A small business owner contributes $3000 at the end of each quarter to a retirement account that earns 8% compounded quarterly.

(a) How long will it be until the account is worth $150,000? (Round your answer UP to the nearest quarter.)

quarters

(b) Suppose when the account reaches $150,000, the business owner increases the contributions to $7000 at the end of each quarter. What will the total value of the account be after 15 more years? (Round your answer to the nearest dollar.)

$

How do I solve this equation? (1/3(5)-1/4(-3))(-3/4(3))

Sketch the graph of the given function. (x^2+x-2) / x^2

Give

a) x intercept

b) y intercept

c) Vertical asymtope

d)Horizontal asymtope

e) first derivative

f)second derivative

g)critical numbers

h)extrema max/min

i) y coordinate of exterma

j) possible point of infletion

h)y coordinate of possible point of inflection

k) table

l)graph

1. In 2003, gross sales at McDonald’s and Burger King totaled $60.9 billion. McDonald’s sales exceeded Burger King’s sales by $22.1 billion.

(a) Write a system of equations whose solution gives the individual sales of each company in billions of dollars.

(b) Solve the system of equations. (c) Is your system consistent or inconsistent? If it is consistent, state whether the equations are dependent or independent.

2. Solve the system symbolically 3x-2y=5 x+5y=13

3. Answer the following questions

a. What is a matrix?

b. What is a matrix Dimension

c. What is a matrix element?

Consider F and C below.

F(x, y, z) =

2xz + y²

i + 2xy j +

x² + 15z²

k

C: x = t²,    y = t + 2,    z = 4t − 1,    0 ≤ t ≤ 1

(a) Find a function f such that F = ∇f.

f(x, y, z) =

(b) Use part (a) to evaluate

∇f · dr along the given curve C.

4(a) Suppose a particle P is moving in the plane so that its coordinates are given by P(x,y), where x = 4cos2t, y = 7sin2t.

x2 y2
(i) By finding a, b ∈ R such that a2 + b2 = 1, show that P is travelling on an elliptical

path. [10 marks] (ii) Let L(t) be the distance from P to the origin. Obtain an expression for L(t).[8 marks] (iii) How fast is the distance between P and the origin changing when t = π/8?[7 marks]

(b) A wire of length 100 centimeters is cut into two pieces. One piece is bent to form a square. The other piece is bent to form an equilateral triangle. Find the dimensions of the two pieces of wire so that the sum of the areas of the square and the triangle is minimized.(25marks)

Two circles intersect at A and B. P is any point on the circumference of one of the circles. PA and PB are joined and produced to meet the circumference of the other circle at C and D respectively. Prove that the tangent at P is parallel to CD.

Assuming that x>0, use the method of reduction of order to find a second solution to

x^2y''−3xy'+4y=0

Given y1(x)=x^2

Find the point of intersection of the tangent lines to the curve r(t) = 5 sin(πt), 2 sin(πt), 6 cos(πt) at the points where t = 0 and t = 0.5. (x, y, z) =

Curvilinear integral of the function f (x, y) = x² + y² on a (3,0) centered and 3 radius circle.
a)Calculate the curvilinear integral by expressing the curve in parametrically.
b)Calculate the curvilinear integral by expressing the curve in polar coordinates.
c)Calculate the curvilinear integral by expressing the curve in cartesian coordinates.

Let V be the set of all ordered triples of real numbers. For u = (u1, u2, u3) and v = (v1, v2, v3), we define the following operations of addition and scalar multiplication on V :

u + v = (u1 + v1, u2 + v2 − 1, u3 + v3 − 2) and ku = (ku1, ku2, ku3).

For example, if u = (1, 0, 3), v = (2, 1, 1), and k = 2 then

u + v = (1 + 2, 0 + 1 − 1, 3 + 1 − 2) = (3, 0, 2) and 2u = (2 · 1, 2 · 0, 2 · 3) = (2, 0, 6).

Complete the following:

(a) Calculate (1, 1, 1) + (2, 2, 2).

(b) Show that (0, 0, 0) 6= 0.

(c) What is 0?

(d) State a vector space axiom that fails to hold. Give an example to justify your claim.

You are looking at a hot air balloon that sinks vertically at a speed of 4.5 m / s.

You stand a horizontal distance A from the point in the ground where the balloon will land.

You choose how far away you are. For calculations, you can choose a distance between 100 m and
700 m from the balloon landing site.

There is a straight line between you and the top of the balloon that forms an angle θ with the ground. The ground is horizontal.

Calculate how fast this angle decreases when the balloon is at a 200 m above the starting point.

Question 1. Solve the following 1. If tan θ = 4 where 0 ≤ θ ≤ π 2 , find sin θ, cos θ,sec θ, csc θ, cot θ.

2. If α = 3π 4 , find exact values for sec α, csc α,tan α, cot α.

Question 2. For each of the following angles, find the reference angle and which quadrant the angle lies in. Then compute sine and cosine of the angle. a. 225◦ b. 300◦ c. 135◦ d. 210◦

Question 3. The point P is on the unit circle. If the x-coordinate of P is 1/5, and P is in quadrant IV, find the y-coordinate.

Question 4. The angle of elevation to the top of a building in New York is found to be 9 degrees from the ground at a distance of 1 mile from the base of the building. Using this information, find the height of the building.

Question 5. A radio tower is located 400 feet from a building. From a window in the building, a person determines that the angle of elevation to the top of the tower is 36◦ and that the angle of depression to the bottom of the tower is 23◦ . How tall is the tower?

What level of output would generate a net income of $15,000 if a product sells for $24.99, has unit variable costs of $9.99, and total fixed costs of $55,005?