Physics I / Fundamental theorem of calculus
Practice question · Match the pairs

Match each FTC concept to its description.

  • FTC Part 1
  • FTC Part 2
  • Antiderivative
  • Net change theorem
  • d/dx[∫axf(t)dt]=f(x)d/dx[\int_a^x f(t)dt] = f(x)
  • ∫abF′(x)dx\int_a^b F'(x)dx = F(b)F(b) − F(a)F(a)
  • Any F such that F′(x)=f(x)F'(x) = f(x)
  • ∫abf(x)dx\int_a^b f(x)dx = F(b)F(b) − F(a)F(a)
Hints
  1. Separate differentiating an integral from evaluating one.
  2. One description reads the theorem physically, as total change from a rate.
Show the answer
  • FTC Part 1 → d/dx[∫axf(t)dt]=f(x)d/dx[\int_a^x f(t)dt] = f(x)
  • FTC Part 2 → ∫abf(x)dx\int_a^b f(x)dx = F(b)F(b) − F(a)F(a)
  • Antiderivative → Any F such that F′(x)=f(x)F'(x) = f(x)
  • Net change theorem → ∫abF′(x)dx\int_a^b F'(x)dx = F(b)F(b) − F(a)F(a)
Why

The net change theorem is FTC Part 2 read physically: total change = integral of rate.

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