Answers:

Answers to Homework Problems

Answers to Selected Problems in Chapter 1

1 : m, b and distance are: (a) 1.5, -3, ; (b) -3, 2, ; (c) 1, 0, 2 ; (d) 0, 3, 6; (e) 1, 1, 5 ; (f) -3, 0.02, 0.02 .

2 : y- and x-intercepts are: (a) 2, -1; (b) 6, 6; (c) 1/2, -1; (d) 3/2, none; (e) -2, -3.

3 : (a) , (b) , (c) , (d) (or ), (e) , (f) .

4 : radius and center are: (a) 5, (0,-5); (b) 7, (5,0); (c) 5, (3,4).

5 : .

6 : .

7 : .

8(b) ; .

9: (a) y = 0 for , for ,
for ; (b) he's introducing a ``negative income tax''
as a rather cheap way of getting good PR --- so that, for example, a
penniless person receives 10 gold coins from the King (since the y-intercept is -10).

10:(a) ; (b) .

11: for , for , y = for .

Answers to Selected Problems in Chapter 2

1: [ ).

2: x not equal -1.

3: x not equal +/- 1.

4: (- ).

5: all x.

6: [ ).

7: [h-r,h+r].

8: x negative or in [ ).

9 : (- 1/3, 1/3) ;

10: x nonnegative but x is not 1.

11: x nonnegative but x is not 1.

Answers to Selected Problems in Chapter 3

1: (a) -5, (b) -2.47106, (c) -2.40679, (d) -2.40068, (e) -2.4.

2: (a) -4/3, (b) -24/7, (c) 7/24, (d) 3/;

3: (a) 5, (b) 4.1, (c) 4.01, (d) 4.001; limit = 4.

4: (a) -10.29, (b) -9.849, (c) -9.8049; -9.8-4.9 t; -9.8 m/sec.

5: (a) -0.107527, (b) -0.110742, (c) -0.111074; -1/ (3*(3+ x)); = -1/9.
6: ( )+1/(1+ ); .

7: (a) 3.31, (b) 3.003001, (c) 3.0000300001; 3+3 + ; .

Answers to Selected Problems in Chapter 4 (first two problems)

1: .

2: (a) , (b) , (c) , (d) .

Answers to Selected Problems in Chapter 5

1: (a) 4.4, (b) 0.9, (c) 1.01, (d) 0.0099, (e) 2.0025, (f) 5.1,
(g) 4.9, (h) 2.001.

2: 3.94

3: 0.9433.

4: 15 ft 2 in.

5: 208.9 ft.

6: +/- .

7: +/- 30, +/-1 and +/- 3 .

8: (a) 5 sec, (b) +/- 0.025 sec, +/- 0.5 , (c) +/- p/2 .

9: (a) r = , , where C = 1/ , (b) +/- 0.5 , (c) +/- p/2.

10: 8 +/- 0.15.

11: +/- 3p .

12: (a) +/- p/2, (b) +/- p/3.

13: (a) +/- p , (b) +/- 3p , (c) +/- 2p .

14: +/- 2p/3 .

15: (a) +/-2p , (b) +/- p , (c) +/- p .

16: 1 >83/84.

17: = -1/12 cm.

Answers to Selected Problems in Chapter 6

1: (a) ,

(b) ,

(c)-(d) ,

(e) ,

(f) ,

(g) , (h) - ,

(i)-(j) - - ,
(k) 1/2 ,

(l) ,

(m) .

(n) .


2: (a) , -9.8;

(b) a, 0;

(c) 1/2 ,-1/4 ;

(d) 1/2 , - 1/4 , which simplifies to 1/4 .

3: , ; the bicycle is moving leftward initially at 5 m/sec, but its speed is decreasing; after 2.5 sec it has slowed to 0 m/sec, at which point it reverses direction and starts moving to the right.

4: 2+ n/4 .

5: +/- 0.96 .

6: +/- .

7: (a) ; (b) - .

Answers to Selected Problems in Chapter 7

1: (a) 1.992, (b) 5.0129, (c) 4.00133.

2: (a) y' = x/y, (b) y' = -( )/( ),
(c) y' = ( )/( ).

3: y' = 1.

4: y = - 1/13 x+ 14/13 , 0.9923.

5: B is 4 times as strong as A.

6: y' = 3/19.

7: (a) dT/dh = - / , (b) h = -0.0259 T+5.493.

8: (a) , (b) 12.492 sec.

Answers to Selected Problems in Chapter 8

1: (a) , (b) , (c) , (d) .

2: (a) 25 rad/sec (b) 6.944 rad/sec.

3: (a) , (b) , (c) .

4: = = = .

6: warmest: on July 12; coolest: - on January 10.

7: , .

8: (a) ;
(b) ;
(c) ;
(d) ;
(e) (or, equivalently, .

Answers to Selected Problems in Chapter 9

1: 0.96509 (exact value = 0.96569).

2: (a) , (b) , .

3: (a) 0.46977, (b) 0.484885, (c) 0.93018.

4: (a) ;

(b) ;

(c) ;

(d) ;

(e) .


5: (a) ,

(b) - ,

(c) ,

(d) .

6: 0.7632.

7: 1.241.

8: (b) 0.5 = 2*sin(Pi*t)*sin(Pi*x/3);

(c) x = 0.5;

(d) ;

(e) = 0.4.

Answers to Selected Problems in Chapter 10

1: (a) , ;

(b) , ;

(c) , .

2: (a) 25 rad/sec;

(b) rev/sec;

(c) m/sec;

(d) 21.6 km/hr.

3: (a) 34.3 mph;

(b) 55 rad/sec, 8.75 rev/sec.

4: (a) 20*t;

(b) ;

(c) .

5: (a) ;

(b) ;

(c) ;

(d) .

6: (a) ;

(b) 2.5 sec;

(c) 5.04 sec.

7: (a) ;

(b) ( )/ ;

(c) ;

(d) 124 ft/sec at below horizontal (hits ground at t = 2).

8: (a) , ;
(b) ;
(c) .

Answers to Selected Problems in Chapter 11

1: (b) .

2 : (a) ;

(b) ;

(c) .

3: .

4: .

5: .

6: .

Answers to Selected Problems in Chapter 12

1: 0.0199 cm/sec.

2: 0.25 m/sec.

3 : mi/min ( = 4.2 mi/sec).

4: 6.71 ft/sec.

5: 2.12 cm/sec.

6: 0.65 ft/sec

7: 7.91 m/sec

8: 1.172 m/min

9: 0.5454 m/min

10: 6 ft/sec, 2.5 ft/sec

11: 69.4 mph

12: 0.1234 /sec.

13 : (a) ;
(b) .


: 81.6 m/sec

15: (a) (or else, using
the law of sines, an equivalent answer is: ;

(b) = (or else: ;

(c) 3.79 cm/sec.

16: (a) a ;

(b) closing at a little over 15 rad/sec;

(c) no violation of relativity, because no part of the scissors is moving
faster than 300 km/sec.

17: (a) -50 m/sec; (b) 68 m/sec; (c)...

18: (a) ;

(b) .

Answers to Selected Problems in Chapter 13

1: Both and when . If n is even, the origin is a minimum, and the curve is always concave up. If n is odd, the origin is a point of inflection, and the curve is concave up for x>0 and concave down for x<0.

2: (a) min at (0.5,-0.25), always concave up, no inflection;

(b) max at (1,4), min at (-1,0), inflection at (0,2), conc up for
x<0, conc down for x>0;

(c) max at (2,20), min at (4,16), inflection at (3,18), conc up for x>3,
conc down for x<3;

(d) max at (0,3), min at ( +/- 1,2), inflections at ( +/- 0.577,2.444),
conc down for -0.577<x<0.577, otherwise conc up, symmetric with respect
to y-axis;

(e) min at (1,-1), inflections at (0,0) and (2/3,-0.5926), conc down
for 0<x<2/3, otherwise conc up;

(f) no max, min or inflections, y-axis is asymptote, conc up for x<0,
conc down for x>0, the line y = x is a slanted asymptote;

(g) no max or min, inflections at ( +/- 1,2), y-axis is asymptote, conc up for x<-1 and x>1, otherwise conc down, symmetric with respect to y-axis;

(h) min when x = / (and x = /12 pmany multiple of ), max when x = /12 (again can add or subtract any multiple of ), inflections when x = /4, /4 ( pm any multiple of ), conc up for Pi/4<x<3Pi/4, general appearance is oscillation around the slanted line y = -x;

(i) max at all points ( /4 +/- ) (where k is any integer), min at ( /4 +/- ), inflections at ( /4 +/- ), you get a sinusoidal function;

(j) asymptotes at odd multiples of , max when x = - /2 +/- , min when x = /2 +/- , inflections at even multiples of , in each section between asymptotes the left half is conc down, the right half is conc up.

5: (a) max at (0,0), min at ( +/- );

(b) and ;

(c) ( +/- );

(d) use 2nd derivative test.

Answers to Selected Problems in Chapter 14

1: (a) max value f(2) = 5; (b) min value f(0) = 1.

2: P/4 by P/4.

3: by by ; ratio = .

4: /8.

5: x = 2000, P = 6, profit = 5000.

6: by , area = .

7: .

8: .

9: 4/27.

10: Take x = a, where f(x) has a corner (if w>v, then is never
zero).

11: h/r = 2; h/r = 3.5.

12 : (a) a/6 on a side; (b) ( )/6 on a
side

13: provided that ; otherwise, the optimal window consists of only the semicircular piece.

14: , max area = approx , bigger than in Problem 3.
15: 0.00139 hours ( approx 5 sec), misses by 0.5726 mi.

16: ratio = a/b.

17: by

18: 1/ ( approx58 %).

19: 18 in times 18 in times 36 in.~~~

20: r = 2.71, h = 10.84.

21: h/r = .

22: h = 1.1798R, r = 0.9837R, 28.54 %

23: from A

24: 1/2.

25: x = 2500, P = 5, profit = 7000.

26: Snell's law: .

Answers to Selected Problems in Chapter 15

1: .

2: (exactly), x = 2.19 to three significant figures ( ..., ...).

3: .

4: Hint : Set the slope of the line joining (- /2,-1) to ( ) equal to the derivative of , then multiply through by the denominator to get the f(x) in Newton's method: ; then , x[1] = 0.7633, = 0.7603.

5: t = 0.5062 sec, height = 6.9938 ft.

6: take , then .

7: .

8: , 0.49254 rad ( = ), 31.89 ft

Answers to Selected Problems in Chapter 16

1:
(a) ,

(b) ,

(c) ,

(d) ,

(e) ,

(f) ,

(g) ,

(h) ,

(i) - ,

(j) .

2: - .

3: - .

4: .

5: (a) k*t^3/6+v[0]*t+s[0],

(b) ,

(c) ,

(d) .

Answers to Selected Problems in Chapter 17

1: (a) 6.25 vs. 16/3 = 5.333,

(b) 5.8125 vs. 5.333,

(c) 4.146 vs. 5.333,

(d) 1.239 vs. 1,

(e) 1.125 vs. 1.

2: (a) 55,

(b) 25/12,

(c) .

3: (a) 28.5, (b) 14/9, (c) 2/3, (d) 0.24, (e) .

4: (a) 2/3,

(b) ,

(c) 4/ + 4/3,

(d) .

5: (a) 13/3,

(b) 56/15,

(c) 0,

(d) /4,

(e) 3/4,

(f) 9.

6: (a) 14600 degree-days,

(b) 4523 degree-days,

(c) 277 degree-days.

7(a)

code

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Answers to Selected Problems in Chapter 18

1: (a) ,

(b) ,

(c) - ,

(d) 2 ;

(e) 0.

2: (a) ,

(b) ,

(c) ,

(d) .

3: (a) ,

(b) .

4: (a)

(b) .

5: (a) ,

(b) .

6: (a) ,

(b) ,

(c) .

Answers to Practice Midterms

Answers to Practice First Midterm #1

1. (a) ; (b) ; (c) all x except 2 and 4; (d) -1<x<1.

2 . (a) ABIJ; (b) DEFG; (c) GHI; (d) FJ; (e) J; (f) F.

3. = = (approx) = 3.85 ft. 4 . 9 = 9.0864 cm.

Answers to Practice First Midterm #2

1. (a) like the -parabola, but shifted to have its vertex at (-2,-4);

(b) bottom half of ellipse, passes through pm 1 on the x-axis and -5 on the y-axis.

2. - /2<x< /2.

3. (a) CDJ; (b) AFGH; (c) ABHI; (d) CGJ; (e) C; (f) G.

4. (a) = 9.5 m/sec;

(b) = , f(5 +/- 0.1) = ( approx) 9.5 +/- = 9.5 +/- 0.79 m/sec.

5. y = f(x) = 1000/x, = (approx) f + = = 92.3 .

Answers to Practice First Midterm #3

1. 0.5< .

2. (a) AFG; (b) CDI; (c) DE; (d) CFI; (e) C.

3. g = f(t) = 22.5/ ;

(a) f(3) = 2.50;

(b) f(3.1) (approx) = 2.33, so g is between 2.33 and 2.50 m/ .

4. y = f(x) = , f(21.6 +/- 0.2) = (approx) 28.8 in

Answers to Practice First Midterm #4

1. (a) -35<x<15;

(b) .

2. (b) when t = 12 (parachute opens); when t = 0 (drop from plane); right after parachute opens.

3. 12.8. 4.

(a) A = /4 ;

(b) ;

(c) s = f(r) = , f(21) = (approx) = 26.93 cm.

Answers to Practice Second Midterm #1

1. 19.861.

2. (a) (-1,-1.5) is max; (b) x<-1 and x>0; (c) y-axis; (d) none; (e) see below.

code for 2(e)

Answers to Practice Second Midterm #2

1. (a) (0,0), ( ), (- ); (b) - <x< - , and 0<x< ; (c) (1,5/6), (-1,5/6); (d) -1<x<1; (e) see below.

>

code for 1(e)

Answers to Practice Second Midterm #3

1. (a) (0,0); (b) (1,2.75) and (3,6.75); (c) see below;

code for 1(c)

Answers to Practice Second Midterm #4

1. (a) max: (0,0), min: (+/- ); (b) - <x<0 and <x< ; (c) ( +/- ,-2.925); (d) - <x< ; (e) see below.

code for 1(e)

Answers to Practice Final Exams

Answers to Practice Final Exam #1

1: (a) -15/4; (b) 5/ , or else - c .
2. -78.125 m/sec.
3: (b) v = /2, ; v = t- 1/2, ; v = 3/2, ; (c) 8/3 m.

4: area of hexagon = , optimal ratio h/r is = 2.2053...

5 : (a) 4.60555 cm; (b) ; (c) dx/dt = ;

(d) (1/21) = -48.654, f(1/20) = (approx) 4.4897 cm.

6: (a) ; (b) 8.18361 cm; (c) sec,
sec, cm.

Answers to Practice Final Exam #2

1: (a) 1/4 ; (b) 2/9.

2: (a) 20+ 10/ rev; (b) rad; (c) 109.25 ft.

3. 16 ft/sec.

4: 2.033 ft.

5: s = 1.59 ft, l = 4.76 ft.

6: (a)

(b) ; (c) ( /12, /144+1/2) and ( /12, /144+1/2); (d) - /4; (e) , ; (f) no other max/min; (g) graph below:

code for diagram

Answers to Practice Final Exam #3

1: (a) -2/7 ; (b) = 0.8619.
2: (b) 2-t for , for ,
0.5 for , for ; (c) 2.92 m.
3: -9.823 ft/sec.
4 : differentiate = const, get = -32/49, answer is 24.013.

5: (a) vol = , area = ;
(b) .

(c) .

6: Choose in Newton's method, so that = ; then = 3, = 3.3163, = 3.3094.

7: (a) x = 1/2 ;
(b) ;
(c) .

Answers to Practice Final Exam #4

1: (a) 28/15; (b) 3B/16 .

2: (a) , ; , ; (b) 9.5 m.

3: (a) Vol = ; (b) = (where ); (c) 2904 .


4: A good initial guess is s_0 = 10 (since vol = 1000, surface area = 600). Choose ; then = 9.757.


5: (a) , , ;

(b) .

6: (a) , ; (b) ; (c) (write T in terms of using (b)); (d) find and then take : -0.5781 (the minus sign means that a positive error in alpha leads to a negative error in X).