1. Calculate:

A) 
10^{4} 
8427^{0} 
0.01^{0.5} 









B) 
5^{2} + 4^{2} 
2^{2} × 2^{5} 









C) 
log 1000 
log 0.01 










D) 
ln e 
ln e^{5} 
e^{ln e} 












E) 
2. Verify that
a) a = e^{ln a} 
b) a = 10^{log a} 
c) for 0.01 < x < +0.01 
d) for 0.5 < x < +0.5 
3. Solve the following expressions applying natural logarithms to both members of the equality:
a) y = a · x^{5} 
b) y = a · e^{b · (x + 2 · c)} 
c) y a = b · e^{c · (x  b)} 
Note: a, b, e c are constants; e is the basis of natural logarithms (e = 2.7183...); x and y are variables.
4. Determine the value of x in the following expressions:
a) e^{x} = 5.2 
b) 10^{x} = 5.5 
c) y a = b · e^{c · (x  b)} 
5. Calculate the derivatives of the following expressions:
a) y = 13 
g) y = 5^{x} 
m) y = (4+2x)^{3} 
b) y = 38x 
h) y = e^{3.x} 
n) y = (x6)^{2} 
c) y = x^{5} 
i) y = ln x 
o) y = a.(3e^{b.x})^{3} 
d) y = x^{2/7} 
j) y = ln(5x+4) 
p) y = (4x+3).(e^{x}4) 
e) y = x^{3} 
k) y = 1/x 

f) y = e^{3.x} 
l) y = (2+4x)/(3x) 

6. Calculate the indefinite integrals of the following functions:

a) f(x) = 0 
f) 
k) f(x) = e^{0.5 · x} 

b) f(x) = 5.34 
g) 
l) f(x) =3 · e^{2 · x + 1} 

c) f(x) = x^{6} 
h) 
m) f(x) = x · e^{x} 

d) f(x) = 1 = 3 · x 
i) f(x) = e^{x} 
n) f(x) = ln x 

e) f(x) = 4 · x^{3} 
j) f(x) = e^{0.2 · x} 
o) f(x) = x · ln x 
7. Calculate the area under the function
a) f(x) = 2 + 5x between x = 1 and x = 4
b) f(x) = e^{3.x} between x = 0 and x = 1
c) between and
d) f(x) = 1 + 3x between x = 2 and x = 2
8. Calculate the value of y_{cumulative} with
a) y = e^{2x} between x = 0 and x = 0.8
b) between x = 0 and x = 2
c) f(x) = 2.x^{3} between x = 0 and x = 1
9. Calculate the Mean Value of y with
a) y = 3 · e^{7x} between x = 0 and x = 1
b) y = 4 · (1  e^{0.2x}) between x = 1 and x = 3
c) y = 2  x between x = 0 and x = 1.2
10. Calculate the integral of
a) f(x) = 2 · e^{0.5x} with the initial condition x=1 ⇒ F(x) = 4 where
b) with the initial condition F(1) = 2
c) with the initial condition x = 0 ⇒ y = 10
d) with the initial condition x = 0 ⇒ y = 0