MTH 161 Midterm: MTH 161 University of Rochester 161 Midterm 2 Fall 2006

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turquoisegnu128

31 Jan 2019

School

University of Rochester

Department

Mathematics

Course

MTH 161

Professor

All

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Document Summary

We use u-substitution, with u = x2 + 3, so du = 2xdx. 2 (cid:18)2 u3/2(cid:19) + c (x2 + 3)5/2 (x2 + 3)3/2 + c. u5/2 3. We use the trigonometric substitution x = (3/2) sin u, so dx = (3/2) cos u du. 18 z sec2 u du cos3 u du. Drawing a triangle, we nd that tan u = Z x2 + 3x x2 + 3x + 2 dx and so the answer is: (8 points) Using long division of fractions, we nd that x2 + 3x x2 + 3x + 2. 2 x + 2 and so: (10 points) Z x2 + 3x x2 + 3x + 2 dx = z dx z. = x 2 ln(x + 1) + 2 ln(x + 2) + c. The density of water is 1000 kg/m3 and the gravitational constant is 9. 8 m/s2.

Related Questions

Trigonometric Substitution Rules

For v(a^2-u^2 ) let u = asin?? and v(a^2-u^2 ) = acos??
For v(a^2+ u^2 ) let u = atan? and v(a^2+u^2 ) = asec?
For v(u^2-a^2 ) let u = asec?? and v(u^2-a^2 ) = atan?

1. Integrate ?dx/v(25+4x^2 ). Identify the following:

a = u = du =

What adjustment, if any, needs to be made to the integral tocomplete the substitution in to u?

2. Rewrite the above integral so that it is all in u.

3. Which of the trig substitution rules above will we use to solvethis integral? Why? Draw and label the relevant right trianglediagram.

4. Use the rule you chose and the diagram to identify what you willreplace u, du and v(a^2+u^2 ) with.

5. Rewrite the integral from step 2 using the replacements fromstep 4. Did you use all the replacements? If not, why not? Can thisintegral be simplified? If so, do it?

6. Integrate the resulting trig function using a basicformula.

7. Use the information from steps 1 and 4 to rewrite the answer instep 6 in terms of the variable x.

u = 2x = 5tan? and v(a^2+u^2 )=v(25+4x^2 )=5sec?

8. How does knowing the trigonometric relationships as defined bythe three sides of a right triangle and the Pythagorean Theoremhelp us solve integrals using trig substitutions?

9. ?dx/(xv(49-9x^2 )) 10. ?dx/v(16x^2-1)

raspberrycrow60

Q18. A size 6 dress in Country C is size 52 in Country D. A function that converts dress sizes in Country C to those in Country D is f(x) = 2(x + 20). Find a formula for the inverse of the function described.
a. f ^-1(x) = (x - 20)/2
b. f ^-1(x) = x - 20
c. f ^-1(x) = (x/2) - 20
d. f ^-1(x) = (x/2) + 20

Q19. If f(x) = 4^x and g(x) = 12, find the point of intersection of the graphs of f and g by solving f(x) = g(x). Give an exact answer.
a. (log ₄ 12, 12)
b. (log ₄ 12, 4)
c. (log ₄ 12, 0)
d. (12, 12)

Q20. Find functions f and g so that f â g = H(x) = (5 - 2x³)².
a. f(x) = 5 - 2x³ ; g(x) = x²
b. f(x) = (5 - 2x)³ ; g(x) = x²
c. f(x) = x² ; g(x) = 5 - 2x³
d. f(x) = x³ ; g(x) = (5 - 2x)²

Q29. Solve the system of equations using Cramer's Rule if it is applicable.

a. x = 9, y = 2, z = 5; (9, 2, 5)
b. x = 8, y = 4, z = 5; (8, 4, 5)
c. x = 4, y = 5, z = 4; (4, 5, 4)
d. x = 8, y = -4, z = -5; (8, -4, -5)

Q30. Solve the system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent.

a. x = 2, y = 3, z = -7; (2, 3, -7)
b. x = 1, y = 3, z = -4; (1, 3, -4)
c. x = 5, y = 13/2, z = -3; (5, 13/2, -3)
d. x = 1, y = -3, z = -16; (1, -3, -16)

Q33. Write the partial fraction decomposition of the rational expression (4x³ + 4x²)/(x² + 5)².
a. (4x - 4)/(x² + 5) + (-20x + 20)/(x² + 5)²
b. (4x + 4)/(x² + 5) + (-20x - 20)/(x² + 5)²
c. (4x + 4)/(x² + 5) + (20x - 20)/(x² + 5)²
d. (4x + 4)/(x² + 5) + (20x + 20)/(x² + 5)²

Q34. Solve the linear programming problem. Minimize z = 11x + 6y + 7 subject to: x = 0, y = 0, x + y = 1.
a. minimum: 7
b. minimum: 18
c. minimum: 24
d. minimum: 13

Q35. Solve the system of equations by elimination.

a. x = 10, y = 0; (10, 0)
b. x = 10, y = 10; (10, 10)
c. x = 0, y = 10; (0, 10)
d. x = 0, y = 0; (0, 0)

Q36. Perform the row operations on the given augmented matrix.

a.
b.
c.
d.

Q37. Solve the system of equations using Cramer's Rule if it is applicable.

a. x = -4, y = 3; (-4, 3)
b. x = -3, y = -4; (-3, -4)
c. x = 4, y = 3; (4, 3)
d. x = 3, y = 4; (3, 4)

Q38. Write the partial fraction decomposition of the rational expression x/(x² - 9x + 20).
a. -4/(x - 4) + 5/(x - 5)
b. -5/(x - 4) + 4/(x - 5)
c. 4/(x - 4) + -5/(x - 5)
d. -4/(x - 4) + -5/(x - 5)

Q39. Solve the system of equations.

a. x = -4, y = 4, z = -3; (-4, 4, -3)
b. x = -3, y = 4, z = -4; (-3, 4, -4)
c. x = -4, y = -3, z = 4; (-4, -3, 4)
d. inconsistent

Q40. Find the value of the determinant.

a. 224
b. 24
c. -72
d. 72