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SampleMT1-4

# SampleMT1-4 - Math 1B Practice material for the First...

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Math 1B, Practice material for the First Midterm Examination Practice exam 1 1.( 15 points ) Evaluate the integral ± ( x x + 1 ) 2 dx 2.( 20 pnts ) Evaluate the integral ± ² 1 + x 1 - x dx 1

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3.( 25 pnts ) Determine whether each improper integral is convergent or di- vergent. Evaluate the integrals which are convergent. (a) ± 2 dx ( x 6 - x 2 ) 1 6 (b) ± 2 1 dx x (ln x ) 2 3 4.( 20 pnts ) We approximate ± 2 1 10(57 x + 1 x ) dx with the Trapezoidal rule, and want to make sure the error is at most 0 . 0001. Which of the following is true? ( Explain your answer) 2
(a) We can take any N 50 (b) We can take any 100 N 400 c) We can take any N 50 d) We can take any N 5 5.( 20 pnts ) Consider the curve given by the equation y = 1 2 (sin x - ± x 0 dx cos y ) = 1 2 (sin x - ln(sec x - tan x )) (1) Find the arclength of this curve for 0 x π/ 4. 3

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1. Using the decomposition into partial fractions we have: x 2 ( x + 1) 2 = 1 - 2 x + 1 + 1 ( x + 1) 2 Thus: ± x 2 ( x + 1) 2 dx = ± dx - ± 2 x + 1 dx + ± 1 ( x + 1) 2 dx = x - 2 ln | x = 1 |- 1 ( x + 1) 2 + C 2. ² x - 1 x + 1 = ³ (1 - x ) 2 1 - x 2 After the substitution x = sin θ , taking into account 1 - sin 2 θ = cos 2 θ we have ² x - 1 x + 1 = 1 - sin θ cos θ Thus, for the integral we have: ± ² x - 1 x + 1 dx = ± 1 - sin θ cos θ cos θdθ = ± - ± sin θdθ = θ + cos θ + C 3(a). x 6 - x 2 < x 6 for x > 2. Therefore 1 ( x 6 - x 2 ) 1 / 6 > 1 ( x 6 ) 1 / 6 = 1 x The integral ± 2 dx x diverges, therefore by the comparison test the integral in question also diverges. 3(b). ± dx x (ln x ) 2 / 3 = ± d (ln x ) (ln x ) 2 / 3 = ± dt t 2 / 3 where t = ln x . Thus, the question becomes: Is the integral ± ln 2 0 dt t 2 / 3 convergent? This is a “ p -integral” for p = 2 / 3 between 0 and ln2. It is conver- gent. 4. We want the error of the approximation to be not greater then 0
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SampleMT1-4 - Math 1B Practice material for the First...

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