Calculus 1
Twenty-nine interactive lessons on limits, derivatives and integrals. Each one starts with a question, gives you a graph to work with, and asks you to commit to an answer before it shows you one.
The lessons are numbered in the order most courses teach them, from the first idea of a limit to the average value of a function. If you are reviewing for a test, go straight to the topic you need. Teaching with them? Start with the instructor guide.
Limits
What do you do when substituting gives you no answer?
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1
The idea of a limit
Getting closeCan a function have a limit where it is not even defined?What a limit means, estimated from graphs and tables. Open lesson -
2
The limit laws
Taking limits apartWhen can you split a limit into simpler pieces, and when does that go wrong?The limit laws and direct substitution. Open lesson -
3
Limits by factoring
Zero over zeroSubstituting gives 0/0. Does that mean there is no limit?Factoring and cancelling, with a graph that shows the hole. Open lesson -
4
Limits by rationalizing
Multiply by the conjugateThe numerator has a square root, so there is nothing to factor. Where is the common factor hiding?Rationalizing with conjugates, and the sign traps that come with it. Open lesson -
5
One-sided limits and absolute values
Left of a, right of aWhat if the function approaches one number from the left and another from the right?One-sided limits, absolute values and piecewise functions. Open lesson -
6
The Squeeze Theorem
Trapped between two curvesA function oscillates too fast to factor or substitute. What else could pin its limit down?The squeeze theorem, including the limit of sin x / x. Open lesson -
7
Limits at infinity
As x growsWhen x is a million, which terms still matter?Limits at infinity and horizontal asymptotes. Open lesson
Continuity and the meaning of a limit
What exactly are you claiming when you write that a limit equals L?
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8
Continuity and the Intermediate Value Theorem
Without lifting the pencilWhat exactly has to be true for a graph to have no break?Continuity, kinds of discontinuity, and the Intermediate Value Theorem. Open lesson -
9
The precise definition of a limit
Within toleranceHowever close someone demands the output be, can you always keep the input close enough to deliver it?The ε-δ definition of a limit, a δ tester and the δ game. Open lesson
The derivative
A slope needs two points. How can a curve have a slope at one?
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10
The definition of the derivative
Slope at a pointWhat happens to the slope of a secant line as its second point slides toward the first?The limit definition of the derivative and a tangent-matching game. Open lesson -
11
The derivative as a function
Drawing f′Could you sketch the derivative from the graph alone, with no formula?The derivative as a function, and matching f with f′. Open lesson -
12
The power, product and quotient rules
Rules from rectanglesIs the derivative of a product just the product of the derivatives?The power, product and quotient rules. Open lesson -
13
The chain rule
Rates in a chainIf u changes three times as fast as x, and y twice as fast as u, how fast does y change?The chain rule for compositions. Open lesson -
14
Implicit differentiation
Curves that are not functionsHow do you find the slope of a circle without solving for y?Implicit differentiation. Open lesson -
15
Logarithmic differentiation
Taking logs firstThe power rule and the exponential rule both fail on xˣ. What works when base and exponent both change?Logarithmic differentiation for variable powers and long products. Open lesson -
16
Differentials
Small changes, estimatedIf you know how a quantity changes, how far off can a small measurement error push the answer?Differentials dy and dx, and how errors spread through a formula. Open lesson
Using derivatives
What can the derivative tell you about the world?
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17
Linear approximation
Straight up closeHow could the slope at 4 help you estimate √4.1?Tangent lines as approximations, and how fast their error shrinks. Open lesson -
18
Related rates
Rates that depend on each otherThe foot of a ladder slides at a steady speed. Does the top?Related rates, with animated scenes. Open lesson -
19
Curve sketching
Shape from signsIf you knew only where f′ and f″ are positive, could you sketch f?Increasing, decreasing, concavity, extrema and inflection points. Open lesson -
20
Optimization
The best choiceWhich corner cut makes the biggest box?Optimization problems, solved by hand and with the derivative. Open lesson -
21
The Mean Value Theorem
Somewhere in betweenIf you averaged 80 km/h, must your speedometer have read 80?The Mean Value Theorem and why its hypotheses matter. Open lesson
Integration
How do you add up infinitely many tiny pieces, and what does that have to do with slopes?
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22
Approximating integrals
Adding up rectanglesHow close can rectangles get to the area under a curve?Left, right and midpoint sums, computed by hand. Open lesson -
23
The definite integral as a limit
Infinitely many rectanglesWhat happens to a Riemann sum when the number of rectangles grows without bound?The definite integral defined as a limit of Riemann sums. Open lesson -
24
The Fundamental Theorem of Calculus
Slope and area, undoneIf differentiating undoes integrating, are slope and area opposites?Both parts of the Fundamental Theorem of Calculus. Open lesson -
25
Antiderivatives and basic integrals
Running derivatives backwardsYou know the derivative. How many functions could it have come from?Antiderivatives, the power rule in reverse, and simplifying before integrating. Open lesson -
26
Integration by substitution
Changing the variableAn integral looks impossible until you rename part of it. Which part?u-substitution for indefinite and definite integrals. Open lesson -
27
Area under a curve
Net area and total areaWhen a graph dips below the axis, does its area count against you?Net area, total area, and splitting an integral at the zeros. Open lesson -
28
Area between curves
Top minus bottomHow do you find an area when the bottom edge is another curve?Area between two curves, with intersections and crossings. Open lesson -
29
The Mean Value Theorem for integrals
Levelling the areaIf the region under a curve were water, how high would it settle?Average value of a function and the Mean Value Theorem for integrals. Open lesson
How to use these lessons
Try each question before you open its answer. The step-by-step solvers count how many steps you get right on the first try, so a wrong guess costs you a point but nothing else. What you work out yourself is what you will remember on the test.