1. Two cars are moving on a road and both start at the same place at time ?=0. The velocity of car number 1 is given by ?1(?)=sin?+1 and the velocity of car number 2 is given by ?2(?)=cos?+1.

a. Which car is ahead at ?=?2 ,?=?,?=3?? [3 marks] b. Write down a definite integral which gives the total distance traveled by car number 1 from ?=0 to ?=10. [3 marks]

b. Write down a definite integral which gives the total distance traveled by car number 1 from ?=0 to ?=10. [3 marks]

c. Are the cars ever at the same place on the road at the same time? If so, at what times? [3 marks]

d. What is the maximum distance between the cars? [3 marks]

A projectile is fired with an initial speed of 150 m/s and angle of elevation 60°. The projectile is fired from a position 80 m above the ground. (Recall g ≈ 9.8 m/s². Round your answers to the nearest whole number.)

(a) Find the range of the projectile

(b) Find the maximum height reached.


(c) Find the speed at impact.

In the function f(x) = 5x⁴-2x2-27 find the following:

a) their critical values

b)local maximum and minimum points

c)The intervals where the concavity is up and down

d) draw the graph and mark on it all the important points; maximum, minimum and inflection points

6. The cost equation for a certain company is given by
C(q) = 0.5q3 − 162q2 + 13500q,

where q is the quantity of items produced and the cost, C, is given in dollars.

1. (a) (1 pt) Find the equation for the average cost and the equation for C′(q).

   a(q) =

   C′(q) =

2. (b) (1 pt) Find the minimum average cost and the quantity at which it occurs. Show work.

Write parametric equations for each of the following curves.

(a) The straight line segment traced from (3,2) to (5,8) as t goes from 0 to 1.

(b) A circle centered at (3,2), of radius 4, traced out two times, counterclockwise, as t goes from 0 to 2pi.

Company MNO generates between 11000 and 15000 million dollars revenue per year. Suppose that the company's revenue stays within this range. Use an interest rate of 9.75% per year compounded continuously. Fill in the blanks in the paragraph below, rounding your answers to the nearest tenth.

The present value of MNO's revenue over a five year period is between _______________ and ______________ million dollars.

Over a twenty-five year period, the present value falls between ____________ and _____________ million dollars

Consider F and C below.

F(x, y, z) = yz i + xz j + (xy + 14z) k

C is the line segment from (3, 0, −1) to (6, 4, 2)

(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.

1) Suppose you invest $200 per month for a period of 30 years. a) How much money would you have at the end of the 30 year period, assuming that the yearly interest rate is 12% compounded monthly? You can assume that the payments are deposited at the end of each month. (Hint: write the amount accumulated in the bank for the first few months, and try to get a pattern our of that. ) b)Using the same approach indicate how much you would have at the end of 35 years?

2) A ball is dropped from a height of 10 ft. and bounces 5/7 of its former distance vertically, at each bounce. Find the total vertical displacement, if this pattern continues indefinitely. (For this question, count the vertical distance that the ball travels up and down as one vertical distance.)

It is clear that the angle of section of a cone will affect the shape of the resulting ellipse or hyperbola. We know that the shape is determined by the eccentricity. Find explicitly a relation between the two.

A particle, initially at rest, moves along the x-axis so that its acceleration at any time t ≥ 0 is given by a(t) = 12t²−4 . The position of the particle when t=1 is x(1)=3 .

Write an expression for the position x(t) of the particle at any time t ≥ 0.

The equation of a curve is 2x⁴ + xy³ + y⁴ = 10.
i) Show that dy/dx=(8x³+y³)/(3xy²+4y³).

ii) Hence show that there are two points on the curve at which the tangent is parallel to the x-axis and find the coordinates of these points.

Please complete this table. I am having trouble converting functions from polar to cartesian in the three dimensional plane.

I understand that x=rcos(?) and y=rsin(?) and r² = x² + y² , but I am having trouble understanding how to apply these functions.

Multiple Means of Representation Learners perceive and comprehend information differently. Your goal in this section is to explain how you would present content in various ways to meet the needs of different learners. For example, you may present the material using guided notes, graphic organizers, video or other visual media, annotation tools, anchor charts, hands-on manipulatives, adaptive technologies, etc. In a bulleted list, describe the materials you will use to differentiate instruction and how you will use these materials throughout the lesson to support learning. Bold any materials you will need to prepare for the lesson.

For example: • I will use a Venn diagram graphic organizer to teach students how to compare and contrast the two main characters in the read-aloud story. • I will model one example on the white board before allowing students to work on the Venn diagram graphic organizer with their elbow partner. Explain how you will differentiate materials for each of the following groups: • English language learners (ELL): • Students with special needs: • Students with gifted abilities: • Early finishers (those students who finish early and may need additional resources/support): Time Needed

8. The price p (in dollars) and the demand x for a particular wool sweater are related by the equation p = 100 − 0.025x a. Find the domain of this function. Show all work and clearly label the answer below. b. Find the revenue R (x) = x · p , from the sale of x wool sweaters and state the domain of the function R (x) . c. Find the marginal revenue at a production level of 1,600 sweaters and interpret the results. Show all work and clearly label the answer below. d. Find the marginal revenue at a production level of 2,500 sweaters and interpret the results. Show all work and clearly label the answer below.