{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# HW3S - Gauthier Joseph Homework 3 Due 3:00 am Inst...

This preview shows pages 1–4. Sign up to view the full content.

Gauthier, Joseph – Homework 3 – Due: Sep 15 2007, 3:00 am – Inst: JEGilbert 1 This print-out should have 20 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. The due time is Central time. 001 (part 1 of 1) 10 points Find all values of t in (0 , π ) for which the tangent line to the graph of x ( t ) = t + sin 2 t , y ( t ) = t - sin 2 t , is vertical. 1. t = π 4 , 3 π 4 2. t = π 3 , 2 π 3 3. t = π 6 , 5 π 6 correct 4. t = 7 π 12 , 11 π 12 5. t = π 12 , 5 π 12 6. t = 5 π 12 , 7 π 12 Explanation: After differentiating, we see that x 0 ( t ) = 1 + 2 cos 2 t , y 0 ( t ) = 1 - 2 cos 2 t . Thus dy dx = y 0 ( t ) x 0 ( t ) = 1 + 2 cos 2 t 1 - 2 cos 2 t . Now the tangent line will be vertical when x 0 ( t ) = 1 - 2 cos 2 t = 0 , hence when cos 2 t = 1 / 2 . For t in (0 , π ), therefore, the tangent line will be vertical when t = π 6 , 5 π 6 . keywords: derivative, tangent line, vertical, horizontal, trig function, parametric curve 002 (part 1 of 1) 10 points Locate the points given in polar coordinates by P - 1 , 1 2 π · , Q 4 , 2 3 π · , R 2 , 1 4 π · , among 2 4 - 2 - 4 2 4 - 2 - 4 1. P : Q : R : 2. P : Q : R : 3. P : Q : R : 4. P : Q : R : 5. P : Q : R : correct 6. P : Q : R : Explanation: To convert from polar coordinates to Carte- sian coordinates we use x = r cos θ , y = r sin θ . For then the points P - 1 , 1 2 π · , Q 4 , 2 3 π · , R 2 , 1 4 π · ,

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

View Full Document
Gauthier, Joseph – Homework 3 – Due: Sep 15 2007, 3:00 am – Inst: JEGilbert 2 correspond to P : Q : R : in Cartesian coordinates. keywords: polar coordinates, Cartesian coor- dinates, change of coordinates, 003 (part 1 of 1) 10 points Which, if any, of A. ( - 4 , 4 π/ 3) , B. (4 , π/ 6) , C. (4 , 7 π/ 3) , are polar coordinates for the point given in Cartesian coordinates by P (2 , 2 3)? 1. B and C only 2. C only 3. none of them 4. A and C only correct 5. all of them 6. A and B only 7. A only 8. B only Explanation: To convert from Cartesian coordinates to polar coordinates we use the relations: x = r cos θ , y = r sin θ , so that r 2 = x 2 + y 2 , tan θ = y x . For the point P (2 , 2 3) in Cartesian co- ordinates, therefore, one choice of r and θ is r = 4 and θ = π/ 3, but there are equivalent solutions for r < 0 as well as values of θ dif- fering by integer multiples of π . For the given choices we thus see that A. TRUE: - 4 cos(4 π/ 3) = 2 , - 4 sin(4 π/ 3) = 2 3 . B. FALSE: θ incorrect. C. TRUE: differs from π/ 3 by 2 π . keywords: T/F, polar coordinaates, Cartesian coordinates 004 (part 1 of 1) 10 points A point P is given in Cartesian coordinates by P ( - 1 , 1). Find polar coordinates ( r, θ ) of this point with r < 0 and 0 θ < 2 π . 1. - 2 , 7 π 4 · correct 2. - 3 , 5 π 4 · 3. - 2 , 3 π 4 · 4. - 2 , 5 π 4 · 5. - 3 , 3 π 4 · 6. - 3 , 7 π 4 · Explanation: Since the relationship between Cartesian coordinates and polar coordinates is x = r cos θ , y = r sin θ , the point P ( - 1 , 1) in Cartesian coordinates can be given in polar coordinates as P - 2 , 7 π 4 · , keywords: polar coordinates, Cartesian coor- dinates, 005 (part 1 of 1) 10 points
Gauthier, Joseph – Homework 3 – Due: Sep 15 2007, 3:00 am – Inst: JEGilbert 3 Find a polar representation for the curve whose Cartesian equation is x 2 + ( y - 2) 2 = 4 .

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 14

HW3S - Gauthier Joseph Homework 3 Due 3:00 am Inst...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online