Exercise Lecture of Mathematics
November 22, 2016
Exercise 1. Compute the derivatives of the following functions, specifying their domain
and that of their derivative.
1) f (x) = x6 − 3x4 − 6x3 + 7x −
5
2
2) f (x) = 4 −
2
5
+ 2
x
x
4) f (x) = sin x + cos x
5) f (x) = sin(4x + 7)
7) f (x) = (5x4 − πx − 1)3
8) f (x) = x2 |x|
2x
10) f (x) = 5e−x
11) f (x) = e x+1
1
cos x
q
p
3
16) f (x) = 1 + x2 − 1
14) f (x) = arctan(x2 )
13) f (x) = tan x +
19) f (x) = x arcsin x +
22) f (x) =
√
2+x
3 − x2
√
x
6) f (x) =
x
√
e
9) f (x) =
2+π
3) f (x) =
17) f (x) = log
p
1 − x2
x − 3 arccos
√
x
20) f (x) = arctan(sin x)
23) f (x) =
25) f (x) = log log(log x)
x2 − 2
x2 + x
sin(x − 5)
(x + 1)2
26) f (x) = log | log(sin x)|
12) f (x) = log2 (3x − 8)
√
5
1 + x3
√
15) f (x) =
5 3
1− x
2
18) f (x) = log x3
21) f (x) = cos earctan x
24) f (x) = sin x−x
27) f (x) = (sin x)cos x
Exercise 2. Write the equation of the tangent to the graph of the function f at the point
whose abscissa is given.
Function
x + e3x
Abscissa of the point
x = log 2
xex
ex − e−x
2
x=0
2
x=1
cos(x3 + π)
p
x= 3 π
2
sin
5
3x +
π
2
x=0
Exercise 3. For each of the following functions, determine the domain, possible symmetries, the sign, possible intersections with the axes, the limits at the extreme points of the
domain. Finally, represent graphically the information obtained.
2
1
x − 10x + 16
1) f (x) = log 2 +
2) f (x) = exp
x
x3 − 8
s
x4 − 1
x3 − 27
3) f (x) =
4)
f
(x)
=
arctan
x2 − 4
x2 + 3x + 2
(
x
log(−x − 1) if x < 0,
5) f (x) = arcsin
6) f (x) = √
x − 1 − 1 if x ≥ 0
x+1
1