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(a) If 1000 is borrowed at $ 8\% $ interest, find the amounts due at the end of 3 years if the interest is compounded (i) annually, (ii) quarterly, (iii) monthly, (iv) weekly, (v) daily, (vi) hourly, and (vii) continuously. (b) Suppose 1000 is borrowed and the interest is compounded continuously. If $ A(t) $ is the amount due after $ t $ years, where $ 0 \le t \le 3, $ graph $ A(t) $ for each of the interest rates $ 6\%, 8\%, $ and $ 10\% $ on a common screen.
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03:33
Wen Zheng
Calculus 1 / AB
Chapter 3
Differentiation Rules
Section 8
Exponential Growth and Decay
Derivatives
Differentiation
Campbell University
University of Nottingham
Idaho State University
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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here we have a compound interest problem, and we're going to use our non continuously compound interest formula for the majority of part A. The amount invested is $1000 for a time period of three years, with an interest rate of 8% and that translates into a decimal value of 0.8 And so we're going to go through several different amounts of calm poundings per year, starting with annually once per year. We use an equals one, and we get $1259.71. Then, for quarterly, we use an equals four. We substitute that into our formula. We get $1268.24 for monthly and equals 12 and that gives us $1270.24 for weekly and equals 52. That gives us $1271.1. We see the amount of money is growing, but the rate of growth is slowing as we accumulate more calm poundings for 365 compound ings per year, that would be daily. We get $1271.22 an hourly that would be 8760 compound ings per year. We get $1271.25 and for the last part, we're going to do continuous compounding, so we use a different formula. The continuous compounding formula is a equals a not e to the R. T. So we still have the same interest rate point await. We still have the same amount of time three years, and we still have the same initial investment, and we end up with $1271.25. Finally, for Part B, we want to graph the continuously compounded function for different amounts of interest rates. So 6%. Let's try that again. We have 6% we have 8% and we have 10%. So let's grab a graphing calculator. We type those in, and then we're going to look at them from time zero to time three so we can go to window and we can change our window dimensions to go from 0 to 3 on the X axis. And then I chose negative 100 to 1600 for my Y axis and then looking at the graphs. The lowest one logically is the one that has a 6% interest rate, and the middle one is the one with 8% interest rate, and the highest one is the one with a 10% interest rate.
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