This document provides a tutorial on calculus that includes 4 problems: 1) Converting improper rational functions to proper rational functions. 2) Simplifying expressions with exponents, logarithms, and natural logarithms. 3) Simplifying expressions with common and natural logarithms. 4) Solving equations with exponents, logarithms, and natural logarithms.

1. PMB 3004: Calculus
Tutorial 2
1. Change the following improper rational functions to the proper rational functions:
a) ( x3 + 5x2 – x – 6) / (x-1)
b) (4x3 – 2x2 - x + 3 ) / x -2
c) (4x3 + 6x2 + 5x +3) / (2x-1)
d) (x3 + 5x2 - 2x -1) / (x+3)
2. Simplify the following expressions:
a )2e3 3e 7
d) e2 x (e2 x  e x  1)  e x (1  e x )
e 1.5
b) e2 x  e x
5e1/ 2 e) x
 1  ex
e
4e3 3e 1 f) e2 x  (1  e x )2
c)
2 e 3 e 2
3. Simplify the following expressions:
a ) ln x  2 ln x  3ln x
e) 4log rs 2 t  2log r 2 st 2
1
b) ln t  ln t 1 x2 1 x3
2 f) log 2  log 6
2 y 3 y
c) log a 100  log a 10  log a 5
10 ln x  ln x  ln x 3
d)
ln x 9  ln x
4. Solve the following equations:

2. a) 25x  1253 h) log10 4 x 2  2
b) x 5  32 i) 2ln(3t  5)  4
1
c) 2 x 1  j) log x  log 2  1
16
k) 3ln(e3 x 4 )  9
d) e ln 2 x  4
e) e2 x  3e x l) 2log(x  2)  log(2x  5)
f)
1
2 m) log(x 2  6)  log(x  1)  1
e 1
x
n) eln x  2x  3
g) 10 x 6  30
PMB/ NS/Jan 09