# 8.1 MATHEMATICAL REVISION

1. Calculate:

 A) 104 84270 0.010.5 B) 52 + 42 22 × 25 C) log 1000 log 0.01 D) ln e ln e-5 eln e E)

2. Verify that

 a) a = eln a b) a = 10log 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) 10x = 5.5 c) y -a = b · ec · (x - b)

5. Calculate the derivatives of the following expressions:

 a) y = 13 g) y = 5x 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 = x2/7 j) y = ln(5x+4) p) y = (4x+3).(ex-4) e) y = x-3 k) y = 1/x f) y = e3.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 · e2 · x + 1 c) f(x) = x6 h) m) f(x) = x · ex d) f(x) = 1 = 3 · x i) f(x) = ex n) f(x) = ln x e) f(x) = 4 · x-3 j) f(x) = e0.2 · 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) = e3.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 ycumulative 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