTC
DAY 1 — Unit 1: Limits & Continuity
Limits basics
• Estimating limits from graphs
• One-sided limits from graphs
• Connecting limit behavior to function behavior
• Approximating limits using tables
• One-sided limits using tables
Limit laws & algebra
• Limits of sums, differences, products, quotients
• Limits of composite functions
• Direct substitution
• When substitution fails
• Limits involving trigonometric functions
• Limits of piecewise functions
Algebraic limit techniques
• Factoring to evaluate limits
• Conjugates
• Trig identities
• Indeterminate forms & next steps
• Squeeze theorem
Continuity
• Classifying discontinuities
• Continuity at a point (graphical & algebraic)
• Continuity on intervals
• Removable discontinuities
• Common continuous functions
Limits at infinity
• Infinite limits (graphical & algebraic)
• Limits at infinity of quotients
• Limits with square roots
• Intermediate Value Theorem (IVT)
SESSION 2
21
Dec
SESSION 2
Differentiation: definition and basic derivative rules
Differentiation: definition and basic derivative rules
Sun 3:00 PM - 5:00 PM UTCDec 21, 3:00 PM - 5:00 PM UTC
DAY 2 — Unit 2: Differentiation: Definition & Basic Rules
Definition of derivative
• Secant lines & average rate of change
• Derivative as slope of tangent line
• Derivative as limit definition
• Estimating derivatives
• Differentiability (graphical & algebraic)
Basic derivative rules
• Power rule (positive, negative, fractional)
• Rewriting expressions before differentiating
• Common errors in basic rules
• Using tables to find derivatives
• Derivatives of polynomials
Trig & exponential derivatives
• Derivatives of sin(x) & cos(x)
• Derivatives of eˣ & ln(x)
Product & quotient rules
• Product rule (with tables)
• Quotient rule (with tables)
• Derivatives of rational functions
• Derivatives of tan, cot, sec, csc
SESSION 3
22
Dec
SESSION 3
Differentiation: composite, implicit, and inverse functions
Differentiation: composite, implicit, and inverse functions
Mon 3:00 PM - 5:00 PM UTCDec 22, 3:00 PM - 5:00 PM UTC
DAY 3 — Unit 3: Composite, Implicit, & Inverse Functions
Chain rule
• Identifying composite functions
• Chain rule introduction
• Chain rule with tables
• Derivatives of aˣ and logₐ(x)
• Multi-step chain-rule problems
Implicit differentiation
• Differentiating implicit equations
• Mixed explicit/implicit forms
• Second derivatives (implicit)
Inverse functions
• Derivatives of inverse functions
• Derivatives of inverse trig functions
Advanced mixed rule problems
• Finding errors in multi-rule differentiation
• Rewriting expressions strategically
• Disguised derivatives
SESSION 4
23
Dec
SESSION 4
Contextual applications of differentiation
Contextual applications of differentiation
Tue 3:00 PM - 5:00 PM UTCDec 23, 3:00 PM - 5:00 PM UTC
DAY 4 — Unit 4: Contextual Applications of Differentiation
Derivatives in context
• Meaning of derivative in real situations
• Motion interpretation: position, velocity, acceleration
• Non-motion rate of change problems
Related rates
• Setting up related-rates expressions
• Converting expressions into equations
• Classic related-rates models (cone, sphere, Pythagorean)
• Multi-rate relationships
• More advanced related-rates word problems
Approximations
• Local linearity
• Linear approximation & tangent line use
L’Hôpital’s Rule
• 0/0 forms
• ∞/∞ forms
• Determining when to apply
SESSION 5
24
Dec
SESSION 5
Applying derivatives to analyze functions
Applying derivatives to analyze functions
Wed 3:00 PM - 5:00 PM UTCDec 24, 3:00 PM - 5:00 PM UTC
DAY 5 — Unit 5: Applying Derivatives to Analyze Functions
Mean Value Theorem
• Using MVT
• Justifying conditions for MVT
First derivative applications
• Critical points
• Increasing/decreasing intervals
• Relative extrema
• Absolute extrema over domains & closed intervals
Second derivative applications
• Concavity
• Inflection points
• Second derivative test
Graphical analysis
• Connecting f, f′, and f″ graphs
• Justification using derivative signs
Optimization
• Identifying optimization structure
• Modeling with derivatives
• Tangents to implicit relations
• Calculator-active optimization
SESSION 6
25
Dec
SESSION 6
Integration and accumulation of change
Integration and accumulation of change
Thu 3:00 PM - 5:00 PM UTCDec 25, 3:00 PM - 5:00 PM UTC
DAY 6 — Unit 6: Integration & Accumulation of Change
Accumulation basics
• Accumulation of change
• Riemann sums (left, right, midpoint)
• Over/under estimation
• Trapezoidal sums
Summation
• Σ-notation intro
• Riemann sums in Σ-notation
• Limits of Riemann sums → definite integrals
Fundamental Theorem of Calculus
• FTC Part 1: derivative of accumulation function
• FTC Part 2: evaluating definite integrals
• Chain rule in accumulation functions
• Interpreting accumulation functions
Definite integrals
• Using geometry & area formulas
• Properties of integrals
• Adjacent intervals
Indefinite integrals (antiderivatives)
• Reverse power rule (simple, negative, fractional)
• Rewriting before integrating
• Integrals of eˣ, 1/x, sin, cos
• Integrals of piecewise functions
Integration techniques
• u-substitution (definite/indefinite)
• Long division
• Completing the square
• Integration by parts (definite/indefinite)
• Partial fractions
• Improper integrals
SESSION 7
26
Dec
SESSION 7
Differential equations
Differential equations
Fri 3:00 PM - 5:00 PM UTCDec 26, 3:00 PM - 5:00 PM UTC
DAY 7 — Unit 7: Differential Equations
Basics of differential equations
• Writing differential equations
• Verifying solutions
Slope fields
• Plotting slope fields
• Reasoning from slope fields
Euler’s method
• Stepwise approximation
• Table generation
Separable equations
• Identifying separable forms
• Separating variables
• Finding general & particular solutions
Models
• Exponential growth/decay
• Logistic growth
• Word-problem models
SESSION 8
27
Dec
SESSION 8
Applications of integration
Applications of integration
Sat 3:00 PM - 5:00 PM UTCDec 27, 3:00 PM - 5:00 PM UTC
DAY 8 — Unit 8: Applications of Integration
Integral applications
• Average value of a function
• Motion problems with integrals
• Definite integrals in context
• Algebraic definite-integral problems
Area
• Area between curve and x-axis
• Area between two curves
• Horizontal slicing
• Multiple intersections (calculator-active)
Volumes
• Cross-sections (squares, rectangles, triangles, semicircles)
• Disc method (standard & non-standard axes)
• Washer method (standard & non-standard axes)
Other applications
• Arc length
• Contextual calculator-active problems
SESSION 9
28
Dec
SESSION 9
Parametric equations, polar coordinates, and vector-valued functions
Parametric equations, polar coordinates, and vector-valued functions
Sun 3:00 PM - 5:00 PM UTCDec 28, 3:00 PM - 5:00 PM UTC
DAY 9 — Unit 9: Parametric, Polar, & Vector Functions
Parametric
• Parametric differentiation
• Second derivatives
• Arc length of parametric curves
Vector-valued functions
• Derivatives of vector functions
• Motion along curves using vectors
• Second derivative & acceleration
• Integrals for displacement & distance
Polar
• Derivatives of polar functions
• Tangent lines in polar form
• Area bounded by polar curves
• Area between two polar curves
• Calculator-active polar problems
SESSION 10
29
Dec
SESSION 10
Infinite sequences and series
Infinite sequences and series
Mon 3:00 PM - 5:00 PM UTCDec 29, 3:00 PM - 5:00 PM UTC
DAY 10 — Unit 10: Infinite Sequences & Series
Sequences & series basics
• Convergence/divergence of sequences
• Partial sums
• Infinite geometric series
Convergence tests
• nth-term test
• Integral test
• p-series
• Direct comparison
• Limit comparison
• Alternating series test
• Ratio test
• Absolute vs conditional convergence
Remainders & error
• Alternating series remainder
• Lagrange error bound
Power series & Taylor/Maclaurin
• Taylor & Maclaurin polynomials
• Function as geometric series
• Maclaurin of eˣ, sin, cos
• Integrating & differentiating power series
• Intervals of convergence
SESSION 11
30
Dec
SESSION 11
Other Topics
Other Topics
Tue 3:00 PM - 5:00 PM UTCDec 30, 3:00 PM - 5:00 PM UTC
Make up day