Calculus 1
Limits, derivatives and their applications, and the first steps of integration. 5985 problems, in the order most textbooks teach them.
Start here
Six core problems from across the course. Pick one, try it on paper, then open the solution.
Interactives
Related Rates Explorer
A scroll-through tutorial: a ripple on a pond, a growing sand pile and a funnel draining into a glass, with each rate worked out as you drag.
Related rates Assumptions statedOptimization Explorer
A scroll-through tutorial: a pen by a river, a box folded from a sheet and a can that holds half a liter, with the best size found as you drag.
Optimization Assumptions statedReview: functions
Composition of functions
Compose two functions both ways, (f∘g)(x) and (g∘f)(x), or evaluate a composition at a point.
59 problemsInverse functions
Find the inverse of a one-to-one function, and check it by composing.
60 problemsExact trigonometric values
Exact values of the six trig functions at multiples of π/6 and π/4, and degrees to radians.
60 problemsExact values of inverse trig functions
Exact values of arcsin, arccos and arctan, inside their restricted ranges.
25 problemsSolving trigonometric equations
Solve trigonometric equations on 0 ≤ θ < 2π, including ones that factor.
60 problemsExponential and logarithmic equations
Solve exponential and logarithmic equations exactly, and reject the extraneous roots.
60 problemsDomain of a function
The domain of a function built from fractions, square roots and logarithms, in interval notation.
60 problemsAmplitude, period and phase shift
Amplitude, period, phase shift and vertical shift of y = A sin(Bx − C) + D.
60 problemsLines and slopes
Slopes, and equations of lines through points, parallel or perpendicular to a given line.
63 problemsLimits
Limits by direct substitution
When the function is continuous at the point, the limit is just the value there.
36 problemsLimits by factoring
Direct substitution gives 0/0; factor, cancel the common factor, and substitute again.
63 problemsLimits by rationalizing
Multiply by the conjugate to clear a square root that makes the expression 0/0.
45 problemsTrigonometric limits
Limits built on sin(u)/u → 1 and (1 − cos u)/u → 0.
69 problemsLimits at infinity
End behaviour of rational and radical functions: divide through by the highest power.
62 problemsL'Hôpital's rule
Indeterminate forms 0/0 and ∞/∞, settled by differentiating top and bottom.
76 problemsLimit laws with given limits
Combine known limits with the sum, product, quotient and power laws.
60 problemsOne-sided limits of piecewise functions
Left- and right-hand limits of piecewise functions, and when the two-sided limit exists.
60 problemsContinuity and types of discontinuity
Find where a function is discontinuous and say whether each break is removable, a jump or infinite.
60 problemsMaking a piecewise function continuous
Choose the constant that makes a piecewise function continuous.
60 problemsVertical and horizontal asymptotes
Vertical asymptotes from the zeros of the denominator, horizontal ones from the end behaviour.
60 problemsEstimating a limit from a table
Estimate a limit from a table of values on both sides, then confirm it exactly.
60 problemsIntermediate Value Theorem: locating a root
Use the Intermediate Value Theorem to show a root exists in an interval, then narrow it by bisection.
58 problemsThe squeeze theorem
Trap an oscillating expression between two bounds with the same limit.
60 problemsDerivatives
Power rule
Derivatives of polynomials, roots and powers of x.
313 problemsProduct rule
The derivative of a product: (fg)′ = f′g + fg′.
264 problemsQuotient rule
The derivative of a quotient: (f/g)′ = (f′g − fg′)/g².
119 problemsChain rule
Derivatives of compositions: differentiate the outside, keep the inside, multiply by the inside's derivative.
422 problemsDerivatives of trigonometric functions
sin, cos, tan, sec, csc and cot, alone and in combination.
233 problemsDerivatives of exponential functions
eˣ and its relatives, usually with the chain rule close behind.
65 problemsDerivatives of logarithms
ln x and ln of anything, where the chain rule gives u′/u.
219 problemsDerivatives of inverse trig functions
arcsin, arccos and arctan, with the chain rule.
53 problemsLogarithmic differentiation
Functions like xˣ, with the variable in the base and the exponent: take logs first.
50 problemsTangent lines
The equation of the line tangent to a curve at a point.
70 problemsSecond derivatives
Differentiate twice: concavity, acceleration, and higher-order derivatives.
47 problemsImplicit differentiation
dy/dx when y is tangled up with x in an equation.
70 problemsThe derivative from the limit definition
f′(x) as the limit of the difference quotient, worked algebraically.
60 problemsAverage rate of change and secant lines
Average rate of change over an interval, the slope of the secant line, and average velocity.
60 problemsRecognising a limit as a derivative
Spot the derivative hiding in a limit: identify f and a, then evaluate f′(a).
32 problemsDerivative rules from a table of values
Product, quotient and chain rules when you only know values of f, g and their derivatives.
60 problemsDerivative of an inverse function
(f⁻¹)′(a) = 1 / f′(f⁻¹(a)), without ever solving for the inverse.
60 problemsParticle motion: position, velocity, acceleration
Position, velocity and acceleration along a line: when the particle stops, which way it moves, and how far it travels.
60 problemsThe chain rule in Leibniz notation
dy/dx = (dy/du)(du/dx), with y given in terms of u and u in terms of x.
60 problemsDerivatives and integrals of hyperbolic functions
Derivatives and antiderivatives of sinh, cosh, tanh and the inverse hyperbolic functions.
60 problemsSecond derivatives by implicit differentiation
Find y″ by differentiating implicitly twice, and evaluate it at a point on the curve.
56 problemsApplications of derivatives
Related rates
Ladders, balloons, tanks and shadows: how one rate of change drives another.
70 problemsLinear approximation
Estimate values with the tangent line: L(x) = f(a) + f′(a)(x − a).
70 problemsCritical numbers
Where f′ is zero or undefined: the candidates for maxima and minima.
70 problemsAbsolute extrema on a closed interval
The closed-interval method: check the critical numbers and the endpoints.
70 problemsIncreasing, decreasing and concavity
Sign charts for f′ and f″: where a graph rises, falls, and bends.
70 problemsOptimization
Fences, boxes and numbers: set up a function, then find its maximum or minimum.
24 problemsDifferentials
dy = f′(x) dx, and using it to estimate a small change.
60 problemsThe Mean Value Theorem and Rolle's theorem
Check the hypotheses, then find every c where the tangent slope equals the average slope.
60 problemsNewton's method
Newton's iteration x₁, x₂, … for a root, done exactly.
60 problemsRelated rates from an equation
Differentiate an equation with respect to time and solve for the unknown rate.
60 problemsAntiderivatives with initial conditions
Find the antiderivative, then use the initial condition to pin down the constant.
60 problemsThe secant method
Two steps of the secant method from two starting guesses.
30 problemsBusiness applications: cost, revenue and profit
Cost, revenue and profit: marginal cost, and the production level that maximizes profit.
60 problemsClassic optimization: wire, window, fence
The classic optimization problems: an open box from a sheet, a Norman window, a can, a cut wire.
25 problemsIntegrals
Basic antiderivatives
Reverse the derivative rules: powers, exponentials and trig functions.
82 problemsu-substitution
Undo the chain rule: spot the inside function and its derivative.
93 problemsDefinite integrals
Evaluate with the Fundamental Theorem of Calculus: F(b) − F(a).
70 problemsFundamental Theorem of Calculus, Part 1
Differentiate an integral whose upper limit is a function of x.
70 problemsRiemann sums: left, right and midpoint
Left, right and midpoint Riemann sums with a handful of rectangles.
60 problemsProperties of definite integrals
Linearity, reversing the limits and splitting the interval, from a few known integrals.
60 problemsAverage value of a function
The average value of f on [a, b], and where f actually takes that value.
60 problemsIntegrals of absolute values
Split the interval where the integrand changes sign, then integrate each piece.
60 problemsSubstitution with a given u
Indefinite integrals with a given substitution u: change variables, integrate, substitute back.
60 problemsDefinite integrals by substitution
Definite integrals by substitution, changing the limits along with the variable.
60 problemsIntegrals giving inverse trig functions
Integrals that land on arctan and arcsin, including completing the square.
60 problemsNet change and accumulation from a rate
Amount now = amount at the start + the integral of the rate of change.
60 problemsSums with sigma notation
Evaluate sums in sigma notation with the formulas for Σi, Σi² and Σi³.
60 problemsLimits of Riemann sums as integrals
Recognize the limit of a Riemann sum as a definite integral, and evaluate it.
60 problemsDefinite integrals from area formulas
Evaluate a definite integral as a signed area: triangles, rectangles, trapezoids and semicircles.
60 problemsAccumulation functions
Express an accumulation function ∫ₐˣ f(t) dt as a function of x, and check it against FTC Part 1.
54 problemsApplications of integration
Area between curves
Integrate top minus bottom between the points where the curves cross.
70 problemsVolumes by disks and washers
Solids of revolution sliced perpendicular to the axis.
18 problemsVolumes by cylindrical shells
Solids of revolution built from nested cylinders: 2πx f(x) dx.
63 problemsArc length
The length of a curve: ∫ √(1 + (dy/dx)²) dx.
57 problemsVolumes with known cross sections
Volumes of solids whose cross-sections are squares, semicircles or triangles.
60 problemsExponential growth and decay
Exponential growth and decay: doubling time, half-life, and solving for the time.
60 problemsArea between curves, integrating in y
Area between curves of the form x = g(y), integrating with respect to y.
10 problemsArc length of x = g(y)
Arc length of a curve given as x = g(y).
60 problems