Calculus curriculum

A single-variable calculus sequence with a BC extension into series, parametric curves, polar curves, and a multivariable preview.

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AP Calculus AB/BC / college bridge

Calculus curriculum

A single-variable calculus sequence with a BC extension into series, parametric curves, polar curves, and a multivariable preview.

Pacing

8 units, 28-36 weeks self-paced

Units

8 unit sequence

Practice

32 checked answers

Support

Self-paced or tutor-guided

Outcomes

Explain limits, derivatives, and integrals as connected ideas.
Use calculus to model change, accumulation, optimization, and motion.
Prepare for AP-style free-response reasoning without treating the exam as the whole course.

Khan Academy companion path

Use these linked courses for video instruction and mastery practice, then return here for PeerTutor original checks and tutor help on the exact unit that got stuck.

AP/College Calculus AB

External video content is linked, not copied. PeerTutor practice is original and cites each source path separately.

Unit sequence

Built for independent progress first, then tutor support where the student gets stuck.

01
Precalculus fluency
- Review functions, trigonometry, logarithms, and composition.
- Use graphs and algebra to predict behavior.
Practice: Diagnose gaps with mixed function tasks before starting limits.
Original practice checks
Answer in PeerTutor, then use the explanation or bring the miss to a tutor.
4 checked answers
Function fluency
If f(x)=ln(x^2), simplify f(e) for e>0.
Composition
If f(x)=x^2 and g(x)=x+3, what is f(g(2))?
7102549
Unit readiness
Which target best matches the Calculus unit "Precalculus fluency"?
Review functions, trigonometry, logarithms, and composition.Classify discontinuities and asymptotic behavior.Estimate, compute, and justify limits.Use graphs and algebra to predict behavior.
Practice artifact
Which practice artifact should you produce for "Precalculus fluency" before asking a tutor for help?
Skip the unit and start a new subject.Memorize vocabulary without checking any examples.Diagnose gaps with mixed function tasks before starting limits.Watch videos without writing or solving anything.
02
Limits and continuity
- Estimate, compute, and justify limits.
- Classify discontinuities and asymptotic behavior.
Practice: Write epsilon-level intuition in plain English before using symbolic rules.
Original practice checks
Answer in PeerTutor, then use the explanation or bring the miss to a tutor.
4 checked answers
Limits
Compute lim x->2 of (x^2 - 4)/(x - 2).
Continuity
A removable discontinuity usually appears when a factor can be canceled and what remains?
a vertical asymptotea holean endpoint onlya jump
Unit readiness
Which target best matches the Calculus unit "Limits and continuity"?
Use graphs and algebra to predict behavior.Classify discontinuities and asymptotic behavior.Review functions, trigonometry, logarithms, and composition.Estimate, compute, and justify limits.
Practice artifact
Which practice artifact should you produce for "Limits and continuity" before asking a tutor for help?
Write epsilon-level intuition in plain English before using symbolic rules.Memorize vocabulary without checking any examples.Watch videos without writing or solving anything.Skip the unit and start a new subject.
03
Derivative definition and rules
- Connect average and instantaneous rate of change.
- Differentiate algebraic, trigonometric, exponential, and logarithmic functions.
Practice: Derive a rule from first principles, then apply it to modeling problems.
Original practice checks
Answer in PeerTutor, then use the explanation or bring the miss to a tutor.
4 checked answers
Derivative rules
Differentiate f(x)=3x^4 - 2x.
Derivative meaning
The derivative at a point gives which quantity?
average valueinstantaneous rate of changearea under a curvetotal accumulation
Unit readiness
Which target best matches the Calculus unit "Derivative definition and rules"?
Connect average and instantaneous rate of change.Use graphs and algebra to predict behavior.Review functions, trigonometry, logarithms, and composition.Estimate, compute, and justify limits.
Practice artifact
Which practice artifact should you produce for "Derivative definition and rules" before asking a tutor for help?
Derive a rule from first principles, then apply it to modeling problems.Watch videos without writing or solving anything.Memorize vocabulary without checking any examples.Skip the unit and start a new subject.
04
Applications of derivatives
- Analyze motion, optimization, and related rates.
- Use first and second derivatives to describe graphs.
Practice: Solve an optimization task with a diagram, domain, derivative, and conclusion.
Original practice checks
Answer in PeerTutor, then use the explanation or bring the miss to a tutor.
4 checked answers
Optimization
A rectangle has perimeter 20 and width x. Write area A(x).
Graph analysis
If f'(x)>0 and f''(x)<0 on an interval, f is:
increasing and concave downdecreasing and concave upincreasing and concave updecreasing and concave down
Unit readiness
Which target best matches the Calculus unit "Applications of derivatives"?
Estimate, compute, and justify limits.Review functions, trigonometry, logarithms, and composition.Use graphs and algebra to predict behavior.Use first and second derivatives to describe graphs.
Practice artifact
Which practice artifact should you produce for "Applications of derivatives" before asking a tutor for help?
Watch videos without writing or solving anything.Memorize vocabulary without checking any examples.Solve an optimization task with a diagram, domain, derivative, and conclusion.Skip the unit and start a new subject.
05
Integrals and accumulation
- Interpret definite integrals as net change.
- Use antiderivatives and substitution.
Practice: Approximate an area numerically, then compare to an exact integral.
Original practice checks
Answer in PeerTutor, then use the explanation or bring the miss to a tutor.
4 checked answers
Antiderivatives
Find an antiderivative of 6x^2.
Net change
If v(t) is velocity, integral from a to b of v(t) dt gives:
speedaccelerationdisplacementposition at a
Unit readiness
Which target best matches the Calculus unit "Integrals and accumulation"?
Interpret definite integrals as net change.Use graphs and algebra to predict behavior.Estimate, compute, and justify limits.Review functions, trigonometry, logarithms, and composition.
Practice artifact
Which practice artifact should you produce for "Integrals and accumulation" before asking a tutor for help?
Memorize vocabulary without checking any examples.Skip the unit and start a new subject.Approximate an area numerically, then compare to an exact integral.Watch videos without writing or solving anything.
06
Differential equations and modeling
- Solve separable differential equations.
- Use slope fields and exponential models.
Practice: Model a population or cooling process and discuss parameter meaning.
Original practice checks
Answer in PeerTutor, then use the explanation or bring the miss to a tutor.
4 checked answers
Separable equations
Solve dy/dx = 3y with y(0)=2.
Slope fields
In a slope field for dy/dx = x - y, the slope at (2, 1) is:
-1013
Unit readiness
Which target best matches the Calculus unit "Differential equations and modeling"?
Use slope fields and exponential models.Estimate, compute, and justify limits.Use graphs and algebra to predict behavior.Review functions, trigonometry, logarithms, and composition.
Practice artifact
Which practice artifact should you produce for "Differential equations and modeling" before asking a tutor for help?
Skip the unit and start a new subject.Watch videos without writing or solving anything.Model a population or cooling process and discuss parameter meaning.Memorize vocabulary without checking any examples.
07
BC extension: series and advanced curves
- Test convergence of series.
- Analyze parametric, polar, and vector-valued motion.
Practice: Build a convergence decision tree and apply it to mixed examples.
Original practice checks
Answer in PeerTutor, then use the explanation or bring the miss to a tutor.
4 checked answers
Series
Does the geometric series 1 + 1/2 + 1/4 + ... converge? Answer yes or no.
Parametric motion
For x=t^2, y=t, dx/dt at t=3 is:
1369
Unit readiness
Which target best matches the Calculus unit "BC extension: series and advanced curves"?
Test convergence of series.Estimate, compute, and justify limits.Review functions, trigonometry, logarithms, and composition.Use graphs and algebra to predict behavior.
Practice artifact
Which practice artifact should you produce for "BC extension: series and advanced curves" before asking a tutor for help?
Build a convergence decision tree and apply it to mixed examples.Skip the unit and start a new subject.Watch videos without writing or solving anything.Memorize vocabulary without checking any examples.
08
Multivariable preview
- Read functions of several variables.
- Connect partial derivatives and gradients to optimization.
Practice: Use contour maps to explain gradient direction and rate of change.
Original practice checks
Answer in PeerTutor, then use the explanation or bring the miss to a tutor.
4 checked answers
Partial derivatives
For f(x,y)=x^2y + y, what is partial f/partial x?
Gradient
The gradient points in the direction of:
steepest decreasesteepest increasezero change onlyaverage value
Unit readiness
Which target best matches the Calculus unit "Multivariable preview"?
Review functions, trigonometry, logarithms, and composition.Use graphs and algebra to predict behavior.Estimate, compute, and justify limits.Connect partial derivatives and gradients to optimization.
Practice artifact
Which practice artifact should you produce for "Multivariable preview" before asking a tutor for help?
Watch videos without writing or solving anything.Skip the unit and start a new subject.Memorize vocabulary without checking any examples.Use contour maps to explain gradient direction and rate of change.