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1.9

Skip to 0 minutes and 12 secondsHello, and welcome back to "A step in practice". Today, we are dealing with money. Well, after all, the study of interest rates is one of the basic motivations to exponentials. We have 10,000 euros, so our initial capital, C of 10 000 euros, and we invested with an interest rate of 3% compounded every 6 months. So, after 6 months, we get the initial capital multiplied by 1 plus 1.5 divided by 100. So this is after 1 semester, that is 6 months. So, after 1 year, we've got 2 semesters, so our initial capital is multiplied by 1 plus 1.5 divided by 100 to the square. This is after 1 year.

Skip to 1 minute and 7 secondsAnd, after 15 years, there are exactly 30 semesters, so our capital is multiplied by 1 plus 1.5 divided by 100 to the 30. And this gives us, with C equal to 10 000 euros, we get, more or less, 15 000 euros, 630.8 euros. So this is exercise 1.

Skip to 1 minute and 53 secondsThe first interest rate is 5.05% per year, compounded every 6 months. So, after one year, we get that initial capital multiplied by 1 plus the 1/2 of 5.05 is 5.05 divided by 2 divided by 100 to the square, because there are 2 semesters in 1 year. And this gives, more or less, C multiplied by 1.05114. In the second case, the interest is compounded continuously. The interest of 5% per year is compounded continuously. This means, as we saw in Francis' talk, that the capital is multiplied by e to the 5%.

Skip to 3 minutes and 0 secondsAnd this is, more or less, C multiplied by 1.05127. So, if we compare the two numbers, while the first three digits, four digits are the same, and then we get something a little bit-- something greater in the second case. So this wins. It is better to have an interest of 5% compounded continuously. And this ends exercise 2.

# It's your turn on the Euler number

Do your best in trying to solve the following problems. In any case some of them are solved in the video and all of them are solved in the pdf file below.

### Exercise 1.

If 10  000 euros is invested at an interest rate of 3% per year, compounded semiannually (1.5% per semester). Find the value of the investment after 15 years.

### Exercise 2.

Which of the given interests rates and compounding periods would provide the better investment after 1 year?

1. $$5.05\%$$ per year, compounded semiannually.

2. $$5\%$$ per year, compounded continuously (so that the capital is multiplied by $$e^{5/100}$$ at the end of each year).

## Get a taste of this course

Find out what this course is like by previewing some of the course steps before you join:

video

Exponentials

• ##### The Euler number
video

The Euler number

• ##### Functions and their inverses
video

Functions and their inverses

• ##### Natural logarithms
video

Natural logarithms

• ##### Trigonometry by triangles
video

Trigonometry by triangles

• ##### Trigonometry by the unit circle
video

Trigonometry by the unit circle

• ##### Graphs of trigonometric functions
video

Graphs of trigonometric functions

• ##### Trigonometric identities
video

Trigonometric identities

• ##### Harmonic motion
video

Harmonic motion

• ##### The function arctan
video

The function arctan

• ##### More inverse trig functions, and a few others
video

More inverse trig functions, and a few others

• ##### Geometry of the plane: points, segments, lines
video

Geometry of the plane: points, segments, lines

• ##### Distance and circles
video

Distance and circles

video

Ellipses

• ##### Parabolas and hyperbolas
video

Parabolas and hyperbolas

video

• ##### Lengths, areas and volumes
video

Lengths, areas and volumes

• ##### Equations with exponentials and logarithms
video

Equations with exponentials and logarithms

• ##### Growth and decay problems
video

Growth and decay problems

• ##### Basic Trigonometric equation
video

Basic Trigonometric equation

• ##### Inequalities with logarithms and exponentials
video

Inequalities with logarithms and exponentials

• ##### Trigonometric inequalities
video

Trigonometric inequalities

• ##### From polar to cartesian coordinates
video

From polar to cartesian coordinates