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1 Mega Review MATH 6 Spring 28 Challenge: Since many stuents fin it helpful to have a solution guie to problems while stuying, we challenge all MATH 6 stuents to create a solution guie to the problems on this worksheet. Sen iniviual solutions to us (6megareview@gmail.com), an we will make them available for all course sections to access by uploaing them to the course webpage. Instructions for Submissions: Write solutions on a clean sheet of paper, line or unline. Be sure to inclue the problem number. Type solutions are appreciate, but not necessary. Submissions shoul be complete, legible, an contain goo explanations. Explain everything as if you were teaching someone the material. Keep in min that mathematicians love concise answers! Leave your signature or a sweet alias at the bottom of your solution so we know who gets creit for the solution. Fun pictures are optional but encourage. Sen all solutions to 6megareview@gmail.com or turn them in to your instructor in person. Here s an example of what I m looking for: 9. Suppose you own an apartment complex with apartments. Every month, you make a total profit given by P = rq 8( q) 86, where r is the price you charge for rent, an q is the number of rooms rente. If the eman for rooms is given by q = r, fin the rent price an quantity of rooms rente that maximize your monthly profit, using calculus.

2 Compute the following limits. (Please submit entire columns of solutions). lim x 8 x 2 6x 6 x 2 + x lim t 3 t 3 t lim x 7x 4 x 4+x 4. lim y y 2 2y lim x 3x 2 + 2x + 9x 2 + 7x 6. lim x 8x 7x lim (9x + ) x h lim h 5 h + 5 sin θ 9. lim θ + cos θ. lim x π/2 tan(x). lim x sin(x) x 2. lim x cos(x) x 3. lim x x + x 2 3x lim x cos(x + sin(x)) 5. lim x 4 (x 4) 4 6. lim h (h ) 3 + h 7. lim x x 2 + x 2 tan(x) 5 x 8. lim t 2 t 2 + 4t 2 t 2 2t y y < 2 9. g(y) = 3 3y y 2 lim g(y) = y 3 lim g(y) = y 2 Fin all horizontal an vertical asymptotes of the following functions. If no horizontal asymptotes exist, explain why x 2 3x 2 x 2 2x + 5x 2 5x 5 (x ) tan(x) x x 5 x2 + 3x 5 (x 5)(x 3)(x + 2) x + x + 5 x 2 3x 2 x2 2x + + 2x + 5x 2 3x 3 sin(x) 2x ( + x) 3 ax 5 (a is a constant) 2(x + a) 5 In each case, fin the values of A an B that make the following functions continuous. Ax + B x < 3. f(x) = 3 x = Ax 2 Bx + 2 x > x 2 + x x < 3. g(x) = Ax + B x sin(πx) x >

3 Compute the following erivatives. (Please submit entire columns of solutions) 32. f(x) = 2x 2 + 2x + x 33. y = 2 3 x 2 3 4x 3 π3 34. g(x) = (3x + 4) 3 2x 35. f(x) = 8 4 2x 2 x 36. y = (x 2 + 3x + 4)(3x + 2) 37. f(x) = x 4 x x 38. g(t) = 4 3t (2t 3) x (sin x) = 4. x (cos x) = 4. x (tan x) = 42. x (cot x) = 43. x (sec x) = 44. x (csc x) = 45. r(y) = sin(y) tan(y) 46. h(x) = tan(4 x) + 2 sin(πx) 47. g(t) = 2t 5 t sec(t) 48. f(x) = 2x + eπ 5 x 3 4 x 49. x(t) = 4 t t v(x) = 7 2 x 3 4x 3 5. f(s) = 3 s(7s 2 πs). 52. h(x) = 3x x + 7x 3 x 53. y = (x 5 + x 4 x 3 x 2 + x + )( x 3 x + 4 x) 54. f(x) = ax 2 + b 3 x3 + c a,b,c are constants 55. s(z) = z m(t) = t g(x) = x k k= k! 58. h(x) = cos(2x) + i sin(2x) 59. y(t) = n= f (n) (a) (x a) n n! 6. For the functions h(x) = 2 x 2 an p(x) = x, fin h (2) an p () from the limit efinition (since that is your only option). If these are not ifferentiable, explain why an how they are ifferent. x x 6. Let g(x) = 4x + 4 x > Is g(x) ifferentiable? If not, explain why using the limit efinition of the erivative. In each case, fin the values of A an B that make the function ifferentiable for all x in the omain. A sin( πx 2 ) x < 62. h(x) = 3 x = Bx + A x > x 2 + Ax + B x < 63. q(x) = Bx x 64. Use g(t) = f(tan(t)) to answer the following. (a) Fin the erivative of g(t). (b) Use the information provie in the table below to fin the value of g (π). f(t) f (t) t = 2 2 t = π 5 2

4 Compute the following antierivatives. (Please submit entire columns of solutions) 65. x 6 x 2 x 66. cos(t) sin(t)t 67. 7x 4 (3 + x 5 ) 6 x 68. 2y(y 4) 3 y 69. x 4 3 2x 5 7 x 7. 2 ( 2 ) x 5 ( 3 x6 ) 2 x 72. 3y 7 3y 2 y 73. 5s + 4 s 74. tan 2 (x) sec 2 (x) x 75. x 2 2 x x Compute the following efinite integrals. (Please submit entire columns of solutions) π 2 3 cos 2 (p) sin(p)p x (x 2 + 2) 2 x t + 2 t2 + 4t + 4 t π 79. θ sin(θ 2 )θ π/ π/4 y + y = tan(x) sec 2 (x) x = t 3 ( + t 4 ) t = Decie whether the following statements are correct r (4 + r 2 ) 2 r = 4x x2 + x = π/4 7 9 π 2 tan 4 (x) sec 2 (x) x 2 + x 3 x4 + 9 x t 4 + 5t t ( cos(3x)) sin(3x) x 9. 2x + x = x2 + x + C 9. x2 + x + 2 x 3 x = 3 x3 + 2 x2 + 2x + C 4 x4 Fin the areas of the following regions. Inclue sketches of the regions. 92. The total area of the regions boune by f(x) = x 3 an g(x) = x. 93. The region boune by x = 2y 2 2y 3 an x = 2y 2 2y, with y. 94. The region boune by y = x an y = x4 4 with y.

5 State whether the following statements are true or false. If a statement is true, explain why it is true. If a statement is false, provie a counterexample to the statement. 95. If f (a) exists, then f(x) is continuous at x = a. 96. If lim x 2 f(x) = f(2), then lim h f(2 + h) f(2) h exists. 97. If the velocity of a particle is increasing, its position must be positive. 98. If f (x) = g (x), then f(x) = g(x). 99. If f(x) is increasing, then 2 2 f(x)x >.. If b a g(x)x =, then g(x) = on [a, b].. If f(x) >, then b a f(x)x > for all a an b. 2. If f(x) is not continuous at x = 5, then 6 3 f(x)x oes not exist. 3. If f(x) an g(x) are ifferentiable, then the total area boune by f an g on the interval [, 4] is given by 4 [f(x) g(x)]x. 4. If h (4) =, then h must have a maximum or minimum at x = If p(x) an q(x) are both increasing functions, then f(x) = p(x) + q(x) must be increasing. 6. f(x) = x has infinitely many tangent lines at x =. 7. If the erivative oes not exist at a point, then a tangent line cannot exist at that point. 8. f(x) efine on [, 6], then it must attain both an absolute maximum an an absolute minimum on [, 6]. 9. f(x) = x + 5 is not ifferentiable at every x in its omain.. If g(x) is continuous on the union of intervals [2, 4] [5, 7], then g(x) must attain an absolute maximum on [2, 4] [5, 7].. If h(x) has a cusp at x = 5, then h(x) has a vertical asymptote at x = Piecewise functions are not ifferentiable. 3. Given that f() = f(3) = an f (2) =, then f(x) must be continuous on the interval [, 3]. 4. A curve that oesn t pass the vertical line test at x = cannot have tangent lines for points where x =. f(2 + h) f(2) 5. If lim = 4, then f(2) = lim f(x) h h x 2 6. If f(3) = 2 an lim f(x) = 2, then lim f(x) = 2 x 3 + x 3 7. If f(x) has a vertical asymptote at x = 2, then f( 2) is unefine. 8. If g(x) is not continuous, then g(x) is not ifferentiable.

6 9. If lim x p(x) = 3, then lim x p(x) = 3 2. A mipoint sum with any number of partitions will fin the exact area uner f(x) = x + 5 on any interval [a, b], a < b. 2. Mipoint sums are always more accurate than left or right han sums. 22. If g (c) =, then there is an inflection point at x = c. 23. If lim x 7 g(x) oes not exist, then g(7) oes not exist. 24. If q (r) < for r < an q (r) > for r >, then q(r) must have an absolute minimum at r =. 25. If a woman travels 2mi in 2.5hrs, she must have exceee the 6mph spee limit at some point uring her rive. 26. It s impossible for a function to have an absolute max on on open interval. 27. Let f (x) > for all x. Then g(x) = f(f(x)) is always increasing. 28. If f(x) is increasing on [a, b], then b a f (x)x >. 29. If a certain type of glass is coole faster than at a rate of 45 F per secon, it will shatter. Suppose the ining hall just put out some freshly-washe glasses at 5 F, an your frien grabs one an fills it with ice col water. If the glass cools to 5 F after two secons, we know for sure that the glass shattere. 3. Suppose we have the same type of glass, an that you ve been having some trouble with your hot water heater. You fill your ishwasher with glasses at room temperature (7 ) an run it, an the glass comes out again at 2 3 minutes later. We are guarantee that none of the glasses shattere ue to temperature changes. 3. If f(x) is ifferentiable on [2, 5], then it must have both a maximum value an minimum value on [2, 5]. 32. If f(a) = f(b) =, then there must be some point c in [a, b] at which f (c) =. 33. If f(x) is continuous an f(a) = f(b) = 5, then there must be some point c in [a, b] at which f (c) =. 34. *Challenge* If two racers start a race at the same time an en at the same time, it is possible that they were never moving at exactly the same spee over the course of the race. 35. *Challenge* If f(x) has jump iscontinuities at every integer, then the integral unefine. 36. *Challenge* lim b b x is unefine for every number p. xp 37. *Extreme Challenge* If h(x) is ifferentiable, then h (x) is continuous. 38. *Extreme Challenge* x x = 9 f(x)x is

7 39. *Challenge* In a beehive, each cell is a hexagonal prism, open at one en with a triheral angle at the other, as in the figure. It is believe that bees form their cells in such a way as to minimize the amount of wax neee to construct the hive. Base on the geometry of the cell, we know that each cell has a surface area of S = 6sh 3 2 s2 cot θ + 3s csc θ Fin the angle that minimizes the surface area of the cell (suppose s an h are given). Actual measurements of this angle show that bees selom iffer from the optimal angle by more than 2 egrees! 4. Fin the maximum prouct of two non-negative numbers such that three times the first number plus the secon is. 4. The rate (in mg of carbon per cubic meter per hour) at which photosynthesis takes place for a I species of phytoplankton is moele by R =, where I is the light intensity (in thousans I 2 + I + 4 of foot-canles). What level of light intensity maximizes R, the rate of photosynthesis? 42. A farmer wants to fence an area of.5 million square feet in a rectangular fiel an then ivie it in half with a fence parallel to one of the sies. What imensions shoul he use to minimize the amount of fence he nees? 43. A box with a square base an no top must have a volume of 32, cm 3. Fin the imensions of the box that minimize the amount of material neee. 44. A piece of wire m long is cut into two pieces. One piece is bent into a square, an the other is bent into an equilateral triangle. How shoul the wire be cut so that the area enclose by the two shapes is a maximum? 45. A fence 8 ft tall runs parallel to a tall builing at a istance of 4 ft from the builing. What is the length of the shortest laer that will reach from the groun over the fence to the wall of the builing? 46. A baseball team plays in a staium that hols 55, spectators. With ticket prices at $, the average attenance is 27,. When ticket prices roppe by $2, attenance increase to 33,. Assuming this pattern hols for any price change, what price shoul the staium charge to maximize its revenue? 47. Your company prouces bouillon cubes. The bouillon cubes are specifie to have a volume of 3375mm 3. (a) What must the sie length measure to result in a volume of 3375mm 3?

8 (b) Suppose that the Packaging epartment says that the volume cannot excee 345mm 3 an the company s Chief Financial Officer says it cannot be less than 3345mm 3. What values must the sie lengths be to ensure both restrictions? (c) Sketch the graph of volume V (x) as a function of sie length x. Then sketch the interval on the V -axis that must contain the cubes volumes. Next, sketch the interval of x-values which ensures that V (x) falls in the interval on the V axis. 48. The physicists at CSU are builing a top-secret machine with a capacitor that, when charge, nees to hol 9 GJ (gigajoules) of energy. If this store energy varies by more than ± 2 GJ, then a black hole will form an estroy the earth. The energy store by this capacitor is given by U(v) = 2.55v2, where v is the applie voltage an U is the store energy. 2 (a) Fin v > such that U(v) = 9 GJ. We will call this value v. Roun to at least 4 ecimal places. (b) What interval aroun v will ensure that corresponing output values are close enough to U = 9GJ? (c) How close to v o we nee to keep v to ensure that U remains close enough to 9 GJ so that the earth is not oome? Roun to at least 4 ecimal places. Explain using wors why you chose this value. () What is the maximum amount of error that can occur on either sie of v. 49. Can the MVT be applie to f(x) = x 3 + 2x 2 x on [, 2]? If so, etermine all the numbers c which satisfy the conclusions of the Mean Value Theorem. 5. Marty McFly ries Dr.Brown s time-travelling Delorean to class, because he s able to teleport the three miles in just two secons. What s Marty s average velocity on the way to class? Can we use the mean value theorem to show that Marty must have been travelling that velocity at some point in his trip? (Hint: Think about what Marty s position function woul look like, an then check the assumptions of the MVT) 5. Suppose you know the function f(x) is ifferentiable on [, ], an that it has a local maximum at x = 3 so that f( 3) = M an a local minimum at x = 7 so that f(7) = m. Which of the following are true? Explain why each statement is true, or give a counterexample. (a) There must be some c ( 3, 7) so that f (c) = m M. (b) There must be some c ( 3, 7) at which f(c) = m+m / 2. (c) f(x) must achieve its absolute max somewhere on the interval [ 3, 7]. () f(x) has both a maximum an a minimum value on the interval [ 3, 7]. (e) We know that m < M. 52. Suppose we are tracking the pace of a short istance runner. We measure the spee, (v(t)), of the runner at minute intervals for the uration of his training run. (We gave him a heastart to get Time (mins) up to spee before time ). Spee (mph) (a) Create an upper estimate for the istance he ran over those 6 minutes. (b) Create a lower estimate for the istance he ran over the 6 minute session.

9 53. Oil is leaking out of a tanker amage at sea. The rate that oil is leaking out of the tanker is recore in the table below: t (hours) a (gal/hr) (a) Fin an upper estimate an lower estimate for the total quantity of oil that has escape after 5 hours. (b) Repeat part (a) for the quantity of oil that has escape after 8 hours. 54. Estimate the area enclose by f(x) = x 2 an g(x) = x 2 using a left enpoint sum with 8 partitions. Sigma notation is not necessary, but sketch a picture of the rectangles use in the sum. 55. *Challenge* If 2 7 f(x)x = 9, 56. Consier the function g(x) is efine by g(x) = 4 2 f(x)x = 2, an f(x) is even, what is x 4 7 f(x)x? f(t)t an the graph of f is given below. (a) Fin g(2) (c) Fin g( 5) (e) Fin g ( ) (g) Fin g (2) (b) Fin g( 2) () Fin g(5) g(4) (f) Fin f(2) (h) Fin g (5) 57. Evaluate the following expressions. (a) x2 x sin(t)t (b) 2 x t 2 + tt 3x (c) x x t 3 t x () cos(x) x t t a

10 58. Let F (x) = x f(t)t where f(x) is the function whose graph is shown below. (a) Where are the critical points of F (x)? (b) Does F (x) have any local maxima? Where are they? Explain how you know they are maxes. Don t forget about the enpoints! (c) Does F (x) have any local minima? Where are they? Explain how you know they are maxes. Don t forget about the enpoints! () Assuming the pattern starte in the graph hols, will F (x) ever be negative for any x >? 59. The graph below is the equation of (x 2 + y 2 8x) 2 = 4(x 2 + y 2 ). Answer the following questions that pertain to it. (a) Verify that the point (3, 3 3) is on the curve. (b) Fin an expression for y x. (c) What is the equation of the tangent line to the curve at the point (3, 3 3)? 6. Fin y x for the following. (Please submit entire columns of solutions) (a) 3y 3 + x 2 = 5 (b) y 4 y = x 3 + x (c) x 2 y + y 4 3x = 8 () x 2 y + 2xy 2 = x + y (e) xy = x + 3y (f) 4 sin(2y) cos(x) = 2 (g) x sec(y) = xy (h) xy 2 tan(πxy) = 2x

11 6. Sketch a graph of a function with the following properties: f(x) f (x) f (x) x < 4 lim (x) > f (x) < x x = 4 f ( 4) = f ( 4) < 4 < x < lim (x) < f (x) < x x = f( ) = 3 f ( ) DNE f ( ) DNE < x < lim < f > x + x = f() = f () DNE f () DNE < x < 3 f (x) > f (x) < x = 3 f (3) = f (3) < x > 3 lim f(x) = x f (x) < 62. Ientify which graph is f(x), which graph is f (x), an which graph is f (x). Explain how you know. 63. There are three functions plotte in the coorinate system below corresponing to the position, velocity, an acceleration of a particle. Determine which label goes with each graph, an explain your answer.

12 64. Suppose that we have a function, f(t). Below is the graph of the erivative of f(t) on the interval [, ]. Be sure to consier the enpoints in your answers. (a) Ientify the critical point(s) of f(t). (b) Ientify the interval(s) on which f(t) is increasing. (c) Ientify the interval(s) on which f(t) is ecreasing. () Ientify the interval(s) on which f(t) is concave up. (e) Ientify the interval(s) on which f(t) is concave own. (f) Ientify where f(t) has local maxima, if applicable. (g) Ientify where f(t) has local minima, if applicable. 65. Let f (x) = 8x + 6 an suppose f(2) = 4 an that f has a horizontal tangent line at x = 3. Fin f(x). 66. A stone was roppe off a cliff an hit the groun with a spee of 2 m s. What is the height of the cliff? 67. A car is traveling 5 mi ft hr when the brakes are applie, proucing a constant eceleration of 22 How far oes the car travel after the brakes are applie? 68. A company estimates that the marginal cost (in ollars per item) of proucing x items is.92.2x. If the cost of proucing one item is $562, fin the cost of proucing items. Fin the volume of the regions escribe below. 69. (Textbook problem, page 32) The base of the soli is the unit isc (x 2 + y 2 ), an the cross sections are isoceles right triangles with one leg in the isc. (picture in book) 7. (Textbook problem 7, page 32) The base of the soli is the region boune by y = 3x, y = 6, an x =. The cross sections are rectangles of height. 7. Fin the volume of the soli generate by revolving the region boune by y = 4 x 2 an y = 2 x aroun the x-axis. 72. Fin the volume of the soli generate by revolving the region boune by x = y y 2 an x = aroun the y-axis. 73. Fin the volume of the soli generate by revolving the region boune by y = x 2 an x = y 2 about x =. Don t forget that you can go to any MATH 6 instructor s help hours. We are all happy to help! s.