This file is /conf/snippets/setheader.pg you can use it as a model for creating files which introduce each problem set.

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1 Yanimov Almog WeBWorK assignment number Sections is ue : 08/3/207 at 03:2pm CDT. Te (* replace wit url for te course ome page *) for te course contains te syllabus, graing policy an oter information. Mat-550--Fall2009 Tis file is /conf/snippets/setheaer.pg you can use it as a moel for creating files wic introuce eac problem set. Te primary purpose of WeBWorK is to let you know tat you are getting te correct answer or to alert you if you are making some kin of mistake. Usually you can attempt a problem as many times as you want before te ue ate. However, if you are aving trouble figuring out your error, you soul consult te book, or ask a fellow stuent, one of te TA s or your professor for elp. Don t spen a lot of time guessing it s not very efficient or effective. Give 4 or 5 significant igits for (floating point) numerical answers. For most problems wen entering numerical answers, you can if you wis enter elementary expressions suc as 2 3 instea of 8, sin(3 pi/2)instea of -, e (ln(2)) instea of 2, (2 +tan(3)) (4 sin(5)) 6 7/8 instea of , etc. Here s te list of te functions wic WeBWorK unerstans. You can use te Feeback button on eac problem page to sen to te professors.. ( pt) Rogawski-ET/3 Differentiation/3. Definition of te Derivative- /3..7.pg From Rogawski ET section 3., exercise 7. Calculate te slope of te secant line troug te points on te grap were x = an x = 3. slope = Determine f (a) for a =,2,4,7. f () = f (2) = f (4) = f (7) = Remember tat te value of te erivative of f at x = a can be interprete as te slope of te line tangent to te grap of y = f (x) at x = a. From te figure, we see tat te grap of y = f (x) is a orizontal line (tat is, a line wit zero slope) on te interval 0 x 3. Accoringly, f () = f (2) = 0. On te interval 3 x 5, te grap of y = f (x) is a line of slope 0.5; tus, f (4) = 0.5. Finally, te line tangent to te grap of y = f (x) at x = 7 is orizontal, so f (7) = 0. Te slope of te secant line is f (3) f () 3 = 0 ( 2) 2 = 2. ( pt) Rogawski-ET/3 Differentiation/3. Definition of te Derivative- /3...pg From Rogawski ET section 3., exercise. Let f (x) be te function wose grap is sown below. 3. ( pt) Rogawski-ET/3 Differentiation/3. Definition of te Derivative- /3..3.pg From Rogawski ET section 3., exercise 3. Let f (x) be te function wose grap is sown below.

2 Wic is larger? A. f (5.5) B. f (6.5) Te line tangent to te grap of y = f (x) at x = 6.5 as a larger slope tan te line tangent to te grap of y = f (x) at x = 5.5. Terefore, f (6.5) is larger tan f (5.5). 4. ( pt) Rogawski-ET/3 Differentiation/3. Definition of te Derivative- /3..5.pg From Rogawski ET section 3., exercise 5. Use te efinition of te erivative to fin te erivative of: x 5. lim 0 f (x+) f (x) lim 0 = lim = 0 x+ 5 x+5 = lim 0 = 5. ( pt) Rogawski-ET/3 Differentiation/3. Definition of te Derivative- /3..9.pg From Rogawski ET section 3., exercise 9. let x. Compute te ifference quotient for f (x) at x = 2 wit = 0.5 ifference quotient = f (x+) f (x) Te ifference quotient is = = ( pt) Rogawski-ET/3 Differentiation/3. Definition of te Derivative- /3..53.pg From Rogawski ET section 3., exercise 53. Te limit below represents a erivative f (a). Fin f (x) an a. (5 + ) 3 25 lim 0 a = Te ifference quotient (5 + ) 3 25 as te form f (a + ) f (a) were x 3 an a = ( pt) Rogawski-ET/3 Differentiation/3. Definition of te Derivative- /3..57.pg From Rogawski ET section 3., exercise 57. Te limit below represents a erivative f (a). Fin f (x) an a lim 0 a = Te ifference quotient as te form f (a + ) f (a) were 4 x an a = ( pt) Rogawski-ET/3 Differentiation/3.2 Te Derivative as a Function- /3.2..pg From Rogawski ET section 3.2, exercise. Use te Power Rule to compute te erivative: t t2/3 t=6 = t t2/3 = 2 3 t /3, so t t2/3 t=6 = 2 3 (6) /3 = ( pt) Rogawski-ET/3 Differentiation/3.2 Te Derivative as a Function- /3.2.3.pg From Rogawski ET section 3.2, exercise 3. Use te power rule to compute te erivative. x x0.6 x x0.6 = 0.6(x 0.6 ) = 0.6x ( pt) Rogawski-ET/3 Differentiation/3.2 Te Derivative as a Function- / pg From Rogawski ET section 3.2, exercise 23. Fin te erivative of te function x 3 + x 2 4. x (x3 + x 2 4) = 3x 2 + 2x.. ( pt) Rogawski-ET/3 Differentiation/3.2 Te Derivative as a Function- / pg From Rogawski ET section 3.2, exercise 27. Fin te erivative of te function 8x 3 + x x (8x 3 + x 2 + 4) = 3 8x 4 + 2x = 24 x 4 + 2x.

3 2. ( pt) Rogawski-ET/3 Differentiation/3.2 Te Derivative as a Function- / pg From Rogawski ET section 3.2, exercise 55. Fin all values of x were te tangent lines to y = x 6 an y = x 7 are parallel. x = Let x 6 an let g(x) = x 7. Te two graps ave parallel tangent lines at all x were g (x). g (x) 6x 5 = 7x 6 ence, x = 0 or x = x 5 7x 6 = 0 x 5 (6 7x) = 0 3. ( pt) Rogawski-ET/3 Differentiation/3.2 Te Derivative as a Function- / pg From Rogawski ET section 3.2, exercise 57. Determine coefficients a an b suc tat p(x) = x 2 + ax + b satisfies p() = 7 an p () =. a = b = Let p(x) = x 2 + ax + b satisfy p() = 7 an p () =. Since p (x) = 2x + a, tis implies 7 = p() = + a + b an = p () = 2 + a; i.e., a = an b = ( pt) Rogawski-ET/3 Differentiation/3.2 Te Derivative as a Function- / pg Domain of From Rogawski ET section 3.2, exercise 77. Fin te points c (if any) suc tat f (c) oes not exist. 6. ( pt) Library/ASU-topics/setDerivativeFunction/ pg Suppose tat x + c = Here is te grap of x +. Its erivative oes not exist at x =. At tat value of x tere is a sarp point. 5. ( pt) Library/UVA-Stew5e/setUVA-Stew5e-C02S09- DerivAsFunct/ pg Use te efinition of te erivative (on t be tempte to take sortcuts!) to fin te erivative of te function 2 + 4x. Ten state te omain of te function an te omain of te erivative. Note: Wen entering interval notation in WeBWorK, use I for, -I for, an U for te union symbol. If te set is empty, enter (0,0). Domain of Fin f (x). f (x + ) 8x 2 3x x Generate by te WeBWorK system c WeBWorK Team, Department of Matematics, University of Rocester 3

4 Yanimov Almog WeBWorK assignment number Sections is ue : 08/4/207 at 09:57am CDT. Te (* replace wit url for te course ome page *) for te course contains te syllabus, graing policy an oter information. Mat-550--Fall2009 Tis file is /conf/snippets/setheaer.pg you can use it as a moel for creating files wic introuce eac problem set. Te primary purpose of WeBWorK is to let you know tat you are getting te correct answer or to alert you if you are making some kin of mistake. Usually you can attempt a problem as many times as you want before te ue ate. However, if you are aving trouble figuring out your error, you soul consult te book, or ask a fellow stuent, one of te TA s or your professor for elp. Don t spen a lot of time guessing it s not very efficient or effective. Give 4 or 5 significant igits for (floating point) numerical answers. For most problems wen entering numerical answers, you can if you wis enter elementary expressions suc as 2 3 instea of 8, sin(3 pi/2)instea of -, e (ln(2)) instea of 2, (2 +tan(3)) (4 sin(5)) 6 7/8 instea of , etc. Here s te list of te functions wic WeBWorK unerstans. You can use te Feeback button on eac problem page to sen to te professors.. ( pt) Rogawski-ET/3 Differentiation/3.3 Prouct an Quotient Rules- /3.3.5.pg From Rogawski ET section 3.3, exercise 5. Use te Prouct Rule to compute te erivative: t ((t2 + )(t + 9)) t=2 = Let y = (t 2 + )(t + 9). Ten y t = ( t (t2 +))(t +9)+(t 2 +) t (t +9) = 2t(t +9)+(t2 + ) = 3t 2 + 8t +. Terefore, y t t=2 = 3(2) = ( pt) Rogawski-ET/3 Differentiation/3.3 Prouct an Quotient Rules- /3.3.9.pg From Rogawski ET section 3.3, exercise 9. Compute te erivative: Te erivative is: Using te Quotient Rule: Terefore, ( x x + ) x= ( ) (x + )(0) () = x x + (x + ) 2 = (x + ) 2 ( = x x + ) x= ( + ) 2 = ( pt) Rogawski-ET/3 Differentiation/3.3 Prouct an Quotient Rules- / pg From Rogawski ET section 3.3, exercise 53. Calculate F (0), were F(x) = 9x6 + 4x 5 0x 4 + 3x 0x 2 (8x 9 + 5x 4 ) + 5. Hint: Do not calculate F (x). Instea, write F(x) = f (x)/g(x) an express F (0) irectly in terms of f (0), f (0),g(0),g (0). F (0) = Taking te int, let an let Ten F(x) = f (x) g(x). Now, an 9x 6 + 4x 5 0x 4 + 3x g(x) = 0x 2 ( 8x 9 + 5x 4) x x 4 0 4x g (x) = 0 2x ( 8 9x x 3) Moreover, f (0) = 0, f (0) = 3, g(0) = 5, an g (0) = 0. Using te quotient rule: F (0) = g(0) f (0) f (0)g (0) (g(0)) 2 = 5 25 = ( pt) Library/UVA-Stew5e/setUVA-Stew5e-C03S02- ProQuotRules/3-2-3b.pg Consier te functions f (x) an g(x), for wic f (0) = 7, g(0) = 9, f (0) = 8, an g (0) = 7. Fin (0) for te function (x) = f (x) g(x). (0) =

5 5. ( pt) Library/UVA-Stew5e/setUVA-Stew5e-C03S02- ProQuotRules/ pg Fin an equation for te line tangent to te grap of at te point (2,.25). y = 5x x ( pt) Rogawski-ET/3 Differentiation/3.4 Rates of Cange/3.4.5.pg From Rogawski ET section 3.4, exercise 5. Calculate te rate of cange V r were V is te volume of a cyliner wose eigt is equal to 3 times its raius. (Te volume of a cyliner of eigt an raius r is πr 2 ). Te rate of cange is To fin te rate of cange V r, first apply te conition tat = 3r to simplify te volume formula. We get V = π r 2 (3r) = 3πr 3. Te erivative is given by te power rule - V r = (3)(3)πr2 = 9πr ( pt) Rogawski-ET/3 Differentiation/3.4 Rates of Cange/3.4.3.pg From Rogawski ET section 3.4, exercise 3. A stone is tosse vertically upwar wit an initial velocity of 30 ft/s from te top of a 23 ft builing. (a) Wat is te eigt of te stone after 0.42 s? (b) Fin te velocity of te stone after s. (c) Wen oes te stone it te groun? (a) (b) (c) We employ Galileo s formula, s(t) = s 0 + v 0 t /2gt 2 = t 6t 2 were te time t is in secons (s) an te eigt s is in feet (ft). (a) Te eigt of te stone after 0.42 secons is s(0.42) = (b) Te velocity at time t is s (t) = 30 32t. Wen t =, tis is -2 ft/s. (c) Wen te stone its te groun, its eigt is zero. Solve 23+30t 6t 2 = 0 to obtain t = 30± or t = (We iscar te oter solution, since it represents a time before te stone was trown.) 8. ( pt) Library/UVA-Stew5e/setUVA-Stew5e-C03S03- RatesofCange/3-3-0.pg Suppose tat a particle moves accoring to te law of motion s = t 2 9t + 22, t 0. (A) Fin te velocity at time t. v(t) = (B) Wat is te velocity after 3 secons? Velocity after 3 secons = (C) Fin all values of t for wic te particle is at rest. (If tere are no suc values, enter 0. If tere are more tan one value, list tem separate by commas.) 2 t = (D) Use interval notation to inicate wen te particle is moving in te positive irection. (If te particle is never moving in te positive irection, enter (0,0).) Answer = 9. ( pt) Library/UVA-Stew5e/setUVA-Stew5e-C03S03- RatesofCange/3-3-6.pg If a tank ols 4200 gallons of water, wic rains from te bottom of te tank in 43 minutes, ten Torricelli s Law gives te volume V of water remaining in te tank after t minutes as ( V = 4200 t ) 2, 0 t Fin te rate at wic te water is raining from te tank after: (A) 4 minutes Rate of cange = (B) 9 minutes Rate of cange = (C) 6 minutes Rate of cange = 0. ( pt) Library/ma22DB/set4/s pg Suppose tat te cost, in ollars, for a company to prouce x pairs of a new line of jeans is C(x) = x + 0.0x x 3. (a) Fin te marginal cost function. Answer: (b) Fin te marginal cost at x = 00. Answer: (c) Fin te cost at x = 00. Answer:. ( pt) Library/UVA-Stew5e/setUVA-Stew5e-C03S02- ProQuotRules/3-2-06a.pg Suppose tat Fin f (). f () = ex x ( pt) Library/UVA-Stew5e/setUVA-Stew5e-C03S02- ProQuotRules/ pg Fin an equation for te line tangent to te grap of at te point (a, f (a)) for a =. y = 2xe x

6 3. ( pt) Library/UVA-Stew5e/setUVA-Stew5e-C03S02- ProQuotRules/ pg Fin f (5). 5. ( pt) Rogawski-ET/3 Differentiation/3.4 Rates of Cange- /3.4.3.pg From Rogawski ET section 3.4, exercise 3. Wat is te velocity of an object roppe from a eigt of 0 m wen it its te groun? Note: Click on grap for larger version in new browser winow. Te graps of te function f (given in blue) an g (given in re) are plotte above. Suppose tat u(x) = f (x)g(x) an v(x) = f (x)/g(x). Fin eac of te following: u () = v () = If 4. ( pt) Library/Rocester/setDerivatives2Formulas/s2 2 2.pg fin f (x). 3 x2 8 + x 2 We employ Galileo s formula, s(t) = s 0 /2gt 2 = 0 4.9t 2 were te time, t, is in secons (s) an te eigt, s, is in meters (m). Wen te ball its te groun its eigt is 0. Solve s(t) = 0 4.9t 2 = 0 to obtain t = s. (We iscar te negative time, wic took place before te ball was roppe.) Te velocity at time t is given by s (t) = t, an so te velocity at impact is v( ) = 9.8( ) = m/s Tis signifies tat te ball is falling at m/s. 6. ( pt) Library/ASU-topics/setProuctQuotientRule/ pg Let x x Fin all te values of x for wic 0. Wat is te prouct of all tese values? (For example, if f (x) was equal to zero at te points, 2, an 3, ten te answer woul be *2*3 = 6.) Prouct = Generate by te WeBWorK system c WeBWorK Team, Department of Matematics, University of Rocester 3

7 Yanimov Almog WeBWorK assignment number Sections is ue : 08/4/207 at :4am CDT. Te (* replace wit url for te course ome page *) for te course contains te syllabus, graing policy an oter information. Mat-550--Fall2009 Tis file is /conf/snippets/setheaer.pg you can use it as a moel for creating files wic introuce eac problem set. Te primary purpose of WeBWorK is to let you know tat you are getting te correct answer or to alert you if you are making some kin of mistake. Usually you can attempt a problem as many times as you want before te ue ate. However, if you are aving trouble figuring out your error, you soul consult te book, or ask a fellow stuent, one of te TA s or your professor for elp. Don t spen a lot of time guessing it s not very efficient or effective. Give 4 or 5 significant igits for (floating point) numerical answers. For most problems wen entering numerical answers, you can if you wis enter elementary expressions suc as 2 3 instea of 8, sin(3 pi/2)instea of -, e (ln(2)) instea of 2, (2 +tan(3)) (4 sin(5)) 6 7/8 instea of , etc. Here s te list of te functions wic WeBWorK unerstans. You can use te Feeback button on eac problem page to sen to te professors.. ( pt) Rogawski-ET/3 Differentiation/3.5 Higer Derivatives- /3.5.3.pg From Rogawski ET section 3.5, exercise 3. Calculate te secon an tir erivatives. y = 6x 4 9x 2 + 5x y = y = Let y = 6x 4 9x 2 + 5x. Ten y = 24x 3 8x + 5, y = 72x 2 8, an y = 44x. 2. ( pt) Rogawski-ET/3 Differentiation/3.5 Higer Derivatives- /3.5.7.pg From Rogawski ET section 3.5, exercise 7. Calculate te erivative inicate. 2 y x 2 x=3, y = 7x 3 + 6x 2 Let y = 7x 3 + 6x 2. Ten y x = ( 2)x 4 + 2x an 2 y x 2 = 84x Hence 2 y x 2 x=3 = 84(3) = ( pt) Rogawski-ET/3 Differentiation/3.5 Higer Derivatives- /3.5.9.pg From Rogawski ET section 3.5, exercise 9. I x ten (4) = Let (x) = x = x /2. Ten, (x) = 2 x /2, (x) = 4 x 3/2, (x) = 3 8 x 5/2, Tus,. (4) = /2 = ( pt) Rogawski-ET/3 Differentiation/3.5 Higer Derivatives- / pg From Rogawski ET section 3.5, exercise 29. Calculate y (k) (0) for 0 k 5, were y = 6x 4 + ax 3 + bx 2 + cx + (wit a,b,c, te constants) y (0) (0) = y () (0) = y (2) (0) = y (3) (0) = y (4) (0) = y (5) (0) = Applying te power, constant multiple, an sum rules at eac stage, we get (note tat y 0 = y by convention) from wic we get: y (0) (0) =, y () (0) = c, y (2) (0) = 2b, y (3) (0) = 6a, y (4) (0) = 44, y (5) (0) = 0. k 0 6x 4 + ax 3 + bx 2 + cx + 24x 3 + 3ax 2 + 2bx + c 2 72x 2 + 6ax + 2b 3 44x + 6a y (k) 5. ( pt) Library/Rocester/setDerivatives3Higer/er.pg Fin te 48 t erivative of te function cos(x).

8 Te answer is function 6. ( pt) Library/UVA-Stew5e/setUVA-Stew5e-C03S07-HigerDerivs- / pg Suppose tat te equation of motion for a particle (were s is in meters an t in secons) is s = t 4 5t (a) Fin te time (oter tan at t = 0 wen te acceleration is 0. Acceleration is 0 at time t = (b) Fin te position an velocity at tis time. Position = Velocity = 7. ( pt) Library/maCalcDB/setDerivatives/nsc2s0p3.pg 0. ( pt) Rogawski-ET/3 Differentiation/3.6 Trigonometric Functions- /3.6.9.pg From Rogawski ET section 3.6, exercise 9. Use te Prouct Rule to fin te erivative of f. x 6 tan(x) Let x 6 tan(x). Ten x 6 sec 2 (x) + 6x 5 tan(x).. ( pt) Rogawski-ET/3 Differentiation/3.6 Trigonometric Functions- /3.6.5.pg From Rogawski ET section 3.6, exercise 5. Use te Prouct Rule to fin te erivative of f. ( 4 + 3x 3x 2) sin(x) Let ( 4 + 3x 3x 2) sin(x). Ten ( 4 + 3x 3x 2) cos(x) + (3 6x)sin(x). 2. ( pt) Rogawski-ET/3 Differentiation/3.6 Trigonometric Functions- /3.6.7.pg From Rogawski ET section 3.6, exercise 7. If sec(x) x 8 ten Applying te Quotient rule: Ientify te graps A (blue), B( re) an C (green) as te graps of a function an its erivatives: is te grap of te function is te grap of te function s first erivative is te grap of te function s secon erivative 8. ( pt) Library/maCalcDB/setDerivatives4Trig/s pg Fin te 29t erivative of sin(x) by fining te first few erivatives an observing te pattern tat occurs. (sin(x)) (29) = 9. ( pt) Rogawski-ET/3 Differentiation/3.6 Trigonometric Functions- /3.6.5.pg From Rogawski ET section 3.6, exercise 5. Use te Prouct Rule to fin te erivative of f. csc(x)tan(x) Let csc(x)tan(x). Ten sec 2 (x)csc(x) csc(x)cot(x)tan(x) ( secx ) (secx) x 8 ( x 8 ) secx = x 8 (x 8 ) 2 = x8 secxtanx 8x 7 secx x 6 = secx(xtanx x 9 3. ( pt) Rogawski-ET/3 Differentiation/3.6 Trigonometric Functions- /3.6.2.pg From Rogawski ET section 3.6, exercise 2. Use te Quotient Rule to fin te erivative of f. x tan(x) + 3 Let f (x) be as above. Ten te Quotient Rule says (tan(x) + 3) x [x] x x [tan(x) + 3] (tan(x) + 3) 2. Since x [x] = an x [tan(x) + 3] = x [tan(x)] = sec2 (x), we see tat tan(x) + 3 xsec2 (x) (tan(x) + 3) 2 2

9 4. ( pt) Rogawski-ET/3 Differentiation/3.6 Trigonometric Functions- / pg From Rogawski ET section 3.6, exercise 39. Te ratio is a constant number. Its value is. x 7 cot(x) csc 2 (x) x 7 cot(x) = 7 cosx sinx( sinx) cosx(cosx) = 7 x sinx sin 2 x Ten, = 7 (sin2 x + cos 2 x) sin 2 x = 7 sin 2 x = 7csc2 x x 7 cot(x) csc(x) 2 = 7csc2 x csc 2 = 7 x 5. ( pt) Library/270/setDerivatives4Trig/s2 4 27b.pg Let f (π) = 7x sinx + cosx 6. ( pt) Library/UVA-Stew5e/setUVA-Stew5e-C03S04-DerivsTrig/ pg Fin te equation of te tangent line to te curve y = 4xcosx at te point (π, 4π). Te equation of tis tangent line can be written in te form y = mx + b were m = an b = Generate by te WeBWorK system c WeBWorK Team, Department of Matematics, University of Rocester 3