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

### mathematics

##### Description

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

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

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?

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.

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?