MATHEMATICS - Sample Test 2 - with solutions 1. Calculate the area of the trangle with vertices A(1, 1, 0), B(0, 1, 0), C(0, 0, 1).
2. Calculate the derivative of the following function:
f (x) = e2x x2 3. Calculate the derivative of the following function:
f (x) =p
sin(ln(2x)) SOLUTIONS 1. Find the vectors:
AB = [−1, 0, 0] ~~ AC = [−1, −1, 1].
Their cross product:
AB × ~~ AC =
~i ~j ~k
−1 0 0
−1 −1 1
= 0 ·~i + 1 · ~j + 1 · ~k = [0, 1, 1].
The length of the obtained vector:
| ~AB × ~AC| =√
02+ 12+ 12 =√ 2.
The area of the triangle ABC:
Area = 1
2| ~AB × ~AC| =√
02+ 12+ 12 =
√2 2 . 2. Use the quotient rule
f0(x) = e2x x2
0
= (e2x)0· x2− e2x· (x2)0
(x2)2 = 2e2x· x2− e2x· 2x
x4 .
3. Use three times the chain rule f0(x) = (p
sin(ln(2x)))0 = 1
2psin(ln(2x)) · (sin(ln(2x)))0 =
= 1
2psin(ln(2x)) · cos(ln(2x)) · (ln(2x))0 =
= 1
2psin(ln(2x)) · cos(ln(2x)) 1
2x· (2x)0 = 1
2psin(ln(2x)) · cos(ln(2x)) 1 2x · 2.
