AP®MathsCollege BoardCalculus ABStudy GuidesUnit 7: Differential EquationsUnit 7 OverviewUnit 7 Summary

Unit 7 Summary (College Board AP® Calculus AB): Study Guide

Roger B

Written by: Roger B

Reviewed by: Jamie Wood

Updated on

Differential equations summary

Key definitions

  • A differential equation involves derivatives

  • The general solution is a family of curves that solve a differential equation

  • A particular solution is a single curve that solves a differential equation and satisfies an initial or boundary condition

  • A slope field is a diagram that shows the tangent of the general solution at different points

  • An exponential model satisfies fraction numerator d y over denominator d t end fraction equals k y (growth) or fraction numerator d y over denominator d t end fraction equals negative k y (decay)

Key formulas

  • Separation of variables can be used to solve fraction numerator d y over denominator d x end fraction equals f open parentheses x close parentheses g open parentheses y close parentheses by solving integral fraction numerator d y over denominator g open parentheses y close parentheses end fraction equals integral f open parentheses x close parentheses d x

  • The solution to an exponential model is of the form y equals y subscript 0 e to the power of k t end exponent (growth) or y equals y subscript 0 e to the power of negative k t end exponent (decay)

    • y subscript 0 is the initial value

Key facts

  • The doubling-time or half-life of an exponential model is t equals fraction numerator ln 2 over denominator k end fraction

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Author
Reviewer
Roger B

Author: Roger B

Expertise: Maths Content Creator

Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

Jamie Wood

Reviewer: Jamie Wood

Expertise: Maths Content Creator

Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.