Download Quiz 5 for MATH 1630 – Internet: Problems on Functions, Logarithms, and Exponential Growth and more Quizzes Mathematics in PDF only on Docsity!
MATH 1630 – Internet Name: KEY 40 possible points
Spring 2009 Quiz 5
You must show all work to receive full credit. This quiz includes material from Sections 4.3, 5.1, 5.2 and 5.
- (3 points) Given that f (^) ( x (^) ) = 3* 4 x , determine f (^) ( − (^10) ). Express your answer in fractional form with no decimals.
( ) (^1010)
10 3* 4 3* 1 3*^1
f − = − = = =
- (4 points) If $4800 is invested for x years at 8%, compounded quarterly, the interest earned is I = 4800 1.02 ( )^4^ x − 4800. Calculate the amount of interest earned in 7 years. ( ) (^ ) ( ) ( )
4 7 28
I
I
I
I
I
- (3 points) Calculate a set of 3 points that are on the graph of f^ ( x^ ) =log 2 x
We’ll rewrite the expression y = log 2 x in exponential form as 2 y^ = x. The simple plan is to pick easy values for y and see what x turns out to be. Let y = 0. Then x = 2 0 = 1. So the point (1, 0) is on this graph. Let y = 1. Then x = 21 = 2. So the point (2, 1) is on this graph. Let y = 2. Then x = 2 2 = 4. So the point (4, 2) is on this graph.
- (4 points) List and label the domain and range of^ f^ ( x^ ) =3*7 x
Domain: Range:
x can be any real number f(x) can be any real number greater than 0.
- (4 points) List and label the domain and range of f (^) ( x (^) ) =log 2 x
We can rewrite this log function as 2 y^ = x. This is basically an exponential function with the x’s and y’s switched. So we can determine the domain and range of this log function by switching the domain and range of exponential functions.
Domain: Range:
- (3 points) Use properties of logarithms to write the expression as a single logarithm.
4 log y −7 log x
We’ll rewrite this in two steps:
4 7 4 7
log log ; Logarithm Property V
log ; Logarithm Property IV
Done
y x
y
x
- (3 points) Use properties of logarithms to write the expression as the sum or difference of two logarithmic functions containing no exponents.
log (^) ( x^8^ y^4 )
We can use log property III (pg 377) to rewrite log (^) ( x^8^ y^4 )= log x^8^ +log y^4
We can use log property V (pg 377) to complete the problem. log (^) ( x^8^ y^4 )= log x^8^ + log y^4 = 8log x +4 log y
The domain is x can be any real number that is greater than 0 (or x > 0).
The range is f(x) can be any real number.
- (4 points) Solve the following equation for x algebraically. Round answer to nearest hundred-thousandth.
log 12 27.63 = x
We’ll first rewrite this equation in exponential form, then solve.
( ) (^ )
log 12 27.
log 12 log 27.
log 12 log 27.
log 12 log 27.
log 12 log 12
x x
x
x
x
x
- (4 points) Page 393 Problem 20. 0 0.
0
Since the problem tells us that the population in the year 2000 was 250000, we'll use this as our starting point (that is, t is number of years past 2000 and P 250000). So our expression i
y = P e^ t
s now
To determine the year in which population reaches 350,000, we solve the equation 250000 350000
Rewrite this expression in log form. Sin
t
t
t
t
y e
e
e
e
ce the base is e, we'll use log base e (natural logs). ln1.4 0. ln1.4 0. 0.03 0.
Since t is number of years after 2000, then the year is 2000 + 11.2157 = 2011.2157 or sometime during 2012.
t t
t
- Extra Credit (10 points) pg 323 problem 28 Show all work. SIMPLEX Method
See page 313 Use Simplex Method to Maximize f = 5 x + 30 y subject to
2 10 96 10 90
x y x y
Since 2 x + 10 y ≤ 96 , then there is some quantity (call it s 1 (a slack variable)) such that 2 x + 10 y + s 1 = 96. Similarly, since x + 10 y ≤ 90 , then there is some quantity (call it s 2 (a second slack variable)) such that x + 10 y + s 2 = 90.
We want to maximize f = 5 x + 30 y. We’ll get all the variables to one side of the equals sign. So we subtract 5x and 30y from both sides we get: − 5 x − 30 y + f = 0
We have a system of equations. 2 x + 10 y + s 1 = 96 x + 10 y + s 2 = 90 − 5 x − 30 y + f = 0
We can now lay out our simplex matrix.
x y 1 2 f 2 10 1 0 0 96 1 10 0 1 0 90 This matrix is our simplex tableau. 5 30 0 0 1 0
3a. Determine the pivot column. The pivot column has the most negative number in the last row. Our
s s ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢⎣ (^) − − ⎥⎦
pivot column is the second column.
3b. Determine the pivot row. Perform the two divisions and select the smallest answer: Row 1 quotient: 96 9.6; Row 2 quotient: 90 9 10 10 The Row 2 quotient is smalle
r than the Row 1 quotient. So Row 2 is our pivot row.
Our pivot entry is the number is the 2nd row, 2nd column of our simplex matrix. 10 is our pivot entry.
