Understanding Anti-differentiation with Boundary Conditions
Reversing the process of differentiation, anti-differentiation involves finding the indefinite integrals of functions with boundary conditions. Learn about rules to find indefinite integrals and apply them to real-world scenarios like calculating displacement and population growth. Explore examples and concepts related to anti-differentiation in mathematics.
Uploaded on Apr 19, 2025 | 0 Views
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19 April 2025 Anti-differentiation with boundary condition LO: To anti-differentiate a function with a boundary condition. www.mathssupport.org
Reversing the process of differentiation Antidifferentiation is also known as indefinite integration And is denoted with an integral symbol, ?? IfF (x) = f(x), we write ? ? ?? = ? ? + ? Constant of integration integrand Variable of integration Is read as: The antiderivative of fwith respect to x The integral of fwith respect to x ? ? ?? www.mathssupport.org
Rules to find the indefinite integrals Power rule 1 ? + 1??+1+ ?,? 1 ???? = Constant rule ??? = ?? + ?, Constant multiple rule ??(?)?? = ? ?(?)?? Sum or difference rule = ? ? ?? ? ? ?? ?(?) ?(?)?? www.mathssupport.org
Indefinite integrals The velocity of a moving object is given by v(t) = 2t + 1 Find an expression for the displacement in terms of t = 2 ??? + 1?? ?(?) = (2? + 1) ?? ?(?) = ?2+ ? + ? This is called the general solution for (2? + 1) ?? If the position of the object at time t = 1 is6we can find C ?(1) = 12+ 1 + ? 6 = 12+ 1 + ? 4 = ? The fact that the position at time t = 1 is6is called a boundary condition ?(?) = ?2+ ? + 4is a particular solution of (2? + 1) ?? www.mathssupport.org
Indefinite integrals Iff (x) = 3x2 + 2x andf(2) = -3, Findf(x) f (x) = 3x2 + 2x ?(?) = (3x2 + 2x) ?? ? ? = ?3+ ?2+ ? This is the general solution for (3x2 + 2x) ?? Use the fact that f(2) = -3 to find C ?(2) = 23+ 22+ ? 3 = 8 + 4 + ? 15 = ? ? ? = ?3+ ?2 15 www.mathssupport.org
Indefinite integrals The rate of growth of a population of fish is given by ?? ??= 150 ? for 0 t 5 years. The initial population was 200 fish. Find the number of fish att = 4 years. ?? ??= 150 ? The initial population was 200 means that at time t = 0, ?(0) = 200 ?(?) = 150 ? ?? 1 2 ?? Rewrite with rational exponents = 150? 3 2+ ? 3 2+ ? 1 2 ?? ?(0) = 100(0) = 150 ? 200 = 100(0) 3 2+ ? 3 2+ 200 3 2+ 200 General solution ?(?) = 100? 200 = ? Particular solution ?(?) = 100? FindPwhent =4 ?(4) = 100(4) ?(4) = 1000 There are 1000 fish when t = 4 years www.mathssupport.org
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