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Clarkson University MA232Final Exam Fall 2016 • Laplace transform: L{ f ( t ) } = R ∞ 0 e  st f ( t ) dt • Convolution: f ( t ) * g ( t )= R t 0 f ( τ ) g ( t  τ ) dτ • Properties of the Laplace Transform: L{ f ( t ) } = F ( s ) L{ c 1 f ( t )+ c 2
The Chain Rule Goal: Determine how to ﬁnd derivatives of compositions of functions. Theorem: (The Chain Rule) Let f and g be diﬀerentiable functions. Then Example: Compute the derivative of the following functions. (1) f ( x ) = cos(3 x 2 + e x ) (2)
Section 21 : Linear Equations The first special case of first order differential equations that we will look at is the linear first order differential equation. In this case, unlike most of the first order cases that we will look at, we can actually
Substitution Rule Goal: Develop a method for integrating certain compositions of functions. Then we can integrate even more! Theorem: (Substitution Rule) If u = g ( x ) is diﬀerentiable and f is continuous, then Examples: Evaluate the following indeﬁ
Firstorder system of linear equations: LINEAR SYSTEMS : We refer to a system of the form : simply as a linear system. MATRIX FORM OF A LINEAR SYSTEM : If ), and denote the respective matrices: 8.1 PRELIMINARY THEORY—LINEAR SYSTEMS Sunday
Courses in MATH
2250  Calculus I for Science and Engineering
2500  Calculus III for Engineering
2700  Elementary Differential Equations
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