In Week 4 we learned about quadratic equations. In physics a quadratic equation can be used to model projectile motion. Projectile motion can describe the movement of a baseball after it has been hit by a bat, or the movement of a cannonball after it has been shot from a cannon. A penny falling from the Empire State Building can even be modeled with this equation!

The projectile motion equation is s(t)=-16t^2+vt+h where s(t) represents the distance or height of an object at time t, v represents the initial speed of the object in ft/s, and h is the initial height of the object, measured in feet.

If an object is starting at rest, then v=0 (such as for a penny being dropped from a building). If the object is starting from the ground, h=0. The baseball or cannonball situations, each have an initial velocity. For example, the initial velocity of the baseball is based on the speed at which the ball comes at you (the speed of the pitch).

Come up with a situation that you can model with this equation. Describe the situation, and tell us what v and h are. Fill in the values so that you have a quadratic equation. If you do research to find initial velocities, include the links to the websites where you found that information. If you would like to make up your own numbers as well, you can (be creative)!

Once you have your equation, find the maximum height as well as the time it takes to reach that maximum. Then use your equation to find when the object hits the ground (i.e. the x-intercepts).

Finally, use those three points as well as the initial height to sketch a graph. You can take a photo of it and include the image, or use an online graphing calculator and take a screenshot if that is easier.

Find the equation of the straight lines and draw the graph.
a) Go through point (2,1) and have slope 5.
b) Go through points (3, -1) and (4,5)

Let A and b be the matrices A = 1 2 4 17

3 6 −12 3

2 3 −3 2

0 2 −2 6

and b = (17, 3, 3, 4) . (a) Explain why A does not have an LU factorization. (b) Use partial pivoting and find the permutation matrix P as well as the LU factors such that PA = LU. (c) Use the information in P, L, and U to solve Ax = b

Determine which of the following sets of vectors form a basis for R 3 . S = {(1, 0, −1),(2, 5, 1),(0, −4, 3)}, T = {(−1, 3, 2),(3, −1, −3),(1, 5, 1)}.

Let Poly3(x) = polynomials in x of degree at most 2. They form a 3- dimensional space. Express the operator T(p) = p'

as a matrix (i) in basis {1, x, x 2 }, (ii) in basis {1, x, 1+x 2 } .

Suppose an ice cream cone has negligible thickness and the tip of the cone is identified with origin (0,0,0)and the circular opening at the top is identified with the equation x^2+y^2=1, z=1. if you fill the cone with ice cream "rounded" so that the surface of the sections forms part of the surface x^2+y^2+z^2=2 and the ice cream has variable density f= x^2+y^2+z^2, what is the mass of the ice cream?

A certain business keeps a database of information about its customers. A. Let C be the rule which assigns to each customer shown in the table his or her home phone number. Is C a function? Explain your reasoning.

Sketch a graph of the polynomials below as demonstrated in the Polynomial Functions video.  

Note: Do not just graph on a calculator and copy the graph! No credit will be given without supporing work or explanation. You should be prepared to do problems like this without a calculator on exams!

h(x)=x³(x-2)(x+3)²

• List of real zeros and their multiplicities:

• End behavior:

resolve the matrix gauss Elimination

Use the simplex method to solve the linear programming problem.

The maxium P= ____ at (x,y) ____

1. Party supply store A rents tables for $10 per day and chairs for $1.50 per day. Party supply store B rents tables for $9 per day and chairs for $1.25 per day, plus a $36 delivery charge. After how many days is it more expensive to rent 3 tables and 24 chairs from store A?

2. The demand function for an electronics company's line of gaming laptops is p = f(q) = 5,000 - 10q, where p is the price (in dollars) per laptop when q laptops are demanded (per week) by consumers. Find (a) the level of production that will maximize the electronics company's total weekly revenue, (b) the maximum revenue at this level of production, and (c) the price per laptop at this level of production.

Describe a situation in real life that can be modeled by an algebraic equation or system of equations. How about one modeled by an inequality or system of inequalities? If it is a situation you read about or saw on television, please give the appropriate citation.

The values of certain types of collectibles can often fluctuate greatly over time. Suppose that the value of a particular limited edition figurine is found to be able to be modeled by the function

?(?) = −0.01?⁴ + 0.47?³ − 7.96?² + 49.18? + 65 for 0 ≤ ? ≤ 20 where ?(?) is in dollars, t is the number of years after the figurine was released, and ? = 0 corresponds to the year 1999.

a) What was the value of the figurine in the year 2009?

b) What was the value of the figurine in the year 2019?

c) What was the instantaneous rate of change of the value of the figurine in the year 2002?

d) What was the instantaneous rate of change of the value of the figurine in the year 2019?

e) Use your answers from parts a-d to estimate the value of the figurine in 2020.

Suppose E and F are two mutually exclusive events in a sample space S with P(E) = 0.34 and P(F) = 0.46. Find the following probabilities.

(1 point) At what point does the normal to y=(−1)+2x2y=(−1)+2x2 at (1,1)(1,1) intersect the parabola a second time?
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Hint: The normal line is perpendicular to the tangent line. If two lines are perpendicular their slopes are negative reciprocals -- i.e. if the slope of the first line is mm then the slope of the second line is −1/m