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Trigonometry
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Chapter 1 / Exercise 129
Trigonometry
Larson
Expert Verified
1THE CHINESE UNIVERSITY OF HONG KONGDepartment of MathematicsExercise for MATH1010 University MathematicsContents1Preliminaries21.1Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21.2Limits and Derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . .21.3Applications of Derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . .31.4Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52Differentiation72.1Graphs of Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72.2Limits and Continuity. . . . . . . . . . . . . . . . . . . . . . . . . . . . .72.3Derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .82.4Mean Value Theorem and Taylor’s Theorem. . . . . . . . . . . . . . . . .102.5L’Hopital’s Rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .123Integration133.1Fundamental Theorem of Calculus. . . . . . . . . . . . . . . . . . . . . .133.2Substitution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .133.3Integration by Parts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .143.4Reduction Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .143.5Trigonometric Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . .153.6Trigonometric Substitution. . . . . . . . . . . . . . . . . . . . . . . . . . .153.7Rational Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .163.8t-method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .163.9Piecewise Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .174Further Problems185Answers23
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Trigonometry
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Chapter 1 / Exercise 129
Trigonometry
Larson
Expert Verified
21Preliminaries1.1Functions1. Graph the functionsf(x) =x2andg(x) =4x-1 together, to identify values ofxfor whichx2>4x-1.Confirm your answer by solving the inequality algebraically. 2. Plot all points ( x, y) on the plane that satisfyx2+y2+ 2x-4y-14 = 0. Explainwhy it is not the graph of any function.4. Determine whether each of the following statement is true or false.3. Find all real numbersxsatisfying2x+ 21-x= 3.Zlnx dx=1x+C1.2Limits and Derivatives 1. Evaluate the following limits 2-x4x-x2x928x928+ 4-212t-1t2+ 2tln(e+t)etx→-2|x|3-8x4-162. Use first principles to finddydxof the following functions.
33. Find the first derivative of the following functions.(a)f(x) = ln(xπ+eπ)(b)g(x) =1πx+x4(c)h(x) =xe1/x2+ln(2x)x(d)u(x) =xpe-x+x(e)v(x) = ln(1 +x2)87x2 4. Ift= (x+ 1)(x+ 2)2(x+ 3)3and lnt=ey, finddydxin terms ofxonly.5. Supposef0(2) = 13,g(7) = 2 andg0(7) = 53. Ify=f(g(x)), finddydxx=7.6. Supposey=f(x) is a smooth function. Determine whether each of the followingstatements is true or false. (a) Iffis increasing, thenf00(x)>0 for allx.(b) Iff0(x)>0 for allx, thenf(x)>0 for allx.(c) Iff0(x)>0 for allx, thenf0is increasing.(d) Iff(x)>0 for allx, thenf0(x)>0 for allx.(e) Iff0(3) = 0, thenfmust have a maximum or a minimum atx= 3.1.3Applications of Derivatives1. IfLis a tangent line to the graphy=x2+ 3, and thex-intercept ofLis 1, find allpossible points at whichLtouches the graph ofy=x2+ 3.2. The positionxof a particle at timetis given byx=t3+at2+bt+cfor some constantsa, b, c. It is known that whent= 0, the particle is at positionx= 0; also, there is a certain timet0such that the velocity and acceleration of theparticle are both zero at timet0, and at timet0the particle is at positionx= 1.Find the values ofa,bandc.

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