Task 1

1.
a) Derive the function h(?)
= ln(4? − 7).

2.
b) Derive the function ?(?)
= ?? − ?
+ ?.

Task 2

Anne will save an annual amount of NOK
15,000 with the first payment on January 1, 2019, and the last payment on
January 1, 2045, a total of 27 payments. We assume that the interest rate will
be 3% per annum throughout this period.

a) What amount did Anne save on January 1,
2046 one year after the last payment?

b) The saved amount will be used by Anne
for an additional pension, which she will take out with a fixed amount in the
year over 10 years, from 1 January 2046 to 1 January 2055. If

the interest rate is still 3% a year, how
big will the annual payment be?

Task 3

Let the function ?(?)
= ?4 − 4?2 be defined for all
real numbers ?.

a) Find the zero points of ? (?).

b) Determine where ? (?) is growing and where ? (?) is
decreasing. Find the extreme points and the extreme values of ? (?).

c) Find the turning points of ? (?). Where is ? (?) convex?

d) Sketch the graph for ? (?).

e) Calculate the integral ∫2 and
give an interpretation of the answer.

Task 4

We have given the function ? (?, ?) = ?2 −?? − ?.

a) Find the partial derivatives of ? (?, ?) of the
first and second order.

b) The function has one stationary point.
Find it.

c) What does the second derivative test
say about the stationary point?

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