##### Description

Task 1

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