An equation of the form \(y = 6000 ( 1.06 ) ^ { x } \) provides an example of interest compounded annually. This means that the full \(6 \% \) of interest is added to the account at the end of one year. This doesn't sound very fair to someone that invests their money for \(11\) months-they get no Interest at all. This became a competitive disadvantage for financial institutions, and some began to divide the annual interest into periodic shares, so that (for example) you could get \(1/12^ { th} \) of that \(6 \% \) each month. When this happens, we say that interest is compounded monthly. Interest can also be compounded weekly (52 times per year), quarterly ( \(4\) times per year), daily ( \(365\) times per year), or really any other period you could think of. 

If interest is compounded monthly, what growth factor would be needed to provide \(1/12^ { th} \) of \(6 \% \) interest each month? (Think about the difference between interest rate and growth factor.) 

The growth factor would be \(\square \) 

Find such that the function \(f( x) = \left \{ \begin{array} { rcl} 2x^ 2 + 4x & \text { if } & x \geq 1 \\  - x + k & \text { if } & x < 1 \end{array} \right . \)is continuous on \(\mathbb { R} \).

Market research for a certain ice cream company indicates that if the price per cup of ice cream is \(P 50.00\) , the demand will be \(1200\) cups, whereas if the price is increased to \(P 60.00\) , the demand will be \(1100\) cups. 

(a) If \(p\) denotes the price per cup (in pesos) and \(x\) denotes the demand for ice cream (in cups), express \(p\) as a linear function of \(x\) . 

(b) Determine the revenue function and its domain. 

(c) Find the marginal revenue at a production level of \(200\) cups of ice cream. Interpret. 

Now, suppose it is known that the total cost of production of \(x\) cups of ice cream is given by ( \(x ) = 3 x + 25,500\) . 

(d) Find the value(s) of \(x\) at the break-even points. 

(e) Use marginal analysis to approximate the profit from the sale of the \(500 ^ { th} \) cup of ice cream. Compare this value with the exact profit from the sale of the \(500 ^ { th} \) cup. 

Write the system of equations corresponding to the given augmented matrix using the variables \(x\) and \(y\) . Then perform the row operation \(R _ { 2 } = - 7 r _ { 1 } + r _ { 2 } \) on the given augmented matrix. 

\(\begin{bmatrix}    1 & 2 &\big m| & 9 \\    7 & 8 &\big m| & - 1 \\    \end{bmatrix} \) 

a) The functions \(p\) and \(q\) are defined as \(p ( x ) = \sqrt { 3 - x } \) and \(q ( x ) = \frac { 2 x - 6 } { x + 4 } , x \neq - 4 .\) 

i) State the domain of \(p ( x ) .\) 

ii) Prove algebraically whether \(p ( x ) \) is a one-to-one function. 

iii) Find the value of \(x\) if \(( p \circ q ) ( x ) = \sqrt { x } \) . 

1. Use the definition of the definite integral to evaluate the integral. Use the right end point of each interval for \(x_ { \space i} ^ *\).

\[\int _ { 1} ^ { 4} { { 2x + 3\space dx} } \]