Linear Algebra

Q1. The pressure at sea level is 1 atmosphere. For every 10 metres that a diver descends into the sea, the pressure increases by 1 atmosphere. (i) What is the independent variable? What is the dependent variable? (ii) Sketch a graph, putting the independent variable on the horizontal axis. (iii) Find the slope of this line and express it as a rate of change. (iv) Write the equation of the line and write the pressure as a function of the depth.

a) Write down the augmented matrix of the system. b) Use elementary row operations to find a and b (or a relation for a and b), such that the given system is always consistent. c) If we write the given system in the form Ax = b, give the matrices A, x, and b, and determine whether we can have the solution of the system through the form \(\bf {x}=A-¹b.

Question (1): Let P(x) and Q(x) be two polynomials and consider the following two cases: Case 1: P(x) x Q(x) Case 2 : 3 Q(x) For each case, find the following: a) The number of terms. b) The degree.

Question (2): Let ax + b = cx + d be a linear equation "given that a, b, c, d are non-zero constants". Discuss the relation between these constants to have each of the following cases: a) The equation has one solution. b) The equation has no solution. c) The equation has all real numbers as solutions.

1. Let P3 be the vector space of polynomials of degree 3 or less. Let V be the subspace of polynomials satisfying f(1) = f(2) = 0. Find a basis for V.

Q 1 If f(x) = a* and f(3) = 64, find f(2). Assume a > 0. Q 2 Solve the given equation for x. Q 3 Solve the given equation for x. Q 4 Solve the equation for x. (5x + 10)³ = 64x³ Q 5 Solve the equation for x. (6x +27)6 = (x+7)6 Q 6 Solve the given equation for x. (Remember that e* *0 for all values of x.) Q 7 Solve the given equation for x. 2xe-9x + x²e-9x = 0 Q 8 Solve the given equation for x. e9x-7-e=0 Q 9 Find the value of an investment of $20,000 for 11 years at an annual interest rate of 3.15% continuously.compounded

You will develop a short (10-minute) "anytime, anywhere" on-line video clip that students in your high school mathematics class can watch to learn about a topic in the course you are teaching. Your video should enhance students' understanding of an important concept(s) and/or skill(s) related to the topic. It should incorporate many of the ideas from this course regarding effective direct instruction.

Problem 18. A door is opened between rooms that hold v(0) = 30 people and w(0) = 10 people. The movement between rooms is proportional to the difference v-w:

1. In each case, find the intervals where f(x) is concave upward or concave downward. Find any inflection points. (a) f(x)= x(x - 3)² (b) f(x) = 5-x² (c) f(x) = xlog |x|

Question 35 Solving Radical Equations Graphically Given the function f(x) = 3√√4 + 2x find x when f(x) = 18 using your graphing calculator. Then follow the steps below to show the intersection graph on the computer. [Hint: Use the following window on your graphing calculator: Xmin = -4, Xmax = 18, Ymin = -2, Ymax =20] 1. Write the point of intersection as an ordered pair below. Round the value of x to two decimal places as needed. If the point of intersection does not exist, enter DNE 18. 2. Use the Line Tool to draw the Horizontal Line y 3. Use the Square Root Tool to draw f(x) = 3√4 + 2x. To draw this function, first plot the point furthest to the left, followed by one more point. 4. Use the Point Tool to identify the point of intersection if it exists. 20+ 19 18 17 16 15 14 Wo A -4 -3 -2 -1 Point of Intersection: 13 12 11 10 9 8 N 7 s6 s6 5 4 3 S < 2 > 27 1 -1 -2 3√4 + 2x 2 3 4 Clear All Draw: = 18 8 9 10 11 12 13 14 15 16 17 18