Section 1 5 Exponential functions An exponential function is of the form f x a x for a positive number a If a 1 f x a x approaches 0 as x approaches negative infinity and grows rapidly upward as x approaches positive infinity Graph 2 x in your calculator Graph 1 2 x and see that the graph is a reflection across the y axis of 2 x This is because 1 2 x 2 x Exponent rules you should know are a x y x a a y a x 1 x a 1 a x a x y a x y When the exponent is a rational number such as 1 4 we write a 1 4 a 3 4 4 a 3 4 Example 8 2 3 3 a 3 8 2 3 64 4 or 8 2 3 x 4 2 Simplify the expression x 4 2 1 4 x 2 2 2 x 1 2 2 x 2 1 4 x 2 2 2 x x 2 2 x 2 2x 2 3x 2 8 3 2 2 2 4 4 a the 4th root of a Also Some applications of exponential functions are exponential growth and decay Example The time required to eliminate half of the amount of a certain drug in the body is 4 days If 15 mg are present how much is left in the body t days later t Amount left in mg 4 15 1 2 8 15 1 2 1 2 12 15 1 2 1 2 1 2 and so on The factor 1 2 is occurs t 4 times A t 15 1 2 t 4 t In general A t A0 1 2 half life for exponential decay Next is an example of exponential growth Example A culture weighs 2mg and the weight triples every 45 minutes Find the amount present after t hours What is the weight after 3 hours Solution The tripling time is 45 minutes 45 60 hour 3 4 hour A t 2 3 t 3 4 2 3 4t 3 A 3 2 3 4 162 mg Compound interest If P dollars is invested at annual interest rate r compounded annually then the accumulated amount at the end of t years is A t P 1 r t is an exponential function of t If the interest is compounded m times per year then r A t P 1 m mt We can imagine that money can flow like a liquid If interest is compounded infinitely often so the interest continuously flows into the account what is the accumulated amount For this we need the number e The number e Graph 1 Y 1 1 x x in windows where xmax is very large You will see that the graph approaches a horizontal line The y value on this line is the number e e 2 71828 with never ending non repeating decimals Algebra then can show that r 1 x 1 1 x r x x r r must approach e r as x approaches infinity Back to continuous compound interest if P dollars is invested at annual interest rate r compounded continuously then A t Pe rt is the accumulated amount after t years When the rate of growth of a substance or population is proportional to the amount present the amount is given as an exponential function similar to this one Exponential functions grow rapidly eventually so that even if r is small the value will eventually exceed any given polynomial in t In fact for any positive r and any n t e n rt approaches 0 as t approaches positive infinity
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