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(a) How long will it lake an investment to double in value if the interest rate is $ 6\% $ compounded continuously? (b) What is the equivalent annual interest rate?

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01:30

Wen Zheng

Amrita Bhasin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 8

Exponential Growth and Decay

Derivatives

Differentiation

Missouri State University

University of Nottingham

Idaho State University

Boston College

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

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(a) How long will it take …

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How long does it take mon…

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Find the amount of time it…

Let's find the time it takes to double and investment when compounded continuously, if the interest rate is 6%. So we're going to translate that 2.6 Okay, if the amount doubles than if you start with a not you're going to end with two times a not so two times, two times a not equals a Not each of the 0.0 60 We're solving this for teeth. Let's divide both sides by a not and we have two equals e to the 20.0 60. Now let's take the natural log of both sides, and we have the natural log of two equals 20.0 60. And then we'll divide both sides by 0.6 and we get natural log of two divided by 20.6 and we're gonna proximate that in the calculator and we get about 11.55 years. So that's the amount of time it would take to double under those circumstances. Now, suppose instead of continuous compounding, you have the same amount of time for non continuous annual compounding. Let's find out the interest rate that would be giving us the same amount of money doubling the money, so we want our way to be doubled. We have two times a not, and we're going to use the same amount of time. So we're using approximately 11.55 years. Let's divide both sides of the equation by a not, and we get two equals one plus R to the 11.55 Now to get rid of that power, let's raise both sides to the 1/11 0.55 power, and then we'll subtract one from both sides. So R equals two to the 1/11 20.55 power minus one. And we can put that in the calculator and we get approximately 0.6185 We can convert that back to a percent, and it's approximately 6.185% to get the equivalent amount of money in that amount of time.

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