Quiz 2 Fall 2006 | 2005 - 2007 | Www.ilearnmath.net
This document was uploaded by user and they confirmed that they have the permission to share it. If you are author or own the copyright of this book, please report to us by using this DMCA report form. Report DMCA
Overview
Download & View Quiz 2 Fall 2006 | 2005 - 2007 | Www.ilearnmath.net as PDF for free.
More details
- Words: 539
- Pages: 2
AP Calculus :: 2006-2007 :: Shubleka
Name______________________________ Page 1
Show all your work for full credit! Problem 1 (10 points) i)
Use the natural logarithm to eliminate the quotients and exponents in the following equation: 3 x 12 x 3 y 1 7 x 2 x 3 3 4
1
ii)
3
3
2
Solve the following equation for x : 1
log 3 ( x 4 ) log 3 ( x 3 ) 2 log 3 ( x 2 ) 5 Problem 2 (10 points) Definition: A real-valued function f (x) is a one-to-one function if and only for any two values x1 , x2 in the domain, the following is true: f ( x1 ) f ( x2 ) x1 x2 i)
Show that f ( x) x 3 is a one-to-one function.
ii)
True or False? All one-to-one functions are invertible (i.e. their inverses are also functions). If true, justify your answer with an informal (or formal!) argument. If false, provide a counterexample.
Problem 3 (20 points) i)
Determine the general solution for the doubling time x D and the half-life xH for the function
y y 0 a x . Comment on the restrictions on a in each case. ii)
Determine the general solution for the doubling time x D and the half-life xH for the function
y y0 e kx . iii)
The half-life of radium-226 is 1620 years. Write a formula for the quantity, Q , left after t years, if the initial quantity is Q0 .
iv)
What percentage of the original amount of radium is left after 500 years?
Problem 4 (10 points) True of False? Briefly justify your answer in each case. No formal proofs are required. 1
i)
If f (x) is strictly increasing, then f
( x) is also strictly increasing.
ii)
If a function is odd, then it does not have an inverse function.
iii)
The function f ( x) | sin x | is even.
A mathematician's reputation rests on the number of bad proofs he has given. Besicovitch, A.S.
AP Calculus :: 2006-2007 :: Shubleka
Name______________________________
iv)
If f (x) is strictly increasing and g(x) is strictly increasing, then f ( x) g( x) is decreasing for all x values.
v)
If f (x) is strictly increasing and g(x) is strictly increasing, then f ( x) g ( x) is decreasing for all x values.
vi)
If f (x) is strictly increasing and g(x) is strictly increasing, then f ( x) g ( x) is decreasing for all x values.
vii)
If f (x) is strictly increasing and g(x) is strictly increasing, then f g (x) is increasing for all x values.
viii)
If g(x) is an even function, then f ( g ( x)) is decreasing always even.
ix)
The inverse of y log x is given by y
x)
An exponential function can be decreasing.
1 log x
.
“I hereb y agree th at I w ill not discuss or disclose the content of this quiz with anyone other than Mr. Shubleka until 10/3/2006.” __ ____ ____ ___ ____ ___ __ (Sign ature)
Page 2
Name______________________________ Page 1
Show all your work for full credit! Problem 1 (10 points) i)
Use the natural logarithm to eliminate the quotients and exponents in the following equation: 3 x 12 x 3 y 1 7 x 2 x 3 3 4
1
ii)
3
3
2
Solve the following equation for x : 1
log 3 ( x 4 ) log 3 ( x 3 ) 2 log 3 ( x 2 ) 5 Problem 2 (10 points) Definition: A real-valued function f (x) is a one-to-one function if and only for any two values x1 , x2 in the domain, the following is true: f ( x1 ) f ( x2 ) x1 x2 i)
Show that f ( x) x 3 is a one-to-one function.
ii)
True or False? All one-to-one functions are invertible (i.e. their inverses are also functions). If true, justify your answer with an informal (or formal!) argument. If false, provide a counterexample.
Problem 3 (20 points) i)
Determine the general solution for the doubling time x D and the half-life xH for the function
y y 0 a x . Comment on the restrictions on a in each case. ii)
Determine the general solution for the doubling time x D and the half-life xH for the function
y y0 e kx . iii)
The half-life of radium-226 is 1620 years. Write a formula for the quantity, Q , left after t years, if the initial quantity is Q0 .
iv)
What percentage of the original amount of radium is left after 500 years?
Problem 4 (10 points) True of False? Briefly justify your answer in each case. No formal proofs are required. 1
i)
If f (x) is strictly increasing, then f
( x) is also strictly increasing.
ii)
If a function is odd, then it does not have an inverse function.
iii)
The function f ( x) | sin x | is even.
A mathematician's reputation rests on the number of bad proofs he has given. Besicovitch, A.S.
AP Calculus :: 2006-2007 :: Shubleka
Name______________________________
iv)
If f (x) is strictly increasing and g(x) is strictly increasing, then f ( x) g( x) is decreasing for all x values.
v)
If f (x) is strictly increasing and g(x) is strictly increasing, then f ( x) g ( x) is decreasing for all x values.
vi)
If f (x) is strictly increasing and g(x) is strictly increasing, then f ( x) g ( x) is decreasing for all x values.
vii)
If f (x) is strictly increasing and g(x) is strictly increasing, then f g (x) is increasing for all x values.
viii)
If g(x) is an even function, then f ( g ( x)) is decreasing always even.
ix)
The inverse of y log x is given by y
x)
An exponential function can be decreasing.
1 log x
.
“I hereb y agree th at I w ill not discuss or disclose the content of this quiz with anyone other than Mr. Shubleka until 10/3/2006.” __ ____ ____ ___ ____ ___ __ (Sign ature)
Page 2
