Prelim 1 - instructor’s name and the time of your class...

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PRELIM 1 Math 112 Write your name, your instructor’s name and the time of your class on the exam booklet immediately. Show all your work. Calculators are not needed or permitted, but you may use the attached sheet of formulas. You do not need to simplify your answer unless the question asks for it, but always SHOW YOUR WORK . To receive full credit, your answers must be neatly written and logically organized. Problem 1 (10 points) Compute the derivative: d dx ( x 2 + 1) Z x 1 e t 2 dt . Note that your answer may contain an definite integral. Problem 2 (15 points) Sketch the region bounded by the curves y = 4 x 2 - 2 and y = 2 x 2 and compute its area. Problem 3 (15 points each) Compute the integrals 1. R dx x 3 + x 2 2. R cos 3 x dx, 3. R 2 - 1 (5 x + 3) e 2 x dx. The exam continues on the next page 1
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Problem 4 (8 points each) Are the following equalities true or false? Justify your answer. 1. R 1 - 1 1 x dx = 0 2. R 1 ( 1 x 2 + 1 ) dx = 3. R 1 - 1 dx ( x - 5)( x +10)( x +12) = Problem 5 (6 points) Suppose that f and g are continuous functions and that f ( x ) > g ( x ) + 3 > 0 on the interval [1,3]. If Z 3 1 g ( x ) dx = 5 , is it possible that R 3 1 f ( x ) dx = 9? Explain your answer . Check before you hand in the exam booklet, if you wrote your name, your
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Unformatted text preview: instructor’s name and the time of your class on the exam booklet. 2 TRIGONOMETRIC IDENTITIES sin2 x = 2 sin x cos x cos2 x = cos 2 x-sin 2 x sin 2 x = 1-cos2 x 2 cos 2 x = 1+cos2 x 2 cos 2 x + sin 2 x = 1 1 + tan 2 x = sec 2 x cot 2 x + 1 = csc 2 x DERIVATIVE FORMULAS d dx sin x = cos x d dx tan x = sec 2 x d dx sec x = sec x tan x d dx cos x =-sin x d dx cot x = csc 2 x d dx csc x =-csc x cot x d dx sin-1 x = 1 √ 1-x 2 d dx tan-1 x = 1 1+ x 2 d dx sec-1 x = 1 x √ x 2-1 d dx cos-1 x =-1 √ 1-x 2 d dx cot-1 x =-1 1+ x 2 d dx csc-1 x =-1 x √ x 2-1 INTEGRAL FORMULAS R x n dx = x n +1 n +1 + C for n 6 =-1 R e x dx = e x + C R 1 x dx = ln | x | + C R sin x dx =-cos x + C R cos x dx = sin x + C R sec x dx = ln | sec x + tan x | + C R csc x dx =-ln | csc x + cot x | + C R sec 2 x dx = tan x + C R 1 √ 1-x 2 dx = sin-1 x + C R 1 1+ x 2 dx = tan-1 x + C 3...
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