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Calculus

Calculus practice problems with step-by-step solutions: derivatives, integrals, series and multivariable topics, with the key ideas summarized below.

Practice pages
6
Problems solved
101
Lessons
1

Practice problems with solutions

Lessons

Calculus: the essentials

Derivatives

The derivative f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h} measures the instantaneous rate of change. Practice: derivatives problems, partial derivatives problems.

Integrals

Integration by parts ∫u dv=uv−∫v du\int u\,dv=uv-\int v\,du and the other techniques are collected in integration techniques.

Exponential growth

A quantity that grows by the same percentage rr each step follows An=A0(1+r)nA_n=A_0(1+r)^n; in continuous time A(t)=A0ektA(t)=A_0e^{kt}. The amount added each step is proportional to the current size, which is why it eventually outgrows any linear growth.

Sequences and series

A geometric series ∑n≥0arn\sum_{n\ge 0}ar^n converges to a1−r\frac{a}{1-r} when ∣r∣<1|r|<1. Practice: power series problems, Fourier series problems.

Sub-topics