FINAL REVIEW FALL 2017

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1 FINAL REVIEW FALL 7 Solutions to the following problems are found in the notes on my website. Lesson & : Integration by Substitution Ex. Evaluate 3x (x 3 + 6) 6 dx. Ex. Evaluate dt. + 4t Ex 3. Evaluate 4 sec 9 (5x) tan(5x) dx. Ex 4. Evaluate Ex 5. Evaluate 5 ln 9 ln 4 x dx. xe x dx. Ex 6. If 6(x 4 + x ) f(x) dx = (x 4 + x ) 3 + C, find f. Ex 7. Find the average value of f(x) = x + over the interval x. Lesson 3: Natural Log Functions ln properties: () a ln b = ln b a () ln(ab) = ln(a) + ln(b) ( a ) (3) ln = ln(a) ln(b) b dx = ln x + C x Ex 8. Evaluate x (ln(x)) 6 dx Ex 9. Evaluate Ex. Evaluate e 4 e x + x dx. x ln(x) dx. Lesson 4 & 5: Integration by Parts Formula: u dv = uv v du To decide what is u, use LIATE:

2 MATH 6 FINAL REVIEW L - ln x I - Inverse trig functions (not for this class) A - Algebraic functions like x, x 3 + x + 7, polynomials (NO ROOTS) T - Trig functions like cos x, tan x, etc E - Exponential functions like e x, x, etc Ex. Evaluate xe x dx. Ex. Evaluate x ln(x) dx. Ex 3. Evaluate π/3 x cos x dx. Ex 4. Suppose a certain plant is growing at a rate of te t inches per day t days after it is planted. What is the height of the plant at the beginning of the third day (assuming it is planted as a seed on the first day)? Ex 5. A factory produces pollution at a rate of pollution does the factory produce in a day? 4 ln(7t + ) (7t + ) 3 tons/week. How much Lesson 6: Solutions, Growth, and Decay A differential equation is called separable if it can be written in the form dy dt = f(t)g(y). The general solution to y = ky is y(t) = y()e kt. Ex 6. Is dy dx = x e 3y x4 a separable differential equation? Ex 7. Find the particular solution to dx dy = x if y = when x =. y Ex 8. Find the particular solution to the differential equation y = ky given y() =, y () = 4. Ex 9. Suppose P (t) is the mass of a radioactive substance at time t. If P (t) = 3 P (t), find the half-life of the substance. Lesson 7 & 8: Separation of Variables Ex. Find the general solution to dy dt + 5y =. Ex. A bacterial culture grows at a rate proportional to its population. Suppose the population is initially, and after hours the population has grown to 5,. Find the population of bacteria as a function of time. Ex. Find the particular solution to the equation da dt A() =, A < for all t. = ( A) such that

3 Ex 3. Find the general solution to x 3 y = y + x e y. MATH 6 FINAL REVIEW 3 Ex 4. A 7-gallon tank initially contains 4 gallons of brine containing 5 pounds of dissolved salt. Brine containing 6 pounds of salt per gallon flows into the tank at a rate of 3 gallons per minute, and the well-stirred mixture flows out of the tank at a rate of 3 gallons per minute. Find the amount of salt in the tank after minutes. Lesson 9 & : First Order Linear Differential Equations (FOLDE) FOLDE are of the form dy + P (t)y = Q(t) with solution given by dt y u(t) = Q(t)u(t) dt where u(t) = e P (t) dt. Ex 5. Find the general solution to dy dx + y x = x. Ex 6. Find the general solution to dy dt + y = t. Ex 7. Find the general solution to (y ) sin x dx dy =. Ex 8. A store has a storage capacity for 5 printers. If the store currently has 5 printers in inventory and the management determines they sell the printers at a daily rate equal to % of the available capacity, when will the store sell out of printers? Ex 9. An 85-gallon tank initially contains 5 gallons of brine containing 5 pounds of dissolved salt. Brine containing 4 pounds of salt per gallon flows into the tank at a rate of 4 gallons per minute. The well-stirred mixture then flows out of the tank at a rate of a gallon per minute. How much salt is in the tank when it is full? Lesson : Area between Curves If f(x) = top function and g(x) = bottom function, then the area between f and g is given by xr x L (f(x) g(x)) dx. If F (y) = right function and G(y) = left function, then the area between F and G is yt y B (F (y) G(y)) dy. Ex 3. Find the area between the curves y = x, y = x + 6 where x 9. Ex 3. Find the area between y = sin x + and y = cos x + where x π. Ex 3. Find the area bounded by x = y y and x + y =. Lesson : Volume of Solids of Revolution Disk Method:

4 4 MATH 6 FINAL REVIEW about x-axis or y =, V = about y-axis or x =, V = xr x L yt y B π(f(x)) dx π(g(y)) dy Ex 33. Find the volume of the solid obtained by revolving the region enclosed by the curves y = x, x =, x =, and y = about the x-axis. Ex 34. Find the volume of the solid obtained by revolving the region enclosed by the curves y = x, x =, and y = 6 in the first quadrant about the y-axis. Ex 35. Find the volume of the solid that results by revolving the region bounded by the curves y = x, y =, and x = about the y-axis. Lesson 3: Volume of Solids of Revolution Washer Method: about x-axis or y =, V = about y-axis or x =, V = xr x L yt y B π[(outer Radius) (Inner Radius) ] dx π[(outer Radius) (Inner Radius) ] dy Ex 36. Find the volume of the solid obtained by revolving the given region about the x-axis: y = x, y = x. Ex 37. Find the volume of the solid that results from revolving the region enclosed by y = 4x, x =, x =, and y = about the y-axis. Lesson 4: Volume of Solids of Revolution Ex 38. Consider the region bounded by the curves y =, y =, x = 5, x =. x Find the volume of the solid generated by revolving the region about the line y = 5. Ex 39. Consider the region bounded by the curves y =, y =, x = 5, x =. x Find the volume of the solid generated by revolving the region about the line x = 5. Lesson 5: Improper Integrals Ex 4. Compute Ex 4. Determine if value. Ex 4. Determine if x dx. 3x e x3 dx converges or diverges. If it converges, find its dx converges or diverges. If it converges, find its value. x

5 MATH 6 FINAL REVIEW 5 Lesson 6 & 7: Geometric Series and Convergence Geometric Series Formula: r < and diverges otherwise. cr n = n= c. A geometric series converges if r Ex 43. Find the 3 rd partial sum of Ex 44. Compute Ex 45. Compute Ex 46. Compute n= n= n= ( ) n. 3 n+ 4 n. 3( ) n 5 n. n +. n= Ex 47. How much should you invest today at an annual interest rate of 4% compounded continuously so that in 3 years from today, you can make an annual withdrawals of $ in perpetuity? Round your answer to the nearest cent. Lesson 8: Intro to Functions of Several Variables Take time to review level curves and how to find x,y-intercepts, asymptotes, etc. Ex 48. If f(x, y) = x ) (, ln(y), find f e3. x Ex 49. Find the domain of f(x, y) = ln(y ) 3. Lesson 9: Partial Derivatives Ex 5. Find f x, f y if f(x, y) = e x + ln y. Ex 5. Find f x, f y if f(x, y) = y cos x. Ex 5. Find f x, f y if f(x, y) = e xy. Lesson : Partial Derivatives Ex 53. Find the second order derivatives of f(x, y) = x 3 y + xy 6. Ex 54. Find the second order derivatives of f(x, y) = x ln(3xy). Ex 55. If f xx (, ) = 4 where f(x, y) = ye ax, find a. Lesson : Differentials of Multivariable Functions Differential Formula: z z z x + x y y

6 6 MATH 6 FINAL REVIEW Ex 56. Use increments to estimate the change in z at (, ) if z x z = 9y given x =. and y =.. y = 3x + y and Ex 57. A company produces boxes with square bases. Suppose they initially create a box that is cm tall and 4 cm wide but they want to increase the box s height by.5 cm. Estimate how they must change the width so that the box stays the same volume. Lesson : Chain Rule for Multivariable Functions Chain Rule for Multivariable Functions: dz dt = z ( ) dx x dt Ex 58. Find dz dt given z = x y, x = cos t, and y = 3t 3. Ex 59. Given z = x + y, x = ln t, and y = dz, find t dt + z y ( ) dy dt evaluated at t =. Ex 6. The width of a box with a square base is increasing at a rate of in/min and a height decreasing at a rate of in/min. What is the rate of change of the surface area when the width is 8 inches and the height is 3 inches? Lesson 3 & 4: Extrema of Functions of Variables The discriminant of f(x, y) is given by D(x, y) = f xx (x, y)f yy (x, y) (f xy (x, y)). The Second Derivative Test is as follows: suppose (x, y ) is a critical point of f. If () D(x, y ) > and f xx (x, y ) <, then (x, y ) is a local maximum point () D(x, y ) > and f xx (x, y ) >, then (x, y ) is a local minimum point (3) D(x, y ) <, then (x, y ) is a saddle point (4) D(x, y ) =, the test is inconclusive Ex 6. Find and classify the critical points of f(x, y) = x3 3 + y3 3 y x. Ex 6. Find the local minima and maxima of f(x, y) = x + y x + y. Ex 63. Find the local minima and maxima of f(x, y) = x3 3 + xy y. Ex 64. We are tasked with constructing a rectangular box with a volume of 64 cubic fee. The material for the top costs 8 dollars per square foot, the material for the sides cost dollars per square foot, and the material for the bottom costs 4 dollars per square foot. To the nearest center, what is the minimum cost for such a box?

7 MATH 6 FINAL REVIEW 7 Ex 65. The post office will accept packages whose combined length and girth is a most 5 inches (girth is the total perimeter around the package perpendicular to the length and the length is the largest of the 3 dimensions). What is the largest volume that can be sent in a rectangular box? (Round your answer to the nearest integer). 5 & 6: Lagrange multipliers Method of Lagrange Multipliers: Suppose we want to minimize or maximize a function f(x, y) subject to the constraint g(x, y) = C. Solve the following system for (x, y): f x (x, y) = λg x (x, y) f y (x, y) = λg y (x, y) g(x, y) = C Ex 66. Find the minimum value of x e y subject to y + x = 6. Ex 67. Find the maximum value of f(x, y) = 3 x3/ y subject to x = y. Ex 68. A rectangular box with a square base is to be constructed from material that costs $5/ft for the bottom, $4/ft for the top, and $/ft for the sides. Find the box of the greatest volume that can be constructed for $6. Round you answer to 4 decimal places. Ex 69. On a certain island, at any given time, there are R hundred rats and S hundred snakes. Their populations are related by the equation (R 6) + (S 6) = 8. What is the maximum combined number of rats and snakes that could ever be on the island at the same time? Round your answer to the nearest hundred. Lesson 7, 8, & 9: Double Integrals, Volume, and Applications f(x, y) da denotes in a general way the volume under the function f(x, y) over R the region R. The da needs to be translated to either dx dy or dy dx, whichever is appropriate depends on how the region R has been described. Ex 7. Compute Ex 7. Evaluate π/ x dx dy. 3y cos x dy dx. Ex 7. Compute π/ π/ y ( sec(y) sin(x)) dx dy. Ex 73. Suppose R is a rectangle with vertices (, ), (, ), (, ), (, ). Find 4x 3 y da. Ex 74. Suppose R is the region bounded by the x-axis, y = x, and x = 3. Find (x + y) da.

8 8 MATH 6 FINAL REVIEW Ex 75. Given Ex 76. Evaluate x f(x, y) dy dx, swap the order of integration. x x + y dy dx., et/ Ex 77. Suppose the function P (x, t) = describes the population of a city + x where x is the number of miles from the center of the city and t is the number of years after the year. Find the average population of the city over the first years within a radius of 5 miles from the city center. Round your answer to the nearest integer. Lesson 3: Systems of Equations, Matrices, and Gaussian Elimination Ex 78. Put the following matrix into row-echelon form: Ex 79. A goldsmith has two alloys of gold with the first having a purity of 9% and the second having a purity of 7%. If x grams of the first are mixed with y grams of the second such that we get grams of an alloy containing 8% gold, find x to the nearest gram. Ex 8. Solve and classify the following system of equations: 3x + y + z = x + y + z = 4x + 3y + 3z = Lesson 3: Gauss-Jordan Elimination Ex 8. Put the following matrix into reduced row-echelon form: Lesson 3: Matrix Operations [ ] [ 3 Ex 8. Let A = and B = 4 Ex 83. If A = Ex 84. Let M = 3 [, find A. ]. Find M 3M. Inverses of Matrices Ex 85. Given A = 3, find A if it exists. ]. Find 3A, 3A B, AB, and BA.

9 Ex 86. Find a solution to coefficient matrix is MATH 6 FINAL REVIEW 9 x + 5y + z = 39 3y + 5z = 39 x + y + z = Lesson 34: Determinants of Matrices [ ] Ex 87. Is A = singular? Ex 88. Find the determinant of A =. 3 Lesson 35 & 36: Eigenvalues and Eigenvectors. given that the inverse of the λ is an eigenvalue of the matrix A if it is a solution to the equation det(λi A) =. v λ is an eigenvector of A associated to λ if it satisfies the equation A v λ = λ v λ. [ ] Ex 89. Let A =. Find the eigenvalues of A and, for each eigenvalue, find 3 an associated eigenvector. [ ] Ex 9. Determine if v = is an eigenvector of A = Ex 9. Find the eigenvalues and eigenvectors of A = Ex 9. An eigenvalue of A = an eigenvector for λ =?, 3, [ 3 ]. 4 4 is λ =. Are any of,.