8.1 MATHEMATICAL REVISION

1. Calculate:

A)	10⁴		8427⁰	0.01^0.5

B)			5² + 4²	2² × 2⁵

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 · x⁵

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 = 3-8x	h) y = e^-3.x	n) y = (x-6)²
c) y = x⁵	i) y = ln x	o) y = a.(3-e^-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)/(3-x)

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) = x⁶	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.x³ 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