UNIVERISTY HELPP IN HERE

NOT for TUTOR: dont submit if you dont do full THEM OR BAD RATE And in full explanation show all your work

1. Solve the following system of nonlinear equations:x^2 + y^2 = 9 x^2 - y^2 = 1
2. Find the maximum value of the function f(x,y) = 3x^2 + 2xy + y^2 subject to the constraint x + y = 6.
3. Evaluate the definite integral of sin(x^2) from 0 to 1 using numerical methods.
4. Find the second derivative of y = tan(x^2).
5. Find the volume of the solid that results from rotating the region bounded by y = x^2 and y = 4 about the x-axis.
6. Determine the equation of the tangent line to the graph of y = cos(x) at x = pi/4.
7. Find the eigenvalues and eigenvectors of the following matrix:
   1. A = [2,1;1,2]
8. Solve the differential equation y' + 2y = x using the method of characteristic equations.
9. Sketch the graph of the complex function f(z) = z^2 + 1 in the complex plane.
10. Use vector calculus to find the surface area of the unit sphere.
12. Find the maximum and minimum values of the function f(x,y) = x^2 - 2xy + y^2 subject to the constraint x + y = 6.
13. Solve the following system of linear differential equations using Laplace transforms:y' + 4y = xe^2x y(0) = 0
14. Use polar coordinates to evaluate the double integral of r^2 sin() over the region bounded by r = 1 and r = 2.
16. Find the Taylor series representation of the function f(x) = sin(x) about x = 0 and use it to approximate sin(0.1).
17. Use partial derivatives to find the local maxima and minima of the function f(x,y) = x^2 + y^2 - 4x - 2y.
18. Find the inverse Laplace transform of the function F(s) = 1/(s^2 + 1) using partial fraction decomposition.
19. Use matrix algebra to find the general solution to the following system of linear differential equations:
20. x' = [0,1;-1,0]x
21. Use Green's Theorem to evaluate the line integral of (2x + y) dx + (x + 3y) dy over the closed path defined by x = cos(t), y = sin(t), where 0 <= t <= 2pi.
22. Find the critical points and determine their nature (local maxima, minima, or saddle points) of the function f(x,y) = x^3 + y^3 - 3xy.
23. Use the method of characteristics to solve the partial differential equation u_t + u_x = 0 subject to the initial condition u(x,0) = sin(x).
24. Use the method of Lagrange multipliers to find the maximum and minimum values of the function f(x,y) = x^2 + y^2 subject to the constraint xy = 1.
25. Solve the following boundary value problem using the method of finite differences:
26. u''(x) = x^2, 0 < x < 1
27. u(0) = u(1) = 0
28. Find the minimum distance between the lines defined by the equations x = 2t + 1 and y = -3t + 2.
29. Find the eigenvalues and eigenvectors of the following matrix:A = [0,1,2;2,1,0;3,1,1]
30. Evaluate the triple integral of z^2 over the region bounded by x^2 + y^2 + z^2 = 4 and z = x.
31. Use the method of characteristic curves to solve the partial differential equation u_t + 2u_x = 0 subject to the initial condition u(x,0) = x^2.
32. Use vector calculus to find the curl of the vector field F(x,y,z) = (x^2 + y^2, 2xy, yz).
33. Solve the following system of nonlinear equations using Newton's method:
    1. x^3 + y^3 = 6 x^2 + y^2 = 4
34. Find the equation of the normal line to the surface z = x^2 + y^2 at the point (1,1,2).
35. Use the method of generating functions to solve the recurrence relation a_n = 2a_{n-1} + 3a_{n-2}.
36. Use the method of separation of variables to solve the partial differential equation u_{xx} + u_{yy} = 0 subject to the boundary conditions u(x,0) = x, u(x,1) = x^2, u(0,y) = y^2, u(1,y) = y^3.
37. Solve the following system of linear equations using Gaussian elimination:
38. x + y - z = 3
39. 2x + 3y + z = 5 -x + y + 2z = -1
40. Find the volume of the solid that is formed by rotating the region bounded by y = x^2 and y = x about the y-axis.
41. Use complex analysis to evaluate the integral of sin(z)/z over the unit circle.
42. Find the gradient, the Hessian matrix, and the critical points of the function f(x,y) = x^4 + y^4 - 6x^2 - 6y^2 + 8x + 8y + 12.
43. Use the method of Laplace transforms to solve the following system of linear differential equations: y' + 3y = 4e^2x y(0) = 0
44. Use vector calculus to find the line integral of F(x,y) = (x^2 + y^2, xy) along the curve defined by x = t^2, y = t^3, where 0 <= t <= 1.
45. Find the equation of the plane that passes through the points (1,2,3), (2,3,4), and (3,4,5).
46. Use partial derivatives to find the global maximum and minimum values of the function f(x,y) = x^2 - 4xy + 4y^2 + 5x - 6y + 9.
47. Use the method of characteristic curves to solve the partial differential equation u_t - u_{xx} = 0 subject to the initial condition u(x,0) = x^2.
48. Use the method of characteristic curves to solve the partial differential equation u_{tx} = u_{xx} subject to the initial condition u(x,0) = x^2.
49. Find the eigenvalues and eigenvectors of the following matrix:
50. A = [1,2,3;4,5,6;7,8,9]
51. Solve the following system of nonlinear equations using the Newton-Raphson method: x^2 + y^2 = 9 y^2 + z^2 = 25 z^2 + x^2 = 16
52. Use the method of Green's functions to solve the partial differential equation u_{xx} + u_{yy} = f(x,y), where f(x,y) is a given function.
53. Use vector calculus to find the surface integral of F(x,y,z) = (y, z, x) over the surface defined by z = x^2 + y^2.
54. Find the minimum distance between the points (2,3,4) and (5,6,7).
55. Use the method of generating functions to solve the recurrence relation a_n = a_{n-1} + 2a_{n-2} + 3a_{n-3}.
56. Solve the following boundary value problem using the shooting method: y'' + y = 0, 0 < x < 1 y(0) = 0, y(1) = 1
57. Find the equation of the normal line to the surface z = x^2 - y^2 at the point (1,1,0).
58. Evaluate the double integral of x^2 + y^2 over the region bounded by y = x and y = x^2.
59. Evaluate the triple integral of x^2 + y^2 + z^2 over the region bounded by x^2 + y^2 + z^2 = 9 and z = 0.
60. Solve the following differential equation using the method of variation of parameters:
61. y'' + y = sin(x), 0 < x < 2 y(0) = 0, y'(0) = 1
62. Use the method of characteristic curves to solve the partial differential equation u_t + u u_x = 0 subject to the initial condition u(x,0) = f(x), where f(x) is a given function.
63. Find the maximum and minimum values of the function f(x,y) = x^2 - y^2 subject to the constraint x^2 + y^2 = 1.
64. Use the method of undetermined coefficients to solve the following nonhomogeneous linear differential equation:
65. y'' + 2y' + y = e^x.
67. Use the method of partial fractions to evaluate the following improper integral:
68. _{-}^{} dx / (x^2 + 9).
70. Use vector calculus to find the line integral of F(x,y) = (x^3 + y^3, x^2 + y^2) along the curve defined by x = t^3, y = t^2, where -1 <= t <= 1.
72. Find the equation of the tangent line to the curve y = x^3 at the point (1,1).
74. Use the method of Fourier series to find the Fourier series expansion of the function f(x) = |x| in the interval - <= x <= .
76. Solve the following initial value problem using the Laplace transform:
77. y'' + y' + y = u(t), 0 < t <
78. y(0) = 1, y'(0) = 2
79. Use the method of partial fractions to find the inverse Laplace transform of the following function: L^{-1}(s / (s^2 + 4s + 13)).
81. Evaluate the surface integral of F(x,y,z) = (x^2 + y^2, y^2 + z^2, z^2 + x^2) over the surface defined by x^2 + y^2 + z^2 = 1.
82. Use the method of Lagrange multipliers to find the maximum and minimum values of the function f(x,y) = x^2 + y^2 subject to the constraint x + y = 1.
84. Use the method of separable variables to solve the following nonlinear differential equation:
85. (y')^2 + y^2 = 1.
87. Use vector calculus to find the curl of the vector field F(x,y,z) = (x^2 + y^2, yz, zx).
89. Find the equation of the plane that passes through the points (1,2,3), (2,3,4), and (3,4,5).
91. Use the method of characteristic curves to solve the partial differential equation u_{tx} = u_{xx} subject to the initial condition u(x,0) = x^2.
93. Find the eigenvalues and eigenvectors of the following matrix:
94. A = [1,1,1;1,1,1;1,1,1]
96. Solve the following system of nonlinear equations using the Newton-Raphson method:
97. x^2 + y^2 + z^2 = 25
98. x + y + z = 5
99. x^2 + y^2 - z^2 = 9
101. Use the method of Green's functions to solve the partial differential equation u_{xx} + u_{yy} + u_{zz} = f(x,y,z), where f(x,y,z) is a given function.
102. Sure thing! Here are a few more challenging math problems to bring us to 100:
104. Find the solution to the following differential equation using Laplace transforms:
105. y'' + 4y' + 4y = cos(2t), 0 < t <
106. y(0) = 0, y'(0) = 1
108. Use the method of partial derivatives to find the critical points and determine their nature of the following function:
109. f(x,y) = x^2 + y^2 - xy.
111. Use the method of characteristic curves to solve the partial differential equation u_t + u u_x = 0 subject to the initial condition u(x,0) = x^3.
113. Use vector calculus to find the line integral of F(x,y) = (y^2 - x^2, 2xy) along the curve defined by y = x^3, where -1 <= x <= 1.
115. Solve the following system of linear differential equations using the method of eigenvalues and eigenvectors:
116. x' = [2,1;1,2]x
118. Use the method of undetermined coefficients to solve the following nonhomogeneous linear differential equation:
119. y'' + 4y' + 4y = cos(2t).
121. Use the method of partial fractions to evaluate the following improper integral:
122. _{0}^{\infty} dx / (x^2 + x + 1).
124. Use the method of characteristic curves to solve the partial differential equation u_t + u u_x = 0 subject to the initial condition u(x,0) = e^x.
126. Find the equation of the plane that passes through the point (1,2,3) and is perpendicular to the vector (1,2,3).
128. Solve the following system of nonlinear equations using the Newton-Raphson method:
129. x^2 + y^2 = 9
130. x^2 - y^2 = 1
132. Use the method of Green's functions to solve the partial differential equation u_{tx} = u_{xx} subject to the initial condition u(x,0) = x^3.
134. Find the eigenvalues and eigenvectors of the following matrix:
135. A = [1,2;3,4]
137. Use vector calculus to find the curl of the vector field F(x,y,z) = (x^2 + y^2, yz, xy).
139. Solve the following initial value problem using the Laplace transform:
140. y'' + 3y' + 2y = u(t), 0 < t <
141. y(0) = 0, y'(0) = 1
143. Use the method of partial derivatives to find the critical points and determine their nature of the following function:
144. f(x,y) = x^3 + y^3 - 3xy.
146. Use the method of characteristic curves to solve the partial differential equation u_{tx} = u_{xx} subject to the initial condition u(x,0) = x^4.
148. Use the method of partial fractions to find the inverse Laplace transform of the following function:
149. L^{-1}(s / (s^2 + 9s + 26)).
151. Evaluate the surface integral of F(x,y,z) = (x^2 + y^2, y^2 + z^
152. Solve the following system of linear differential equations using the method of eigenvalues and eigenvectors:
153. x' = [3,2;2,3]x
155. Use the method of undetermined coefficients to solve the following nonhomogeneous linear differential equation:
156. y'' + 9y' + 20y = sin(3t).
158. Use vector calculus to find the line integral of F(x,y) = (x^2 + y^2, xy) along the curve defined by y = x^2, where -1 <= x <= 1.
160. Find the equation of the plane that passes through the point (2,3,4) and is perpendicular to the vectors (1,2,3) and (4,5,6).
162. Solve the following system of nonlinear equations using the Newton-Raphson method:
163. x^2 + y^2 = 4
164. x^2 - y^2 = 2
166. Use the method of Green's functions to solve the partial differential equation u_{tx} = u_{xx} subject to the initial condition u(x,0) = cos(x).
168. Find the eigenvalues and eigenvectors of the following matrix:
169. A = [2,3;4,5]
171. Use the method of partial derivatives to find the critical points and determine their nature of the following function:
172. f(x,y) = x^4 + y^4 - 4xy.
174. Use the method of characteristic curves to solve the partial differential equation u_t + u u_x = 0 subject to the initial condition u(x,0) = e^(-x).
175. Use vector calculus to find the curl of the vector field F(x,y,z) = (xz, yz, xy).
176. Solve the following initial value problem using the Laplace transform:
177. y'' + 5y' + 6y = t^2, 0 < t <
178. y(0) = 0, y'(0) = 1
179. Use the method of partial fractions to find the inverse Laplace transform of the following function:
180. L^{-1}(s / (s^2 + 4s + 7)).

Unit 1 Activity 1: Characteristics of functions

Instructions: homework based on the topics that were covered in this activity. If you have any questions, please post your questions to the Unit 1 Discussion.

1. For each of the following functions complete the chart and sketch the function using Desmos.

Function	degree	Sign of leading coefficient	End behaviour x	End behaviour x -

2. Based on the above chart, find the relationship between the end behaviour

of a function and the degree.

3. Using Desmos, fill in the following chart.

Function	degree	Number of zeroes
y = -x3 + 7x + 6
y = x4 + x3 - 7x2 - x + 6

4. Based on the chart in question #3, what is the minimum and maximum number of zeroes possible for each of the following polynomial functions with the following degrees?:

1. Start with a quartic polynomial function, in standard form. Make sure that it is factorable. Hint: Make one up by starting with the factors and expand to get your polynomial.
2. Factor your polynomial. Use the factor theorem and repeated division. Explain each step and use proper math notation and vocabulary. Use one of the methods from class (Box method, Synthetic Division, Long Division). Hint: Use Desmos at each step to verify your work
3. Include a Desmos sketch of your polynomial function, and list as many key features / characteristics as you can think of.

1) The full process of factoring the polynomial function that is part of the inequality, including using the factor theorem

2) Using the zero product property to determine zeros of the polynomial function

3) Creating an appropriate interval chart that shows intervals where the function is positive and where it is negative

4) The solution to the inequality, including a number line representation

Finally, make connections to the graph of the corresponding polynomial function (include a Desmos graph or hand sketch).

Use Desmos to explore various polynomial functions and reflect on the following questions:

• What is therelationship between the end behaviour of a polynomial function and the Domain andRange of the function?
• Whatcharacteristic(s) of a polynomial function affect the Domain and Range? (Consider degree of the function, number of terms, leading coefficient)

Come to ageneral conclusion, and justify your reasoning usingselected examples and your own words.

Consider the important characteristics of polynomials that help you make connections between the equation and the graph in factored form:

• even vs odd degree
• sign of the leading coefficient (positive or negative)
• end behaviours
• different orders of roots (e.g. order 1 - linear, order 2 - double, order 3 - cubic)

1. polynomial functionin factored form that you haven't used or seen before in this course.
   1. Identify important characteristics of your polynomial.
   2. What do these characteristics tell you about the graph of your function?
2. Review other posts. Respond to two other posts:
   1. One post where a student has described a polynomial that has characteristics in common with your function. Make direct reference to the characteristics to describe the similarities
   2. One post where a student has described a polynomial that has characteristics that are different from those of your function. Make direct reference to the characteristics to describe the differences.

a) 2 (b) 3 (c) 5 (d) 6 (e) 7 (f) n

5. Identify the sign of the leading coefficient and describe the end behaviour. Using this information, decide if each function is cubic or quartic

6. Without using any graphing technology, graph the following polynomial functions:

Function: Degree: Sign: Quadrants: Roots: y-intercept: domain: range: end behaviours:	Function: Degree: Sign: Quadrants: Roots: y-intercept: domain: range: end behaviours:
Function: Degree: Sign: Quadrants: Roots: y-intercept: domain: range: end behaviours:	Function: Degree: Sign: Quadrants: Roots: y-intercept: domain: range: end behaviours:
Function: Degree: Sign: Quadrants: Roots: y-intercept: domain: range: end behaviours:	Function: Degree: Sign: Quadrants: Roots: y-intercept: domain: range: end behaviours:

Solve for x:

1. -(2x + 1)(x - 4)(x +3) 0

2. x³ - x² - 20x >0

3. x³ +10x² + 21x<0

4. 2x³ - x² - 15x +180

5. x⁴ - x³ - 11x² + 9x +180

6. x³ - 7x² + 6 >0

7. -x⁴ + 3x³ - 2x² -16x+ 16<0

8. x⁴ - 5x³ - 17x² + 3x +180

Check your solution set by graphing each of the polynomial functions.

Graph each of the following polynomial functions either by hand or with graphing software and solve the polynomial equation algebraically.

Polynomial Function	Polynomial Equation
	(x - 2)(x + 3) = 0
Graph:	Solve:
Polynomial Function	Polynomial Equation
	x2 - 3x - 4 = 0
Graph:	Solve:
Polynomial Function	Polynomial Equation
	x(x - 1)(x + 4) = 0
Graph:	Solve:
Polynomial Function	Polynomial Equation
	x3 + 6x2 - x - 30 = 0
Graph:	Solve:

Generalization: What do you notice about the graph of a polynomial function and the solutions to a polynomial equation?