Let L₁ be the line passing through the point P₁=(−1, −2, −4) with direction vector →d=[1, 1, 1]^T, and let L₂ be the line passing through the point P₂=(−2, 4, −1) with the same direction vector.
Find the shortest distance d between these two lines, and find a point Q₁ on L₁ and a point Q₂ on L₂ so that d(Q₁,Q₂) = d. Use the square root symbol '√' where needed to give an exact value for your answer.

Determine the solution to the initial value differential equation;

y′=0.0015y(1100−y), y(0)=32

1. y(x) = ?

2. What is the maximum value of this function. In other words, evaluate: lim x-> inf y(x)

3. Determine x for which y(x) reaches 86% of its maximum value.

Find a possible formula for the trigonometric function whose values are in the following table.

Question 1. Let V and W be finite dimensional vector spaces over a field F with dimF(V ) = dimF(W) and let T : V → W be a linear map. Prove there exists an ordered basis A for V and an ordered basis B for W such that [T] A B is a diagonal matrix where every entry along the diagonal is either a 0 or a 1.

Hint 1. Suppose A = {~v1, . . . , ~vn} and B = { ~w1, . . . , ~wn}. If the k th column of [T] A B consists of all zeros, what can you deduce?

Hint 2. Suppose A = {~v1, . . . , ~vn} and B = { ~w1, . . . , ~wn}. If the k th column of [T] A B has a one in the k th entry and all other entries are zero, what can you deduce?

Hint 3. Now construct bases with the properties found in Hint 1 and Hint 2.

Hint 4. Theorem 18 part 5 is your friend.

Hint 5. The proof of the Rank-Nullity Theorem is your best friend.

Let f(x, y) =x^2+ 3y^2−2x−12y+ 13 on the domain A given by the triangular region with vertices (0,0),(0,6), and (2,0).

Find the maximum of f on the boundary of A.

Let PQ be a focal chord of the parabola y² = 4px. Let M be the midpoint of PQ. A perpendicular is drawn from M to the x-axis, meeting the x-axis at S. Also from M, a line segment is drawn that is perpendicular to PQ and that meets the x-axis at T. Show that the length of ST is one-half the focal width of the parabola.

Find a cubic function y = ax3 + bx2 + cx + d whose graph has horizontal tangents at the points (−2, 8) and (2, 2).

Find an equation of the normal line to the parabola y = x² − 8x + 7  that is parallel to the line x − 2y = 2.

The function f(x)=3x+2 is one-to-one

a) find the inverse of f

b) State the domain and range of f

c) State the domain and range of f^-1

d) Graph f,f^-1, and y=x on the same set of axes

a. consider the plane with equation -x+y-z=2, and let p be the point (3,2,1)in R^3. find the distance from P to the plane.

b. let P be the plane with normal vector n (1,-3,2) which passes through the point(1,1,1). find the point in the plane which is closest to (2,2,3)

The COV-19 virus has virtually changed the way we work, study and shop. People are scared and frightened of contracting the deadly virus. Traditional grocery stores such as Loblaws, Longo’s, etc. have seen fewer shoppers, who also spent less time in the stores. A survey of 500 Canadian grocery shoppers was conducted last week to find out if the scare of contracting the COV-19 virus, (very scared , VS) or (somewhat scared, SS) has affected their grocery shopping habits: whether they purchase groceries online (OL) or they purchase at a conventional grocery store (TS).The survey reported that 400 shoppers purchased online and that 150 of all shoppers were somewhat scared. The survey also reported that 75 Canadians shopped online given that the shopper was somewhat scared.

a) Construct a cross-classification (or contingency) table with joint and marginal probabilities. [or you can construct a tree diagram].

b) What is the probability that a randomly selected shopper is very scared and shops online?   

c) What is the probability that a randomly selected very scared shopper buys from a traditional grocery store?

d) What is the probability that a shopper who shops online is somewhat scared?

e) What is the probability that a randomly selected shopper is somewhat scared or shops online?

f) Are “somewhat scared” and “online” independent events or dependent events or mutually exclusive events? Your answer must include supporting probability calculations.

Show that if P;Q are projections such that R(P) = R(Q) and N(P) = N(Q), then P = Q.

A gift shop expects to sell 400wind chimes during the next year. It costs $1.20 to store one wind chime for one year. There is a fixed cost of $⁢15 for each order. Find the lot size and the number of orders per year that will minimize inventory costs.

T̶w̶o̶ ̶l̶i̶n̶e̶a̶r̶l̶y̶ ̶i̶n̶d̶e̶p̶e̶n̶d̶e̶n̶t̶ ̶s̶o̶l̶u̶t̶i̶o̶n̶s̶ ̶o̶f̶ ̶t̶h̶e̶ ̶f̶o̶l̶l̶o̶w̶i̶n̶g̶ ̶e̶q̶u̶a̶t̶i̶o̶n̶ ̶(̶1̶ ̶−̶ ̶x̶)̶ ̶y̶″̶ ̶ ̶+̶ ̶ ̶x̶ ̶y̶′̶ ̶ ̶−̶ ̶ ̶y̶ ̶ ̶=̶ ̶ ̶0̶ ̶

a̶r̶e̶ ̶ ̶y̶1̶(̶x̶)̶ ̶ ̶=̶ ̶ ̶3̶e̶^̶x̶ ̶a̶n̶d̶ ̶ ̶y̶2̶(̶x̶)̶ ̶ ̶=̶ ̶ ̶6̶x̶.̶ ̶

a̶)̶ ̶F̶i̶n̶d̶ ̶t̶h̶e̶ ̶W̶r̶o̶n̶s̶k̶i̶a̶n̶ ̶W̶(̶y̶1̶,̶ ̶y̶2̶)̶ ̶o̶f̶ ̶y̶1̶ ̶a̶n̶d̶ ̶y̶2̶.̶ ̶

(̶b̶)̶ ̶U̶s̶i̶n̶g̶ ̶t̶h̶e̶ ̶m̶e̶t̶h̶o̶d̶ ̶o̶f̶ ̶v̶a̶r̶i̶a̶t̶i̶o̶n̶ ̶o̶f̶ ̶p̶a̶r̶a̶m̶e̶t̶e̶r̶s̶,̶ ̶f̶i̶n̶d̶ ̶a̶ ̶p̶a̶r̶t̶i̶c̶u̶l̶a̶r̶ ̶s̶o̶l̶u̶t̶i̶o̶n̶ ̶o̶f̶ ̶(̶1̶ ̶−̶ ̶x̶)̶ ̶y̶″̶ ̶ ̶+̶ ̶ ̶x̶ ̶y̶′̶ ̶ ̶−̶ ̶ ̶y̶ ̶ ̶=̶ ̶ ̶2̶(̶x̶ ̶−̶ ̶1̶)̶^̶2̶ ̶(̶e̶ ̶^̶−̶x̶)̶