Download 5.3 B Notes

5.3 "The Natural Exponential Function"
(ex) Let's say you invest $10,000 @ 10% APR for 1 year.
Compound it quarterly:
monthly:
weekly:
daily:
hourly:
minutely:
often used equation in calculus
If p=1, r=1, t=1:
e
2.71828 (as n )
natural exponential function
very useful function in advanced math
f(x) = ex
Continuously Compounded
Interest Formula
A = Pert
* Try our 1st example: P=$10,000, r=10%, t=1
(ex) $20,000 @ 9% for 10 yrs compounded continuously
(ex) $10,000 grew to $26,116.96 in 12 years. If it was compounded continuously,
find the interest rate.
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Use the graph of y = ex to help sketch the graph of f.
(ex) a) f(x) = e2x , b) f(x) = 2ex (ex) a) f(x) = e­2x , b) f(x) = ­2ex y=ex
y=ex
(ex) Solve (ex) Find the zeros of f(x) = ­x2e­x + 2xe­x
(ex) Find the zeros of f(x) = x2(2e2x) + 2xe2x + e2x +2xe2x
(ex) Simplify the expression:
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Crop Growth. An exponential function W such that W(t) = W0ekt for k>0 describes the first month of growth for crops. The function value W(t) is the total weight in mg, W0 is the weight on the day of emergence, and t is the time in days. If, for a species of cotton, k=0.21 and the weight after 10 days is 575 mg, estimate W0.
Population Growth in India. The 1985 population estimate for India was 766 million, and the population has been growing continuously at a rate of about 1.82% per year. Assuming that this rapid growth rate continues, estimate the population N(t) of India in the year 2015.
Law of Growth (or Decay)
Formula:
q(t) = q0ert
The radioactive tracer 51Cr can be used to locate the position of the placenta in a pregnant woman. Often the tracer must be ordered from a medical laboratory. If A0 units (microcuries) are shipped, then because of the radioactive decay, the number of units A(t) present after t days is given by A(t) = A0e­0.0249t .
a) If 35 units are shipped and it takes
2 days for the tracer to arrive, approximately how many units will
be available for the test?
b) If 35 units are needed for the test, approximately how many
units should be shipped?
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Halibut Growth. The length (in centimeters) of many common commercial fish t years old can be approximated by a von Bertalanffy growth function having an equation of the form f(t) = a(1 ­ be­kt), where a, b, and k are constants.
a) For Pacific halibut, a=200, b=0.956, and k=0.18. Estimate the length of a 10­year­old halibut.
b) Use the graph of f to estimate the
maximum attainable length of the
Pacific halibut.
Polonium isotope decay. If we start with c milligrams of the polonium isotope 210
Po, the amount remaining after t days may be approximated by A = ce­0.00495t . If the initial amount is 50 mg, approximate, to the nearest hundredth, the amount remaining after:
a) 30 days
b) 180 days
c) 365 days
Land Value. In 1867 the United States purchased Alaska from Russia for $7,200,000. There is 586,400 square miles of land in Alaska. Assuming that the value of the land increases continuously at 3% per year and that land can be purchased at an equivalent price, determine the price of 1 acre in the year 2010. (One square mile is equivalent to 640 acres)
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