Calculus 1
Limits, derivatives and their applications, and the first steps of integration. 1140 problems, in the order most textbooks teach them.
Limits
Limits by direct substitution
When the function is continuous at the point, the limit is just the value there.
0 problemsLimits by factoring
Direct substitution gives 0/0; factor, cancel the common factor, and substitute again.
0 problemsLimits by rationalizing
Multiply by the conjugate to clear a square root that makes the expression 0/0.
0 problemsTrigonometric limits
Limits built on sin(u)/u → 1 and (1 − cos u)/u → 0.
0 problemsLimits at infinity
End behaviour of rational and radical functions: divide through by the highest power.
0 problemsL'Hôpital's rule
Indeterminate forms 0/0 and ∞/∞, settled by differentiating top and bottom.
0 problemsDerivatives
Power rule
Derivatives of polynomials, roots and powers of x.
57 problemsProduct rule
The derivative of a product: (fg)′ = f′g + fg′.
61 problemsQuotient rule
The derivative of a quotient: (f/g)′ = (f′g − fg′)/g².
53 problemsChain rule
Derivatives of compositions: differentiate the outside, keep the inside, multiply by the inside's derivative.
297 problemsDerivatives of trigonometric functions
sin, cos, tan, sec, csc and cot, alone and in combination.
76 problemsDerivatives of exponential functions
eˣ and its relatives, usually with the chain rule close behind.
52 problemsDerivatives of logarithms
ln x and ln of anything, where the chain rule gives u′/u.
50 problemsDerivatives of inverse trig functions
arcsin, arccos and arctan, with the chain rule.
38 problemsLogarithmic differentiation
Functions like xˣ, with the variable in the base and the exponent: take logs first.
39 problemsTangent lines
The equation of the line tangent to a curve at a point.
30 problemsSecond derivatives
Differentiate twice: concavity, acceleration, and higher-order derivatives.
30 problemsImplicit differentiation
dy/dx when y is tangled up with x in an equation.
30 problemsApplications of derivatives
Related rates
Ladders, balloons, tanks and shadows: how one rate of change drives another.
30 problemsLinear approximation
Estimate values with the tangent line: L(x) = f(a) + f′(a)(x − a).
30 problemsCritical numbers
Where f′ is zero or undefined: the candidates for maxima and minima.
30 problemsAbsolute extrema on a closed interval
The closed-interval method: check the critical numbers and the endpoints.
30 problemsIncreasing, decreasing and concavity
Sign charts for f′ and f″: where a graph rises, falls, and bends.
30 problemsOptimization
Fences, boxes and numbers: set up a function, then find its maximum or minimum.
24 problemsIntegrals
Basic antiderivatives
Reverse the derivative rules: powers, exponentials and trig functions.
0 problemsu-substitution
Undo the chain rule: spot the inside function and its derivative.
0 problemsDefinite integrals
Evaluate with the Fundamental Theorem of Calculus: F(b) − F(a).
30 problemsFundamental Theorem of Calculus, Part 1
Differentiate an integral whose upper limit is a function of x.
30 problemsApplications of integration
Area between curves
Integrate top minus bottom between the points where the curves cross.
30 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.
30 problemsArc length
The length of a curve: ∫ √(1 + (dy/dx)²) dx.
15 problems