Presentation on theme: "The Natural Exponential Function (4.3) An elaboration on compounded growth."— Presentation transcript:
The Natural Exponential Function (4.3) An elaboration on compounded growth
A POD to warm up Find the amount of money you’d have at the end of one year in an account if you invested $1000 at 6% interest, compounded annually monthly weekly hourly by the minute
A POD to warm up Find the amount of money you’d have in an account if you invested $1000 at 6% interest, compounded annually monthly weekly hourly by the minute Notice the limit to growth as we approach continuous compounding.
Consider another limit What is the value of if n=1 if n=10 if n=100 if n=1000 if n=100,000 if n=1,000,000
Consider another limit, As n approaches infinity, the value of the expression approaches a limit. That limit is defined as e, the base for the natural exponential and logarithmic functions. e is an irrational, transcendental number, much like pi. It’s numerical value can be rounded to 2.71828…
Consider another limit In other words: e and the natural exponential function is f(x)=e x. Graph y=e x on your calculators. How is it similar to the graph of y=3 x ?
Continuous compounding There is such a thing as continuous compounding; its formula looks somewhat similar to the formula we used for compounded interest.
Continuous compounding Let’s derive the formula. Start with our discrete compounding formula Then, a little deliberate manipulation through replacement. Let 1/k = r/n. Then n=kr. And if we substitute, we get if k is infinitely large. This is the formula we use if we have continuous compounding.
Continuous compounding Find the amount of money you’d have in the POD with continuous compounding. How much would you have if you’d started with $10,000?
Natural exponential function Just as with other bases of exponential functions, we can solve for the exponent as long as the bases are the same. Solve for x: e 3x =e 2x-1.
Natural exponential function We can also use it in applications (word problems). In 1978, the population of blue whales in the southern hemisphere was thought to number 5000. With whaling outlawed and assuming an abundant food supply, the population N(t) is expected to grow exponentially according to the formula N(t)=5000e.0036t. where t is in years, and t=0 corresponds to 1978. Why would t=0 correspond to 1978? Predict the population in the year 2010.