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Problem 2 Consider the following 11 powers of 3: (a) (4 points) Show that there exist...
3). 12. Recall that the centroid of a triangle with vertices (11; yı), (22:42), (13,) €...
3). 12. Recall that the centroid of a triangle with vertices (11; yı), (22:42), (13,) € RP is the point (1 , Given a set of 20 points in RP with integer coordinates, prove that three of them will form a triangle whose centroid has integer coordinates. Hint: work mod 3 and use the pigeonhole principle. Distribute the points ( yi) into 9 boxes depending on r, mod 3 and yi mod 3.
Problem 3. (15 points) Given the following data: 3, 2, 3, 4, 5, 3, 4, 6,...
Problem 3. (15 points) Given the following data: 3, 2, 3, 4, 5, 3, 4, 6, 4, 10, 7, 11 () (2 points) Calculate the sample mean by hand using its definition. Round your in results to two decimal places and your final answer to one decimal place. Same for the remai questions. (2) (2 points) Calculate the median of the data. (3) (3 points) Calculate the sample variance by hand using its definition. 4) (3 poi ning ints) Calculate...
14 points LarApCalc10 3.1.048 12. Consider the following. y--+4, +2x x0 Find the critical numbers. (Hint:...
14 points LarApCalc10 3.1.048 12. Consider the following. y--+4, +2x x0 Find the critical numbers. (Hint: Check for discontinuities.) (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) Find the open intervals on which the function is increasing or decreasing. (Enter your answers using interval notation If an answer does not exist, enter DNE.) increasing decreasing Sketch the graph of the function to verify your results. 5 -3 -2 -1 -6-5-43-2-11 1 23 Type...
Problem 1 Consider the matrix Problem 1 Consider the matriz a 2 5 3 11 08...
Problem 1 Consider the matrix Problem 1 Consider the matriz a 2 5 3 11 08 a Find the cofactors C11,C2,C3 of A. b Find the determinant of 1, det(A) [ 2 4 61 Problem 2 Consider the matriz A=008 | 2 5 3 a Use the ero's to put A in upper triangular form 5 Pinul the determinant of A. (A) by keeping track of the row operations in part a and the properties of determinant Problem 3 Consider...
Duality Axiom 1. There exist exactly 4 distinct points. Axiom 2. There exist exactly 5 distinct...
Duality
Axiom 1. There exist exactly 4 distinct points. Axiom 2. There exist exactly 5 distinct lines. Axiom 3. There is exactly 1 line with exactly 3 distinct points on it. Axiom 4. Given any 2 distinct points, there exists at least 1 line passing through the 2 points. Which of the following is the dual of Axiom 4? O a. Every line has at least 2 points on it. b. There exists at least 1 point with at least...
Problem 2 Consider the system of equations 2 1. Show that the z and t are determined as a functio...
Problem 2 Consider the system of equations 2 1. Show that the z and t are determined as a function of x and y near the point (0, 1,-1). Can we apply the Implicit Function theorem? 2. Compute the partial derivatives of z and t with respect to z, y at (0,1) 3. Without solving the system, what is approximate value of 2(0.001,1.002) (Hint: Use the first order Taylor approximation about the point (1,0) to find the approximation) 4. Compute...
(2 points) Consider the following initial value problem, in which an input of large amplitude and...
(2 points) Consider the following initial value problem, in which an input of large amplitude and short duration has been idealized as a delta function. yy1+(t-4), y(0)0. a. Find the Laplace transform of the solution. Y(s) = L {y(t)) = b. Obtain the solution y(t) C. Express the solution as a piecewise-defined function and think about what happens to the graph of the solution at t = 4. if 0st<4, y(t) if 4t< o0.
Consider the following matrix A= -3 4 4 3 (a) (8 points) Diagonalize A. (b) (4...
Consider the following matrix A= -3 4 4 3 (a) (8 points) Diagonalize A. (b) (4 points) Using your result of part (a) compute A^20 . You must perform the multiplication to receive a single matrix as a result but you don’t have to simplify the high powers in the entries. Your result should look like A^20 = 5^b × B for some matrix B and power b.
Problem #2: Consider the following vectors, which you can copy and paste directly into Matlab. x=[3 4 4 3 5 5 1 2 3...
Problem #2: Consider the following vectors, which you can copy and paste directly into Matlab. x=[3 4 4 3 5 5 1 2 32); y [2 4 4622 4 2 4] Use the vectors x and y to create the following matrix. 3 2 0 0 0 0 0 0 0 o Such a matrix is called a tri-diagonal matrix. Hint: Use the diag command three times, and then add the resulting matrices. To check that you have correctly created...
Powers of 11 2. a. Use the Binomial Theorem on powers of (10+1) to demonstrate that the numbers b. At what point does the pattern not continue? Why does the pattern no longer c. Compute 11 x 11 using...
Powers of 11 2. a. Use the Binomial Theorem on powers of (10+1) to demonstrate that the numbers b. At what point does the pattern not continue? Why does the pattern no longer c. Compute 11 x 11 using the standard algorithm for multiplication. Do likewise d. Explain how the standard algorithm for multiplication of powers of 11 relates to in the first few rows of Pascal's Triangle resemble the digits of the powers of 11 work? with 121 x...
3). 12. Recall that the centroid of a triangle with vertices (11; yı), (22:42), (13,) € RP is the point (1 , Given a set of 20 points in RP with integer coordinates, prove that three of them will form a triangle whose centroid has integer coordinates. Hint: work mod 3 and use the pigeonhole principle. Distribute the points ( yi) into 9 boxes depending on r, mod 3 and yi mod 3.
Problem 3. (15 points) Given the following data: 3, 2, 3, 4, 5, 3, 4, 6, 4, 10, 7, 11 () (2 points) Calculate the sample mean by hand using its definition. Round your in results to two decimal places and your final answer to one decimal place. Same for the remai questions. (2) (2 points) Calculate the median of the data. (3) (3 points) Calculate the sample variance by hand using its definition. 4) (3 poi ning ints) Calculate...
14 points LarApCalc10 3.1.048 12. Consider the following. y--+4, +2x x0 Find the critical numbers. (Hint: Check for discontinuities.) (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) Find the open intervals on which the function is increasing or decreasing. (Enter your answers using interval notation If an answer does not exist, enter DNE.) increasing decreasing Sketch the graph of the function to verify your results. 5 -3 -2 -1 -6-5-43-2-11 1 23 Type...
Problem 1 Consider the matrix Problem 1 Consider the matriz a 2 5 3 11 08 a Find the cofactors C11,C2,C3 of A. b Find the determinant of 1, det(A) [ 2 4 61 Problem 2 Consider the matriz A=008 | 2 5 3 a Use the ero's to put A in upper triangular form 5 Pinul the determinant of A. (A) by keeping track of the row operations in part a and the properties of determinant Problem 3 Consider...
Duality
Axiom 1. There exist exactly 4 distinct points. Axiom 2. There exist exactly 5 distinct lines. Axiom 3. There is exactly 1 line with exactly 3 distinct points on it. Axiom 4. Given any 2 distinct points, there exists at least 1 line passing through the 2 points. Which of the following is the dual of Axiom 4? O a. Every line has at least 2 points on it. b. There exists at least 1 point with at least...
Problem 2 Consider the system of equations 2 1. Show that the z and t are determined as a function of x and y near the point (0, 1,-1). Can we apply the Implicit Function theorem? 2. Compute the partial derivatives of z and t with respect to z, y at (0,1) 3. Without solving the system, what is approximate value of 2(0.001,1.002) (Hint: Use the first order Taylor approximation about the point (1,0) to find the approximation) 4. Compute...
(2 points) Consider the following initial value problem, in which an input of large amplitude and short duration has been idealized as a delta function. yy1+(t-4), y(0)0. a. Find the Laplace transform of the solution. Y(s) = L {y(t)) = b. Obtain the solution y(t) C. Express the solution as a piecewise-defined function and think about what happens to the graph of the solution at t = 4. if 0st<4, y(t) if 4t< o0.
Problem #2: Consider the following vectors, which you can copy and paste directly into Matlab. x=[3 4 4 3 5 5 1 2 32); y [2 4 4622 4 2 4] Use the vectors x and y to create the following matrix. 3 2 0 0 0 0 0 0 0 o Such a matrix is called a tri-diagonal matrix. Hint: Use the diag command three times, and then add the resulting matrices. To check that you have correctly created...
Powers of 11 2. a. Use the Binomial Theorem on powers of (10+1) to demonstrate that the numbers b. At what point does the pattern not continue? Why does the pattern no longer c. Compute 11 x 11 using the standard algorithm for multiplication. Do likewise d. Explain how the standard algorithm for multiplication of powers of 11 relates to in the first few rows of Pascal's Triangle resemble the digits of the powers of 11 work? with 121 x...
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