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This 12 page Class Notes was uploaded by Ms. Ally Koelpin on Thursday October 22, 2015. The Class Notes belongs to MATH 1351 at Texas Tech University taught by Victoria Ellen Howle in Fall. Since its upload, it has received 28 views. For similar materials see /class/226461/math-1351-texas-tech-university in Mathematics (M) at Texas Tech University.
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Date Created: 10/22/15
Math 1351 011 September 26 2007 1 Announcements 0 Homework 5 due Friday 92807 0 Exam 1 will be returned Friday TTU Department of Mathematics amp Statistics Math 1351 011 September 26 2007 2 Derivatives The derivative of f at E is given by fxAx fx 11320 Ax provided the limit exists TTU Department of Mathematics amp Statistics Math 1351 011 September 26 2007 3 Techniques of Integration Theorem The derivative of a constant is zero d Wk Theorem For any real number n the power function f x n l has derivative f x mg ie TTU Department of Mathematics amp Statistics Math 1351 011 September 26 2007 4 Basic rules for combining derivatives Rule Function Notation Cfx Cf x Constant multiple Sum Wit 900 f 9 Difference 9IE ffE 995 Linearity afx bgx elfx Dyx Product f969 f969 96 f 9 Quotient 9frff1 frglfr TTU Department of Mathematics amp Statistics Math 1351 011 September 26 2007 5 Rule Leibniz Notation Constant multiple 601 6 d d Sum 6fg Difference 6 9 Z jg Linearity 6in b9 2 a 93 Product 6f9 f i 1 gd i 3 2 Quotient dxg g2 TTU Department of Mathematics amp Statistics Math 1351 011 September 26 2007 6 Extended linearity rule If f1 f2 fn are differentiable functions and a1a2 an are constants then d dfl df2 dxia1f1a2f2 a1 dx 00 dx i Cbn dz TTU Department of Mathematics amp Statistics Math 1351 011 September 26 2007 Higherorder derivatives lst derivative 3 f x 2nd derivative y f E 6 327 1605 2 3 3rd derivative 3 f r ng 37 1195 4th derivative 94 f4x 34771 1605 nth derivative gm fm E Cir n TTU Department of Mathematics amp Statistics Math 1351 011 September 26 2007 8 Derivatives of trigonometric functions Theorem The functions sinx and cosx are differentiable for all x and sinxcosx cosxsinx dx dx TTU Department of Mathematics amp Statistics Math 1351 011 Recall September 26 2007 More Trig review sinac cosac tanx cow cotx Sm 1 1 SEC IE CSC IE cosac sinx COS2ZE l sin2 25 1 1tan2x sec2 16 cot2 16 1 csc2 x sina l sinozcos l cosozsin cosa cos oz cos sina sin TTU Department of Mathematics amp Statistics Math 1351 011 September 26 2007 10 Derivatives of trigonometric functions The functions sin costan csc sec cot are all differentiable Wherever they are de ned and isinxcosx icosx sinx dac dac itanxsec2x icotx csc2x dac dac secxsecxtanx cscx cscxcotx TTU Department of Mathematics amp Statistics H F 9 7 U 03 5 00 3 REVIEW FOR THE FINAL EXAMINATION FALL 2007 Evaluate the limits x272z1 lim za1x273z2 Using l7Hopital7s theorern7 compute the limits I 1 lim 7n n m mace z lin1sin s In x Find the constants a and b so that the function 15 if z 7 2 b if z 2 Using the de nition of the derivative di erence quotient7 nd the derivative of f x3 2x f96 is continuous for all x Using the rules of di cerentiation7 nd the derivative of g cosz xcotz x2 tanx 3x 2 tan z h em tanil x A or V 9 Using the rules of di cerentiation7 nd the derivatives of f sin3z2 2x gt 673 2 cos 2t 2t110 Find the equation of the tangent line to the graph of the function f z23zsin x at z 0 Find 2 by irnplicit di erentiation if e y2 x2 z I If x2y2 100 and 17 107 nd wheny 6 H H H D H r H 03 H 5 18 A ball is dropped from a height of 160 ft A light is located at the same level7 10 ft away from the initial position of the ball How fast is the balls shadow moving along the ground one second after the ball is dropped Find the absolute extrema of the function de ned by the equation f 3 7 3x2 on the closed interval 713 Sketch the graph of f zg 7 9x Sketch the graph of 72z3 7 71 f 96 A person plans to fence off a rectangular garden whose one side is the wall of the house What is the maximal amount of area that can be enclosed given that the length of the fence cannot exceed 100 ft Find the antiderivatives of the functions we z 1 2 g 3cosx Estimate using the right endpoints the integral 1 3 3dz 0 by using a Riemann sum Sn 21 fkAAx for n 4 Evaluate the integrals 1 2x 14dz 0 3x 2 dx g f z lnidx z Find the area of the subgraph of the parabola y 2 1 that lies over the interval 72 2 dx
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