105-Mid2Prac1Sols

# 105-Mid2Prac1Sols - Math 105 Practice Midterm 1 for Midterm...

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Unformatted text preview: Math 105 - Practice Midterm 1 for Midterm 2 Solutions This practice midterm may be harder and/or longer than the real midterm. Not all question will be worth the same number of points. 1. Evaluate Z ∞ e- √ x √ x dx , or show that it doesn’t exist. Note: this was corrected on March 16, the earlier version missed the fact that the function is not defined at x = 0 . Let’s first evaluate the indefinite integral with the substitution u = √ x , 2 du = 1 √ x dx : Z e √ x √ x dx = Z 2 e- u du =- 2 e- u =- 2 e- √ x . The improper integral is actually improper in two ways: one limit of integration is ∞ and the other is an asymptote of the function. To be able to handle that, we should split the integral up, at some arbitrary number like x = 1, and do the two resulting improper integrals. Z ∞ e- √ x √ x dx = Z 1 e- √ x √ x dx + Z ∞ 1 e- √ x √ x dx = lim c → +- 2 e- √ x 1 c + lim d →∞- 2 e- √ x d 1 =- 2 lim c → + ( e- 1- e- c ) + lim d →∞ ( e- √ d- e- 1 ) =- 2 ( ( e- 1- 1) + (0- e- 1 ) ) = 2 . 2. Solve the initial value problem y = 1 √ xy , y (1) = 4 . dy dx = 1 √ x √ y ⇒ Z √ ydy = Z 1 √ x dx ⇒ 2 3 y 3 / 2 = 2 √ x + C 1 ⇒ y 3 / 2 = 3 √ x + C 2 ⇒ y = (3 √ x + C 2 ) 2 / 3 . y (1) = 4 ⇒ 4 = (3 + C 2 ) 2 / 3 ⇒ 3 + C 2 = 4 3 / 2 = 8 ⇒ C 2 = 5 ⇒ y = (3 √ x + 5) 2 / 3 ....
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105-Mid2Prac1Sols - Math 105 Practice Midterm 1 for Midterm...

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