x - E-SGH

Mathematics - list 3
1. Compute derivatives
√
2
(a) x3 −3x2 +2x−2 (b) 2 3 x−x 5 (c)
√
x3 +2x x
x2
(f) (x2 + 3) cos x (g) (2ex + x2 )(ln x12 + x) (h)
(j)
3x2 +2x−1
4x2 +5
(o)
q
(k)
√
x
x−1
(l)
cos x
1−sin x
2. Compute limits
2
(a) lim x +3x−1
(b) lim
ex
xα
x
x→∞ e
x→∞
(d) lim
x ln x
x→∞ x+ln x
x→0
(m) (2x4 + 3x2 − 1)10 (n)
(f) lim
x→0
ex −1−x
x2
(k) lim
1−x+ln x
x→1 1+cos πx
1−2 sin x
cos 3x
(n) lim+ x ln x (o) lim+ xe x
3x2 + 2x − 2
3e2x +e−x
ex +1
(g) lim
(h) lim
x5 −1
4
x→1 x −1
ln x
3
x→1 x −1
(l) lim
(m) lim
e2x −1
x→0 ln(1+2x)
x→0
tan x−sin x
x−sin x
(p) lim+ (arctan x − π2 ) ln x (r) lim (x − ln x)
1
x→0
√
ln x
α
x→∞ x
ln ln x
x→∞ ln x
x→ 6
(e) 2ex +3 ln x−sin x
, where α ∈ R is a constant (c) lim
(e) lim
(j) limπ
ex −e−x
x
√
x−
x
√
3 x
(i) (ex − 1) arctan x
ln x
x
(p) e2 sin(3x−1) (r) ln(x + arctan x) (s)
x2 +1
x2 +2
(i) lim
(d)
x→0
x→∞
x→0
(s) lim+ (ln(x3 − 1) − ln(x2 − 1)) (t) lim+ ( x1 −
x→1
x→0
1
ex −1 )
(u) limπ − ( cos1 x − tan x)
x→ 2
1
(v) lim+ ( x−1
− ln1x ) (w) lim+ xx (x) lim+ (tan x)sin x (y) lim x 1−x
1
x→0
x→0
x→∞
x→0
(z) lim (1 + sin x1 )2x
x→∞
3. Determine points in R at which the following functions are dierentiable. Give the
equation of the tangent line to the graph of these functions at a chosen point.
√
(a) | x − 1|
(b) f (x) = 2x−4
x−1
(c)
(
f (x) =
(d) f (x) = |ex
2
(e)
−4
√
2
e x +1
e cos(ln(x + 1))
for x ≤ 0
for x > 0
− 1|
(
x2 sin x1
f (x) =
0
(f)
(
f (x) =
x arctan x1
0
1
for x 6= 0
for x = 0
for x 6= 0
for x = 0
4. Find parameters a, b, c, d ∈ R for which the functions are dierentiable in their
domain
(a)


for x ≤ 0
ax + b
2
f (x) = cx + dx for x ∈ (0, 1]


x−
(b)
f (x) =
1
x
x>1
(
ax + b
ln(x+1)
x
for x ≤ 0
for x > 0
5. Find the image or the inverse image of the function:
(a) f (x) = 2x3 + 3x2 − 12x + 6, the image f ([−1, 2]) and the inverse image
f −1 ([−∞, 6]).
(b) f (x) = x2 e−x , the image f ([−1, 3]).
(c) f (x) =
x2 +3
x
− 2 ln x, the image f ([1, 4]).
6. Find the absolute minimum and maximum of the following functions in the interval
I . How many zeros do these functions have in I ?
2
(a) f (x) = x3 − 15
2 x + 12x + 7, I = [2, 5].
(b) f (x) = |ex −4 − 1| − 1, I = [−1, 3].
(c) f (x) = |2x3 − 3x2 + 5| − 1, I = [−2, 12 ].
2
7. Find local extremes of the function f (x) = (x2 − 2x) ln x − 32 x2 + 4x.
8. Sketch the graphs of the following functions.
2
x
(a) 3x4 − 4x3 − 12x2 + 1 (b) 1+x
(c) e−x (d)
2
(f) ln2 x − ln(x2 ) (g) xe x
(h) |x|e−x
(l)
x3
(x−1)2
1
x2 −x−4
x−1
(m) x ln x (n)
2
(i)
e−x
x2 −1
ln
√x
x
(e) x2 e−x
(j) ln x + ln1x (k)
x2 +x+1
x2 −1
9. Find asymptotes of the functions
2x
(a) x − 2 arctan x (b) x ln( x1 + e) (c) x arctan x (d) x ln x−2
(e)
10. Check continuity of the functions and determine their asymptotes
(a)
( 3
x +2
for x ≤ 1
2 +1
f (x) = 2x
ln x
for x > 1
x−1
(b)
(
f (x) =
2x2 +3
x−2
x
ex −1
2
for x ≤ 0
for x > 0
2xex +ln(x+1)
ex −1
x
11. Prove the inequality ln(x + 1) ≥ x+1
for x > −1. How many solutions does have
x
the equation ln(x + 1) = x+1 + 7 for x > −1?
12. Show that arctan x1 + arctan x =
π
2
for x > 0
13. How many solutions in R does have the equation ex =
3
2−x
x+1
?