Math141SolsFinalF11

Math141SolsFinalF11 - SOLUTIONS TO THE FINAL (CHAPTERS 7,...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
SOLUTIONS TO THE FINAL (CHAPTERS 7, 8, AND 9) MATH 141 – FALL 2011 – KUNIYUKI 250 POINTS TOTAL Show all work, simplify as appropriate, and use “good form and procedure” (as in class). Box in your final answers! No notes or books allowed. A scientific calculator is allowed. To maximize chances for partial credit, please be neat and indicate any elementary row operations (EROs) you use! Clarity is important. I might not grade “messes.” 1) Find the intersection point(s) of the graphs of x 2 ± y = 0 and 3 x 2 ± 8 x + y = 0 in the usual xy -plane by solving a system, as we have done in class. Do not rely on graphing, “trial-and-error,” guessing, or point-plotting as a basis for your method. Show all work! Write all solutions as ordered pairs of the form x , y () . If there are none, write “NONE.” (14 points) We want to find all real solutions of the system: x 2 ± y = 0 3 x 2 ± 8 x + y = 0 ² ³ ´ µ ´ Let’s use the Substitution Method. We can, for example, solve the first equation for y in terms of x : y = x 2 We can then substitute our expression for y into the second equation and solve for x : 3 x 2 ± 8 x + y = 0 3 x 2 ± 8 x + x 2 = 0 4 x 2 ± 8 x = 0 4 xx ± 2 = 0 x = 0 or x = 2
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Plug (substitute) these x -values into the equation y = x 2 to find the corresponding y -values. x = 0 ± y = 0 () 2 = 0 x = 2 ± y = 2 2 = 4 The solution set is: 0 , 0 , 2 , 4 {} These correspond to intersection points for the graphs of the given equations. The graph of the first equation, x 2 ± y = 0 , or y = x 2 , is in black. The graph of the second equation, 3 x 2 ± 8 x + y = 0 , or y = ± 3 x 2 + 8 x , is in blue.
Background image of page 2
2) Write the PFD (Partial Fraction Decomposition) for 7 x ± 3 x 2 ± 3 x ± 4 . You must find the unknowns in the PFD Form. Show all work, as we have done in class! (17 points) Fortunately, the given rational expression is proper, since the numerator has degree 1 and the denominator has degree 2. Factor the denominator over ± : 7 x ± 3 x 2 ± 3 x ± 4 = 7 x ± 3 x ± 4 () x + 1 Set up the PFD Form: 7 x ± 3 x ± 4 x + 1 = A x ± 4 + B x + 1 Multiply both sides by the LCD, which is the denominator on the left, x ± 4 x + 1 . 7 x ± 3 = Ax + 1 + Bx ± 4 This is the basic equation. Plug in x = ± 1 and solve for B : 7 ± 1 ± 3 = 0 + B ± 1 ± 4 ² ³ ´ µ ± 10 = ± 5 B B = 2 Plug in x = 4 and solve for A : 74 ± 3 = A 4 + 1 ² ³ ´ µ + 0 25 = 5 A A = 5
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
PFD: 7 x ± 3 x ± 4 () x + 1 = A x ± 4 + B x + 1 7 x ± 3 x ± 4 x + 1 = 5 x ± 4 + 2 x + 1 3) Write the PFD (Partial Fraction Decomposition) for 3 x 3 ± x 2 + 12 x + 2 x 2 + 4 2 . You must find the unknowns in the PFD Form. Show all work, as we have done in class! (22 points) Fortunately, the given rational expression is proper, since the numerator has degree 3 and the denominator has degree 4. Factor the denominator over ± : (already done!) Set up the PFD Form: 3 x 3 ± x 2 + 12 x + 2 x 2 + 4 2 = Ax + B x 2 + 4 + Cx + D x 2 + 4 2 Multiply both sides by the LCD, which is the denominator on the left, x 2 + 4 2 .
Background image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 18

Math141SolsFinalF11 - SOLUTIONS TO THE FINAL (CHAPTERS 7,...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online