ANSWERS, Homework Problems, Fall 2014: Lectures Now You Try It, Supplemental problems in written homework, Even Answers. 24x + 72 (x 2 6x + 4) 4

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1 ANSWERS, Homework Problems, Fall 014: Lectures Now You Try It, Supplemental problems in written homework, Even Answers Lecture d [ 4 ] dx x 6x + 4) 3 = 4x + 7 x 6x + 4) 4. a) P 0) = 800 b) dp = 800 ; population is increasing by 700/3 or about 33 people t) 1/3 per year 3. a) f x) = 4xx + 1) x 1)5x ); HT at x = 0, x = ±1 and x = ± 5 b) f x) = c) f x) = d) f x) = 4. a) 4 b) x ; HT at x = 4 x x + 1 ; No horizontal tangent lines 3x + 1) 4/3 4 3x) 3x + 1) 3 ; HT at x = π cubic inches per minute 6. Find dm when t = : the carbon monoxide level is increasing by ppm per year. Answer to Textbook problems, Sec ) f[g[x]] =, domain:, 3) 8 x 3x 8 g[fx)] =, domain:, 0) x [ ) 8 3, 0) fx) = x + x + 5 and gx) = x 3 is one option 5) 4 3x + 1) 5 54) a) $ d) Rx) = 4x + x) /3 x 1 e) R x) = 8x 1) xx + x) 1/3

2 Answer to Textbook problems, Sec. 4.3, continued 56) dq dp = 30 p + 1) 3/ 64) a. dr dq = Lecture 0 6 Q 1. a) dy dx = x3 1 x C Q 3 + C Q 3 b..83 c. increasing b) dy dx = 3x 4x 3 x3, and from a) use the fact that y = to show that 1 x) 1 x the two forms are equivalent substitute into your expression for dy dx ). y = 5 x a) dy dx = 1 6 3y b) 0, 6), 0, 0), and 0, 6) c) 0, 6) and 0, 6): m = 1 1, 0, 0): m = 1 6 d) none e) CORRECTION: find each point at which the curve has a vertical tangent line: 4, ) and 4, ) 4. horizontal tangent lines occur at 1, 1) and 1, 5): equations y = 1 and y = 5 vertical tangent lines occur at, ) and 4, ): equations x = and x = 4 graph is a circle with center 1, ) and radius 3 5. a) dy dx = x y b) height is decreasing by 0.75 foot per one foot increase in horizontal distance x c) height is decreasing by 6 feet per one foot increase in horizontal distance x 6. number of daily hours of unskilled labor must decrease by 1 or approximately hour 7 minutes)

3 Answer to Textbook problems, Sec ) y = 11 1 x 5 6 4) dq dp = p is the rate of change of demand with respect to price q dp dq = q p Lecture is the rate of change of price with respect to demand dy = 1 3 so the y-coordinate is decreasing by 1 3 in/min. ds = 1400 so weekly sales are increasing by $1400 per week. dr = 5 16 dp = so the radius is increasing by mm/min. = 560 so weekly profit is increasing by $560 per week. 5. if x represents the distance between the observer and helicopter, dx = The distance between the helicopter and observer is 34 increasing by about 5.7 ft/sec. 6. If V represents the volume of the sand, dv 8π cubic ft/sec. = 8π, so volume is increasing by 7. If A represents the area of the triangle, da = so area is decreasing by sq. ft/sec. 8. dx = 40 so quantity demanded is increasing by 40 per month. Answer to Textbook problems, Sec ) 5 14) revenue is increasing by $1650 per day 0) energy expended is decreasing by kcal/kg/km per day 4) a) 50 mph b) about mph 8) volume is increasing by 54π cubic in/min 3) 5 3 ft/min 3

4 Lecture 1. y = 3 x + 3. ) 1, e, 1, e ) equations: y = e and y = e 3. f 0) = 3 ln 3 4. f x) = e x + e x 5. m = 3 6. MR = R x) = 50e 0.0x [ 0.0x + 1]; R 100) so revenue is decreasing by $6.77 per unit when 100 units are sold 7. a) P 0) = 16 students b) k = 1 ) 19 ln 49 c) P 10) 80 students/day d) lim t P t) = 800 students Answer to Textbook problems: Sec ) dy dx = 15x3 + 9x + 0x 4)e 5x 30) ds t 5 ln ) = t 44) a) around 118 million people per year b) around 18 million people per year 56) a) 36.8 b) c) close to 0 d) H N) = 100 > 0 for all N; the habit is strengthened with each repetition e0.1n 4

5 Lecture 3 1. m = 4 3 ln3). f ln 3) = f e) = e 4. y = 1 3 x 3 5. f x) = 1 3x ; Horizontal tangent line at x = 4 x 5)x + ) 6. a) At) = r ) t 100 ln b) T = ln ) years 1 + r 100 c) dt dr = ln r) [ ln )] 1 + r 100 dt dr = ln 104[ln1.04)] 4.33: doubling time decreases by 4.33 years per percent increase in the interest rate 7. optional) dy [ dx = x 6 3x 1 ] e 3x 6 + 3x 3x 1) ); m = e optional) y = 4 + ln 4)x ln 4 = 8[ln + 1]x 3 ln 16 Answer to Textbook problems, Sec ) e x 1 lnx 1) + ex 1 x 1 46) Note that d 1 ln ax = dx ax 54) h x) = x x 1 + ln x) 38) [ d dx ax) 54x 1) ln )x x) ] = 1 ax a) = 1 x = d ln x dx 6) a. 4 kj/day b. when a fawn is 5 kg in size, the rate of change of the energy expenditure of the fawn is about kj/day per gram 5

6 Lecture 4 1. increasing on, 3) and 0, ), decreasing on 3, 0). a) Critical Numbers: x = ±3 Increasing on, 3) 3, ) Decreasing on 3, 0) 0, 3) b) No critical numbers Increasing nowhere, Decreasing on c) Critical Numbers: x = 1, 4, 1 ) and 1 ), Increasing on 0, 1) 4, ), Decreasing on 1, 4) d) Critical Number: x = 1 Increasing on, 1 ) ) 1, Decreasing on, 3. Intercepts: 0, 0), 3, 0), 3, 0) Critical Points:, 5 1 ) and, 5 1 ) Increasing on 5. Increasing on 0, 40) 0, 5 ) ) 5, Decreasing on, 1 6. Increasing on, 1), ), Decreasing on 1, ) 6

7 Answer to Textbook problems Sec. 5.1): 10) increasing: 3, 5) and decreasing:, 3) and 5, ) 0) a) CN: x = 3, 0 and 1 b) increasing: 3, 0) and 1, ) c) decreasing:, 3) and 0, 1) 6) a) CN: x = ± 3, ±3 b) increasing: 3, 3 ) ) c) decreasing: 3, 3 and 3 ), 3 30) a) CN: x = 0, b) increasing: 0, ), c) decreasing:, 0) 36) a) CN: x = 1 4 and x = 0, b) increasing: 1 4, ), c) decreasing:, 1 4 ) 38) vertex: b a, f b )) a increasing:, b ) and decreasing: b ) a a, 46) increasing over its domain 58) 0, ) 6) a) f x) < 0, b) mpg/lb Lecture 5 1. a) Increasing:, 1) and 0, 1), Decreasing: 1, 0) and 1, ) relative minimum: 10 = f0) and relative maximum: 5 = f 1) = f1) ) 1 b) Increasing: e,, Decreasing: 0, 1 ) e relative minimum: 1 ) 1 e = f, no relative maximum e c) Increasing: 1, 1), or 1, 0) and 0, 1)), Decreasing:, 1) and 1, ) relative maximum: = f1) and relative minimum: = f 1) d) Increasing:, ), 1) and, ) Decreasing: 1, ), ) relative maximum: 1 e4 = f 1) and relative minimum: e = f) 7

8 . a) f x) = 1 x b) critical number at x = 1 x + 1) 3 c) relative maximum: 1 ) 1 8 = f, no relative minimum 3. a) critical numbers: x = 0 and x = 4 b) x = 4 c) x = 0 d) relative maximum at x = 4 and relative minimum at x = 0 4. He should charge $300 to sell 400 tablets 5. relative maxima at x = and x = 4, relative minimum at x = 0 Answer to Textbook problems Sec. 5.): 10) relative minimum at x = 3 and relative maximum at x = 5 4) local maximum; f0) = 0 and local minimum: f) = 9 /3 ) 3) local minimum only: f e) = e 48) they should sell 5 items at a price of $4600 8