Science:Math Exam Resources/Courses/MATH104/December 2014/Question 01 (j)
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Question 01 (j) 

An investment earns at an annual interest rate compounded continuously. How fast is the investment growing when its value is ? 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

First, set up the model that describes the value of the investment as a function of time in years. 
Hint 2 

Recognize P as satisfying . 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 1 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. Since the annual interest rate is compounded continuously, the model that we use is , where is the value of the investment after time in years, and is the initial amount invested. We want to find when . Taking the derivative of yields and solving for the time when we find that
Putting those two results together we obtain

Solution 2 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. Since the annual interest rate is compounded continuously, the model that we use is , where is the value of the investment after time in years, and is the initial amount invested. We want to find when . Take of both sides in the model and then differentiate with respect to :
When the investment is , . 