The normal monthly precipitation at the Seattle-Tacoma airport can be approximated by the model shown below, where R is measured in inches and t is the time in months, with t = 0 corresponding to January 1. (Round your answers to two decimal places.) R = 2.876 + 2.202 sin(0.576t + 0.847) (a) Determine the extrema of the function over a one-year period. relative minimum (t, R) = ( ), relative maximum (t, R) = ( ), (b) Use integration to approximate the normal annual precipitation. (Hint: Integrate over the interval [0, 12].) in (c) Approximate the average monthly precipitation during the months of June, July, and August. in
Part (a) The sales S (in thousands of units) of a seasonal product are given by the model below, where t is the time in months, with t = 1 corresponding to January. Find the average sales for the time period. S = 76.81 + 45.85 sinπt6 The first quarter (0 ≤ t ≤ 3) Part (b) The sales S (in thousands of units) of a seasonal product are given by the model shown below, where t is the time in months, with t = 1 corresponding to January. Find the average sales for the time period. S = 76.81 + 45.85 sinπt6 The second quarter (3 ≤ t ≤ 6) Part (c) The sales S (in thousands of units) of a seasonal product are given by the model shown below, where t is the time in months, with t = 1 corresponding to January. Find the average sales for the time period. S = 76.81 + 45.85 sinπt6 The entire year (0 ≤ t ≤ 12)
Use a graphing utility to graph the function. Then use the Horizontal Line Test to determine whether the function is one-to-one on its entire domain and therefore has an inverse function. f(θ) = 4 sin(θ) The function have an inverse function.
Use a graphing utility to graph the function. Determine whether the function is one-to-one on its entire domain and therefore has an inverse function. g(t) = 4 t2+4 The function is one-to-one on its entire domain and therefore has an inverse function. The function is not one-to-one on its entire domain and therefore does not have an inverse function.
Use a graphing utility to graph the function. Determine whether the function is one-to-one on its entire domain and therefore has an inverse function. f(x) = 5x x−1 The function is one-to-one on its entire domain and therefore has an inverse function. The function is not one-to-one on its entire domain and therefore does not have an inverse function.
Consider the following function. f(x) = x5 − 5 (a) Find the inverse function of f. f−1(x) = (b) Graph both f and f−1 on the same set of coordinate axes. (c) Describe the relationship between the graphs of f and f−1. The graph of f−1 is the reflection of f in the x-axis. The graph of f−1 is the reflection of f in the line y = −x. The graph of f−1 is the reflection of f in the line y = x. The graph of f−1 is the reflection of f in the y-axis. The graph of f−1 is the same as the graph of f. (d) State the domain and range of f. (Enter your answers using interval notation.) domain range State the domain and range of f−1. (Enter your answers using interval notation. ) domain range
Consider the function. f(x) = x3 /5 (a) Find the inverse function of f. f−1(x) = (b) Use a graphing utility to graph f and f−1 in the same viewing window. (c) Describe the relationship between the graphs. The graphs of f and f−1 are reflections of each other about the line y = (d) State the domain and range of f and f−1. Domain of f : x > 3 x > 5 x > 0 all x≠0 all real numbers Range of f : y > 3 all y≠0 all real numbers y > 5 y > 0 Domain of f−1 : all x ≠ 0 x > 3 all real numbers x > 0 x > 5 Range of f−1 : y > 5 y > 0 all real numbers all y ≠ 0 y > 3
Consider the function. f(x) = x+1 x (a) Find the inverse function of f. f−1(x) = (b) Use a graphing utility to graph f and f−1 in the same viewing window. (c) Describe the relationship between the graphs. The graphs of f and f−1 are reflections of each other in the line y = (d) State the domain and range of f and f−1. Domain of f : all real numbers all x ≠ 0 all x ≠ 1 x > 0 Range of f : y > 0 all real numbers all y ≠ 1 all y ≠ 0 Domain of f−1 : all x ≠ 0 all real numbers x > 0 all x ≠ 1 Range of f−1 : all y ≠ 0 y > 0 all real numbers all y ≠ 1
Consider the function. g(x) = 8x2 x2+9 (a) Use a graphing utility to graph the function. (b) Use the drawing feature of a graphing utility to draw the inverse relation of the function. (c) Determine whether the graph of the inverse relation is an inverse function. The inverse relation an inverse function.
Consider the following. f(x) = (x+8)2 (a) Sketch a graph of the function f. (b) Determine intervals on which f is one-to-one. (−∞, −8] and [−8, ∞) (−∞, ∞) (−∞, 0] and [0, ∞) [−8, 0] (−∞, 8] and [8, ∞) (c) Find the inverse function of f on the interval found in part (b). (d) Give the domain of the inverse function. [0, ∞) (−∞, ∞) (−∞, −8]∪[8, ∞) (−∞, 0)∪(0, ∞) [−8, 0]
Use the functions below to find the given value. f(x) = 1 6x − 4 g(x) = x3 (f−1∘g−1)(1) =
Question 2 [6 points] Consider the matrix defined below and answer the following questions. A = [−1 3 0 2 5 1 0 1 −7] (a) [4 points] The inverse of the matrix above can be expressed as A−1 = 1 78[a b c d 7 f g h −11] Use the Inversion Algorithm to find the missing entries. (b) [2 points] Use your answer from part (a) to solve the system Ax = b where x = (x1, x2, x3) and b = (1, −1, 0).
(6 marks) A quadratic function f(x) = ax2 + bx + c can be represented by a vector [a b c]T in R3. Hence f(x) = 2x2 + x − 5 corresponds to the vector [2 1 −5]T while f(x) = 2x−1 corresponds to the vector [0 2 −1]T. (a) Find a matrix equation M[a b c] = z so that [a b c]T solves the equation exactly if f(x) = ax2 + bx + c satisfies f(1) = 2 and f′(0) = 3. (b) Which columns in M contain leading entries? How many parameters are there in the set of solutions for the equation? (c) Find a non-zero vector in the null space of the matrix.
(a) For a Markov chain on the state space {1, 2, …} give the definition of a stationary distribution. (b) Suppose π denotes the stationary distribution associated with a transition matrix P. Show that πP2 = π. (c) Calculate the stationary distribution for the Markov chain {Xn} on state space {1, 2, 3} with the following transition matrix P = (1/4 1/2 1/4 1/4 1/4 1/2 0 1/2 1/2). (d) Suppose X0 = 1, and I ran this Markov chain for a very long period of time. Approximately what proportion of time would you expect this chain to be in state 3? How about if X0 = 2?
Given the following matrix: A = [5 0 3 0 −1 0 0 α β] a) Study for what real values of α and β the matrix diagonalizes. b) Calculate the diagonal and step matrix for α = 0 and β = −1. c) Find the expression of A raised to the nth power.
Compute the matrix (BA⊤)⊤, if it exists, when the following matrices are given. A = [0 −1 1 0 1 −1]B = [0 −2 0 −1 2 0 −1 0 1] Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. (BA⊤)⊤ = (Simplify your answer. ) B. The computation is not possible.
Consider the following matrices: A = [−5 −4 8 4 9 1], B = [−2 −4 2 −3 −3 8], C = [−5 −8 5 −7] D = [−1 5 4 −3] and E = [−7 9 −1 −2 −8 −3] From the following statements select those that are true. B+C is well defined. The matrix product A×E is well defined and is of order 2×2.3C−7D is well defined and is of order 2×2 The matrix product ExD is not well defined. The matrix product B×C is well defined. A+B is well defined and is of order 2×3.
Matrix dimensions. Suppose A is an m×n matrix, and B is a p×q matrix. For each of the following equations, give the conditions on m, n, p, q under which it makes sense. For example, A = B requires that m = p and n = q. (a) ATB = I. (b) AB = 0. (c) AB = BA. (d) AT = [B I]. (e) [A B] = I.
Suppose A is a 5×10 matrix, B is a 20×10 matrix, and C is a 10×10 matrix. Determine whether each of the following expressions make sense. If the expression makes sense, give its dimensions. (a) ATA + C. (b) BC3. (c) I + BCT. (d) BT − [C I]. (e) B[A A]C
(a) A and B are any matrices with the same number of rows. What can you say (and explain why it is true) about the comparison of rank of A rank of the block matrix [A B] (b) Suppose B = A2. How do those ranks compare? Explain your reasoning. (c) If A is m by n of rank r, what are the dimensions of these nullspaces? Nullspace of A Nullspace of [A A]
(3) Let A be a 2×2 matrix with E0(A) = Span([1 1]), E1(A) = Span([3 1]). (Recall Eλ(A) denotes the eigenspace of A with eigenvalue λ. ) What are the eigenvalues and eigenspaces of B = A2 − 3A? Justify your answer.
Consider the following matrix. A = [3 4 5 −1 −1 −1 3 5 6] Choose the correct description of A. Find A−1 if it exists. A is nonsingular. That is, it has an inverse. A−1 = A is singular. That is, its inverse doesn't exist.
Determine whether such a matrix A exists. If yes, find an example of A, if no, justify your answer. (a) ATA has an eigenvector (2, 1) corresponding to the eigenvalue 6 , and AAT has two eigenvectors (2, −1, 1) and (1, 1, 0) corresponding to eigenvalues 6 and 5 , respectively. (b) ATA has an eigenvector (2, 1) corresponding to the eigenvalue 6 , and AAT has two eigenvectors (2, −1, 1) and (1, 2, 0) corresponding to eigenvalues 6 and 5 , respectively.
Consider the matrix P = [1 5 0 a 0 b 0 2 5 0 1 5]. Find a value for a and a value for b that together make P orthogonal. a = b =
Given A = [−1 1 3 3 0 −9 −2 1 6] Select all true statements about the matrix A. Note: there may be more than one true statement. A. A is singular B. A is nonsingular C. N(A) contains only the trivial solution D. The system LS(A, b) has a unique solution for every vector b in C3 E. N(A) contains infinitely many vectors F. A is row-equivalent to the three-by-three identity matrix
Suppose the average daily metabolic rate for certain types of animals can be expressed as a function of mass by r = 147.2m 0.5, where m is the mass of the animal (in kilograms) and r is the metabolic rate (in kcal per day). a. If the mass of an animal is changing at a rate of dm dt, find dr dt. b. Determine dr dt for a 230−kg animal that is gaining mass at a rate of 4 kg per day. a. If the mass of an animal is changing at a rate of dm dt, find dr dt. dr dt =