1. Find the derivatives of each of the following functions:

1. $y = 3x + 5$
2. $y = -5x - 12$
3. $y = 3x^2 + 5 $
4. $y = -4x^2 + 5x - 7$
5. $y = 6x^5 + 5x^4 - 8x^3 + 7x^2 - 8x + 5$

2. Suppose your profit when you sell 1000 units is \$5000, and that the derivative of ...

1. For the equation $y = x^2 + 3 $ :

1. Find the average slope between $x = 2 $ and $ x = 4$
2. Find the average slope between $x = -4 $ and $ x = -2$
3. Find the average slope between $x = -2 $ and $ x = 2$
4. Determine what value of x will give a slope of zero.

Solution:

1. $\frac ...

1. Rearrange the following to determine allowed values for y:
   1. $$ 2y + 3 < 10 $$
   2. $$ -3y -5 > 6 $$
2. Rearrange to find allowed values for y in terms of x:
   1. $$ 2y + 3x < 10 $$
   2. $$ -3y - 5x + 7 > 2 $$
3. Sketch graphs of the two inequalities given in #2, and shade in the region of the plane ...

1. You have three employees, whose salaries are 8, 10, and 15 dollars per hour. Working alone, they can bake and decorate cakes at rates of 3, 7, and 12 cakes per hour (respectively). An order comes in for 184 cakes, to be delivered tomorrow (in 24 hours), and you will ...

1. Construct a matrix representation of the following system of equations, and solve it using the matrix:

$$ \begin{eqnarray} -3x + 6y - 9z &=& 3\\ x - y - 2z &=& 0\\ 5x + 5y - 7z &=& 63 \end{eqnarray} $$

Solution: $$ x = 8, y = 6, z = 1 $$

1. Write clear sentences explaining how you can tell:
   1. Whether it is possible to multiply two matrices together.
   2. What the size of the resulting matrix will be.
2. Solve the following system of equations (find the intersection of the two lines):

$$ \begin{eqnarray} 2x + 3y & = & 7 \\ -2x - 5y & = & 11 \end{eqnarray} $$

3. Using ...

1. The following data represents the distance an object falls after being thrown downward from a very high height. I propose that the equation $d(t) = 5 + 4t + 5t^2$ models this data well. Determine which parts of the equation are incorrect, and fix them.