1. No category

Calculus 1 – Test 2 (Sample)
Date: December 10, 2013, Time: 12:00, Classroom: IK-102
Total points: 30 , time: 90 minutes. Answers with arguments or calculations are required.
Problems
1. Let
φ(x) = (x5 − 2x3 ) · 32x ,
g(x) =
p
x2 + 2
.
x4 + 4 and h(x) = log2 2
x +4
Determine φ0 (x) , g 0 (x) and h0 (x) (for all x ∈ R).
(2+3+3=8 points)
2. Investigate the function f defined by the formula
f (x) = x3 e−x
(x ∈ R).
Investigation involves the following tasks:
(a) Determine the limits limx→+∞ f (x) and limx→−∞ f (x) .
(b) Calculate f 0 (x) and f 00 (x) (for all x ∈ R).
(c) Considering subintervals according to the zeros of f 0 and f 00 , establish whether f is increasing or decreasing, convex or concave, and find
all points of inflection or local extremum (minimum or maximum).
(d) Sketch the graph of f .
(e) Determine the range Rf of f .
(3+2+6+2+2=15 points)
3. Determine
√
lim
x→1+
x−1
x−1
and
x sin x
x→0+ ln(1 + x2 )
lim
(3+4=7 points)