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Unformatted text preview: Review of Differentiation and Integration for MATH 3200 UCD Department of Mathematical and Statistical Sciences This is a packet of prerequisite material necessary for understanding material covered in ordinary differential equations. Many students take this course after having taken their previous course many years ago, at another institution where certain topics may have been omitted, or have not seen this material in a long time or just feel uncomfortable with it. Because understanding this material is so important to being successful in this course, we have put together this review packet. In this packet you will find sample questions and a brief discussion of each topic. If you find the material in this pamphlet is not sufficient for you, it may be necessary for you to look in the appropriate sections of a calculus textbook and understand the material on your own. Because this is considered prerequisite material, it is ultimately your responsibility to learn it. The topics to be covered include Differentiation and Integration. 1 Differentiation In this course you will be expected to be able to differentiate and integrate quickly and accurately. What follows is a very brief review . If you find this is not enough of a review, please refer to your calculus textbook. Exercises: 1. Find the derivative of y = x 3 sin( x ). 2. Find the derivative of y = ln( x ) cos( x ) . 3. Find the derivative of y = ln(sin( e 2 x )). Discussion: It is expected that you know, without looking at a table, the following differentiation rules: d dx [( kx ) n ] = kn ( kx ) n − 1 (1) d dx bracketleftBig e kx bracketrightBig = ke kx (2) d dx [ln( kx )] = 1 x (3) d dx [sin( kx )] = k cos kx (4) d dx [cos( kx )] =- k sin x (5) 1 d dx [ uv ] = u ′ v + uv ′ (6) d dx bracketleftbigg u v bracketrightbigg = u ′ v- uv ′ v 2 (7) d dx [ u ( v ( x ))] = u ′ ( v ) v ′ ( x ) . (8) We put in the constant k into (1) - (5) because a very common mistake to make is something like: d dx e 2 x = e 2 x 2 (when the correct answer is 2 e 2 x ). Equation (6) is known as the product rule, Equation (7) is known as the quotient rule, and Equation (8) is known as the chain rule. From these, you can derive the derivative of many other functions, such as the tangent: tan( x ) = sin( x ) cos( x ) d dx [tan( x )] = cos( x ) cos( x )- sin( x )(- sin( x )) (cos( x )) 2 = cos 2 ( x ) + sin 2 ( x ) cos 2 ( x ) = 1 cos 2 ( x ) = sec 2 ( x ) where we have used the quotient rule and simplified. The chain rule is applied when there is a function of a function , i.e. f ( g ( x )). The idea is to take the derivative of the outside function first, leaving its argument alone. Then multiply that by the derivative of the next outermost function, leaving it’s argument alone....
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