Math · Calculus

High school Calculus, unit by unit

Q: What do you learn in high school Calculus?

Limits and continuity, derivatives and their rules, applications of derivatives to rates, motion and optimization, integrals and the Fundamental Theorem of Calculus, differential equations, applications of integrals, parametric and polar calculus, and infinite series. In Mathflow, that’s 10 units and 69 skills.

Q: Does Mathflow cover AP Calculus AB or BC?

Mathflow’s ten Calculus units follow the order of the ten units of AP Calculus BC published by the College Board. AP Calculus AB covers the first eight units; parametric and polar calculus and infinite series are BC only.

Q: What should I know before taking Calculus?

Pre-Calculus: function families, their graphs and transformations, logarithms and trigonometry on the unit circle. Solid algebra matters too, because factoring, fractions and exponents come up in almost every problem.

Q: What is the difference between a derivative and an integral?

A derivative measures an instantaneous rate of change, like the slope of a tangent line. A definite integral measures accumulation, like the area under a curve. The Fundamental Theorem of Calculus links them: integration undoes differentiation.

Q: Is Calculus hard?

It adds new ideas, limits, rates of change and accumulation, on top of everything you know about functions. Taking it one skill at a time, with regular practice on derivatives and integrals, makes it manageable.

Q: How can I practice Calculus at home?

Redo the worked problems on this page without looking, then practice skill by skill. Mathflow covers the whole course in three levels per skill, with Alphi, an AI math tutor, for hints when you’re stuck. It’s free to start on iOS and Android.

Limits, derivatives, integrals, differential equations, parametric and polar curves, and infinite series: here is everything Calculus covers, with worked problems and the mistakes to avoid. Then practice each skill in Mathflow, from intuition to solving.

Start for free See the 10 units

The shaded area under

y = 4x - x^2

, from

x = 0

x = 4

, is

\dfrac{32}{3}

. The green line is the tangent at

x = 1

, with slope

f'(1) = 2

What you’ll learn

The 10 units of Calculus

The course as it is organized in Mathflow, in order. Each unit is split into precise skills that you practice one at a time.

Unit 17 skills
Limits & Continuity
Read limits from graphs, tables and formulas, evaluate them algebraically or with a squeeze argument, study continuity and asymptotes, and use the Intermediate Value Theorem.
- Read and interpret limits across representations
- Evaluate limits with an appropriate algebraic method
- Establish a limit using a squeeze argument
- Analyze continuity and repair removable breaks
- Describe unbounded behavior near a vertical asymptote
- Determine end behavior using limits at infinity
- Justify an intermediate value or a root
Unit 26 skills
Derivatives
Build the derivative from a difference quotient, estimate it from data or a graph, decide where a function is differentiable, and use the derivative rules for powers, products, quotients, exponentials, logarithms and trigonometric functions.
- Build the derivative from a difference quotient
- Estimate a derivative from data or a graph
- Decide whether a function is differentiable
- Differentiate powers and linear combinations
- Differentiate elementary transcendental functions
- Differentiate products and quotients
Unit 35 skills
Composite & Implicit Differentiation
Use the chain rule, differentiate implicitly defined relations, find the derivatives of inverse and inverse trigonometric functions, and compute higher-order derivatives.
- Differentiate nested functions using the chain rule
- Differentiate an implicitly defined relation
- Find the derivative of an inverse function
- Differentiate inverse trigonometric expressions
- Compute and interpret successive derivatives
Unit 45 skills
Rates & Local Approximation
Interpret derivatives in context, analyze motion along a line, solve related-rates problems, approximate a function with its tangent line, and evaluate indeterminate limits with L’Hospital’s rule.
- Interpret derivatives in a measured context
- Analyze straight-line motion using derivatives
- Solve a related-rates problem
- Approximate a function with its tangent line
- Evaluate indeterminate limits with L’Hospital’s rule
Unit 57 skills
Derivative-Based Function Analysis
Apply the Mean Value Theorem, find absolute and local extrema, study concavity with the second derivative, rebuild a function from its derivatives, and solve optimization problems.
- Apply the Mean Value Theorem with its hypotheses
- Locate absolute extrema on a closed interval
- Use derivative signs to justify variation and local extrema
- Analyze concavity and apply the second derivative test
- Reconstruct a function from its derivative information
- Model and justify an optimal value
- Analyze tangent directions on an implicit curve
Unit 69 skills
Integration & Accumulation
Approximate accumulation with Riemann sums, connect it to the definite integral and the Fundamental Theorem of Calculus, and integrate by substitution, by parts and with partial fractions, improper integrals included.
- Approximate accumulation and connect sums to an integral
- Analyze an accumulation function and its derivative
- Use the structure and properties of definite integrals
- Find antiderivatives and evaluate definite integrals
- Integrate a composition using substitution
- Rewrite an integrand before choosing a technique
- Integrate a product by parts
- Integrate rational functions using simple partial fractions
- Decide convergence and evaluate an improper integral
Unit 76 skills
Differential Equations
Model a rate of change with a differential equation, read slope fields, solve separable equations, and work with exponential and logistic growth and Euler’s method.
- Model a rate law and check differential-equation solutions
- Read and sketch solution behavior with slope fields
- Solve separable differential equations with initial values
- Derive and use continuous growth and decay models
- Approximate a solution with Euler’s method
- Interpret logistic growth and its constraints
Unit 87 skills
Integral Applications
Use integrals for average values, motion, net change, areas between curves, volumes by cross-sections, disks and washers, and the length of a curve.
- Find and interpret a function average over an interval
- Recover motion from velocity and acceleration integrals
- Calculate a net amount from changing inflow and outflow
- Find a bounded area by integrating curve differences
- Find volume from known cross-sectional areas
- Find a solid of revolution volume using disks or washers
- Measure the length of a smooth function graph
Unit 96 skills
Parametric & Polar Calculus
Find slopes, concavity and lengths on parametric curves, analyze motion in the plane with vectors, and find slopes and areas in polar coordinates.
- Find slopes and concavity on a parametric curve
- Compute the length of a parametrized path
- Differentiate and integrate vector-valued functions
- Analyze planar motion with position and velocity vectors
- Differentiate a curve described in polar coordinates
- Calculate areas enclosed by polar curves
Unit 1011 skills
Infinite Series & Approximation
Decide whether a series converges with the geometric, integral, comparison, alternating series and ratio tests, then build Taylor polynomials, bound their error and find power series.
- Distinguish sequence limits from series convergence
- Determine when a geometric series has a finite sum
- Use integral comparisons and benchmark series
- Justify series convergence using comparison
- Classify convergence of an alternating series
- Use the ratio test and select a valid convergence argument
- Control the error of an alternating partial sum
- Build a local Taylor polynomial
- Bound a Taylor approximation error
- Find a power series convergence interval
- Represent functions with Taylor and transformed power series

Calculus at a glance

Calculus usually comes after Pre-Calculus, often in 12th grade. It answers two questions: how fast does a quantity change at an instant, and how much does it accumulate over time?

Before you start. You need every function family of Pre-Calculus, with their graphs and transformations, logarithms and the unit circle, and solid algebra from Algebra 1 and Algebra 2: factoring, fractions and exponents come up in almost every problem.

AB or BC? The ten units follow the order of the ten units of the College Board’s AP Calculus BC course. AP Calculus AB covers the first eight, from limits to applications of integration. Parametric and polar calculus and infinite series are BC only.

In Mathflow. The course has $10$ units and $69$ skills. Each skill is a series in three levels, and each unit ends with a Chapter Challenge.

Worked problems

Two problems, solved step by step

An optimization problem and an area under a curve: one classic for each half of Calculus. Try each one on paper before you read the solution.

Problem 1
The biggest open box
From a $12$ -inch by $12$ -inch sheet of cardboard, you cut a square of side $x$ inches from each corner, then fold up the sides to make an open box. Which value of $x$ gives the largest volume, and what is that volume?
Solution
The base is a square of side $12 - 2x$ and the height is $x$ , so the volume is
$V(x) = x(12 - 2x)^2, \qquad 0 < x < 6$
Differentiate with the product rule and the chain rule, then factor:
$\begin{aligned} V'(x) &= (12 - 2x)^2 - 4x(12 - 2x) \\ &= 12(6 - x)(2 - x) \end{aligned}$
On $0 < x < 6$ , $V'(x) = 0$ only at $x = 2$ . Before $2$ , $V'(x) > 0$ , so the volume increases; after $2$ , $V'(x) < 0$ , so it decreases. The maximum is at $x = 2$ :
$V(2) = 2 \times 8^2 = 128$
Cut $2$ -inch squares: the box holds $128$ cubic inches. Check: $V(1) = 100$ and $V(3) = 108$ , both smaller.
Problem 2
Area under a parabola
Find the area of the region between the curve $y = 4x - x^2$ and the $x$ -axis.
Solution
First find where the curve meets the $x$ -axis: $4x - x^2 = x(4 - x) = 0$ , so $x = 0$ or $x = 4$ . Between them, $4x - x^2 \ge 0$ , so the area is a definite integral.
An antiderivative of $4x - x^2$ is $2x^2 - \dfrac{x^3}{3}$ . By the Fundamental Theorem of Calculus:
$\begin{aligned} \int_0^4 (4x - x^2)\,dx &= \left[2x^2 - \frac{x^3}{3}\right]_0^4 \\ &= 32 - \frac{64}{3} = \frac{32}{3} \end{aligned}$
The area is $\dfrac{32}{3} \approx 10.67$ square units. Check: the region fits inside the $4$ by $4$ rectangle under the vertex $(2, 4)$ , whose area is $16$ , so an area of about $10.67$ is plausible.

How Mathflow teaches it

Calculus in three levels

Every skill of the course is a series of short missions in three levels. Here is what you actually do in three of them.

Level 1 · Explore
Explore and play with the concepts to build your intuition.
Level 2 · Explain
Turn your intuitions into the right words and uncover hidden mistakes.
Level 3 · Solve
Reason it out, crunch the numbers, and find your way to the answer.

You don’t start with a list of rules. In the first level, you act on the math: you fill a hole in a graph, slide a ladder, build Riemann sums rectangle by rectangle and watch the numbers respond. In the second level, you put words on what you saw: you decide which of two students is right, find the first wrong line of a solution, or explain which condition fails. In the third level, you solve problems on your own, like the ones you meet in class and on tests.

If you get stuck, Alphi, the AI math tutor, gives you a hint instead of the answer, at any hour. When the skills of a unit are done, a Chapter Challenge checks the whole unit.

Unit 1 · Limits & Continuity
1Analyze continuity and repair removable breaks
- Level 1 · Explore
  Drag a point to define $f(1)$ so that $f(x) = \dfrac{x^3 - 1}{x - 1}$ is continuous at $x = 1$ , and change $k$ until a graph has no break at $x = 2$ .
- Level 2 · Explain
  $g(x) = \dfrac{x^2 - 4}{x - 2}$ for $x \neq 2$ , and $g(2) = 3$ . You say which condition of continuity fails, and you tap the first wrong line in Maria’s check of a piecewise function.
- Level 3 · Solve
  A function is defined by $kx - 3$ for $x < 2$ and by $x^2 + 1$ for $x \ge 2$ . You find $k$ so that it is continuous at $x = 2$ .
Unit 4 · Rates & Local Approximation
2Solve a related-rates problem
- Level 1 · Explore
  Drag the foot of a sliding ladder until its top slides down at $0.75$ foot per second, and set the depth at which water rises in a cone-shaped tank at $1$ foot per minute.
- Level 2 · Explain
  A balloon’s radius is $3$ centimeters and grows at $0.5$ centimeter per second. You tap the first wrong line of Liam’s work, and you choose which values you may substitute before differentiating.
- Level 3 · Solve
  A $6$ -foot person walks away from a $15$ -foot lamppost at $3$ feet per second. You find how fast the shadow gets longer.
Unit 6 · Integration & Accumulation
3Approximate accumulation and connect sums to an integral
- Level 1 · Explore
  Drag the top of each rectangle to build a left Riemann sum, then double $n$ until the right sum differs from the integral by less than $1$ .
- Level 2 · Explain
  The speed $v$ is increasing and concave down on $[0, 6]$ , and you decide who is right about its left Riemann sum. You also tap the first wrong line of Riley’s left sum.
- Level 3 · Solve
  You evaluate $\lim_{n\to\infty}\sum_{k=1}^{n}\left(2 + \dfrac{4k}{n}\right)\dfrac{4}{n}$ by writing it as a definite integral.

Practice Calculus for free

Common mistakes

Five mistakes to avoid

These slips come up again and again in Calculus. Learn to spot them, and you’ll catch them in your own work.

Forgetting the chain rule
The derivative of $\sin(3x)$ is $3\cos(3x)$ , not $\cos(3x)$ . When a function is nested inside another, multiply by the derivative of the inside function.
Substituting values before differentiating
In a related-rates problem, differentiate the equation first, then substitute the values of the instant. If you replace $x$ with $6$ too early, its rate $\dfrac{dx}{dt}$ disappears from the equation.
Dropping the constant $C$
An indefinite integral is a whole family of functions: $\int 2x\,dx = x^2 + C$ . Without $C$ , you can’t match an initial condition such as $y(0) = 5$ .
Taking a signed integral for an area
$\int_0^{2\pi} \sin x\,dx = 0$ , yet the region between the curve and the $x$ -axis has an area of $4$ . Split the interval where the function changes sign, and add the absolute values.
Thinking terms that go to $0$ make a series converge
The terms of $\sum \dfrac{1}{n}$ go to $0$ , but this harmonic series diverges. Terms going to $0$ is necessary for convergence, not sufficient.

Frequently asked questions

What do you learn in high school Calculus?

Limits and continuity, derivatives and their rules, applications of derivatives to rates, motion and optimization, integrals and the Fundamental Theorem of Calculus, differential equations, applications of integrals, parametric and polar calculus, and infinite series. In Mathflow, that’s 10 units and 69 skills.

Does Mathflow cover AP Calculus AB or BC?

Mathflow’s ten Calculus units follow the order of the ten units of AP Calculus BC published by the College Board. AP Calculus AB covers the first eight units; parametric and polar calculus and infinite series are BC only.

What should I know before taking Calculus?

Pre-Calculus: function families, their graphs and transformations, logarithms and trigonometry on the unit circle. Solid algebra matters too, because factoring, fractions and exponents come up in almost every problem.

What is the difference between a derivative and an integral?

A derivative measures an instantaneous rate of change, like the slope of a tangent line. A definite integral measures accumulation, like the area under a curve. The Fundamental Theorem of Calculus links them: integration undoes differentiation.

Is Calculus hard?

It adds new ideas, limits, rates of change and accumulation, on top of everything you know about functions. Taking it one skill at a time, with regular practice on derivatives and integrals, makes it manageable.

How can I practice Calculus at home?

Redo the worked problems on this page without looking, then practice skill by skill. Mathflow covers the whole course in three levels per skill, with Alphi, an AI math tutor, for hints when you’re stuck. It’s free to start on iOS and Android.

Start for free

Standards

Calculus and the standards

The Common Core State Standards stop before Calculus. They describe the mathematics every student needs to be college and career ready, and mark with (+) the additional mathematics needed for advanced courses such as calculus.

Many high school Calculus courses follow the College Board’s AP Calculus framework. AP Calculus BC has ten units, from limits and continuity to infinite sequences and series; AP Calculus AB has the first eight. Mathflow’s ten units follow the same order.

High school Calculus, unit by unit

Limits & Continuity

Derivatives

Composite & Implicit Differentiation

Rates & Local Approximation

Derivative-Based Function Analysis

Integration & Accumulation

Differential Equations

Integral Applications

Parametric & Polar Calculus

Infinite Series & Approximation

Calculus at a glance

The biggest open box

Area under a parabola

Level 1 · Explore

Level 2 · Explain

Level 3 · Solve

1Analyze continuity and repair removable breaks

2Solve a related-rates problem

3Approximate accumulation and connect sums to an integral

Forgetting the chain rule

Substituting values before differentiating

Dropping the constant CCC

Taking a signed integral for an area

Thinking terms that go to 000 make a series converge

Math is all about flow.

Dropping the constant $C$

Thinking terms that go to $0$ make a series converge