Download the App!
Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.
Question
Answered step-by-step
If an initial amount $ A_0 $ of money is invested at an interest rate $ r $ compounded $ n $ times a year, the value of the investment after $ t $ years is$$ A = A_0 \left( 1 + \frac{r}{n} \right)^{nt} $$If we let $ n \to \infty $, we refer to the continuous compounding of interest. Use l'Hospital's Rule to show that if interest is compounded continuously, then the amount after $ t $ years is $$ A = A_oe^{rt} $$
Video Answer
Solved by verified expert
This problem has been solved!
Try Numerade free for 7 days
Like
Report
Official textbook answer
Video by Carson Merrill
Numerade Educator
This textbook answer is only visible when subscribed! Please subscribe to view the answer
03:27
Wen Zheng
01:12
Amrita Bhasin
Calculus 1 / AB
Calculus 2 / BC
Chapter 4
Applications of Differentiation
Section 4
Indeterminate Forms and l'Hospital's Rule
Derivatives
Differentiation
Volume
Missouri State University
University of Nottingham
Boston College
Lectures
04:35
In mathematics, the volume of a solid object is the amount of three-dimensional space enclosed by the boundaries of the object. The volume of a solid of revolution (such as a sphere or cylinder) is calculated by multiplying the area of the base by the height of the solid.
06:14
A review is a form of evaluation, analysis, and judgment of a body of work, such as a book, movie, album, play, software application, video game, or scientific research. Reviews may be used to assess the value of a resource, or to provide a summary of the content of the resource, or to judge the importance of the resource.
01:04
If an initial amount $A_{0…
03:07
02:51
If an initial amount A0 of…
02:19
Compound Interest The form…
06:26
The formula for the amoun…
04:59
Suppose you make a deposit…
04:12
01:37
The formula for the amount…
So here we have um an interest problem. So A equals A. Not Times one Plus R. Divided by and to the empty. Um So is constant. So we don't really care about it in terms of the limit, we can remove it when we take the limit as N. Goes to infinity. So based on this we can take the natural log of both sides to simplify things. When we rewrite it, we end up getting the natural log of one plus R over N divided by one over Mt. Then when we plug in and going to infinity we need to use towels real again. So we end up getting R. P. by one plus are over 10. So now when we plug an ongoing to infinity we get artie over one. So it's just going to be our t remember we took the natural log of both sides of the natural log of Y equals RT. Which means that why is going to equal E. To the R key? However, we did remove that A not So we have to put that back into the original equation, meaning that A. Is going to equal a. Not. E. To the RT.
View More Answers From This Book
Find Another Textbook
03:46
Problem 1. Use Newton's Law of Gravitation to compute the work W = 5F(r…
08:42
Exercise 1.2.1 Find the point (x1,Y1) which lies on both lines, x+3y = 1 and…
02:14
Exercise 1.2.10 Suppose system of equations has fewer equations than variab…
03:13
Exercise 2.1.1 For the following pairs of matrices, determine if the sum A+ …
04:50
By using the method of Lagrange multipliers, find the relative extrema of f(…
08:53
(1 point) Find the Taylor polynomial of degree n =4 for x near the point a =…
02:36
Construct the cumulative frequency distribution that corresponds to the give…
00:01
24) (6 marks) A couple finds out they're having twins| a) Find the sam…
01:09
It is the end of the budgeting period for John Smith,and he has $200 left in…
