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Feb 11, 2023 •159 likes •357 views

Section 4.1 d i E Exponential Functions a l l u d Dr. Abdulla Eid b A College of Science . r D MATHS 103: Mathematics for Business I Dr. Abdulla Eid (University of Bahrain) Exponential 1 / 18 Topics d i E a 1 Exponential

  1. Section 4.1 d i E Exponential Functions a l l u d Dr. Abdulla Eid b A College of Science . r D MATHS 103: Mathematics for Business I Dr. Abdulla Eid (University of Bahrain) Exponential 1 / 18

  2. Topics d i E a 1 Exponential Function, graph, and its properties. l l u 2 Compound Interest (Application to Finance). d b 3 The Euler Number e . A . r D Dr. Abdulla Eid (University of Bahrain) Exponential 2 / 18

  3. 1- The Exponential Function Definition The function f ( x ) = a x , a > 0, a � = 1 d i E is called an exponential function . The number a is called the base and x is called the exponent ( power ). a l l u Recall: Rule for the exponent d b 1 a x · a y = a x + y . A a x a y = a x − y . 2 . r 3 ( a x ) y = a xy . D 4 ( ab ) x = a x b y . 5 a − x = 1 a x . 6 a 0 = 1. 7 a 1 = a . Dr. Abdulla Eid (University of Bahrain) Exponential 3 / 18

  4. Graphing Exponential Function with a > 1 Example Graph the function f ( x ) = 3 x d i E Solution:Using the calculator, we fill the following table a l l u x − 2 − 1 0 1 2 3 d y b A Domain = ( − ∞ , ∞ ) . . r D Co–domain= ( − ∞ , ∞ ) . Range= ( 0, ∞ ) . y –intercept = ( 0, 1 ) . x –intercept = none. Dr. Abdulla Eid (University of Bahrain) Exponential 4 / 18

  5. Exercise Graph f ( x ) = 7 x and observe the difference with the previous example. d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Exponential 5 / 18

  6. Graphing Exponential Function with 0 < a < 1 Example Graph the function � 1 � x d f ( x ) = 3 i E a Solution:Using the calculator, we fill the following table l l u d x − 2 − 1 0 1 2 3 b y A . r Domain = ( − ∞ , ∞ ) . D Co–domain= ( − ∞ , ∞ ) . Range= ( 0, ∞ ) . y –intercept = ( 0, 1 ) . x –intercept = none. Dr. Abdulla Eid (University of Bahrain) Exponential 6 / 18

  7. Exercise � 1 � x and observe the difference with the previous example. Graph f ( x ) = 7 d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Exponential 7 / 18

  8. Summary d i y = f ( x ) = a x E a l Domain = ( − ∞ , ∞ ) . l u d Co–domain= ( − ∞ , ∞ ) . b A Range= ( 0, ∞ ) . y –intercept = ( 0, 1 ) . . r D x –intercept = none. Dr. Abdulla Eid (University of Bahrain) Exponential 8 / 18

  9. Graphing Exponential Function with a > 1 Example Graph the function f ( x ) = 3 x + 1 − 2 d i E Solution:Using the calculator, we fill the following table a l l u x − 3 − 2 − 1 0 1 2 3 d y b A Domain = ( − ∞ , ∞ ) . . r D Co–domain= ( − ∞ , ∞ ) . Range= ( − 2, ∞ ) . y –intercept = ( 0, 1 ) . x –intercept = later in Section 4.4. Dr. Abdulla Eid (University of Bahrain) Exponential 9 / 18

  10. 2 - Compound Interest Example Suppose you save 100 BD in a saving account that pays 1% annually. Find the total money in your account every year. d i Solution: Let A n be the amount in the account in year n , i.e., A 2 is the E amount in the account after 2 years. a l Year 0: A 0 = 100 BD. l u Year 1: A 1 = 100 + 100 ( 0.01 ) = 101 BD. d b Year 1: A 1 = A 0 + A 0 ( r ) = A 0 ( 1 + r ) . A Year 2: A 2 = 101 + 101 ( 0.01 ) = 102.1 BD. . Year 2: r D A 2 = A 1 + A 1 ( r ) = A 1 ( 1 + r ) = A 0 ( 1 + r )( 1 + r ) = A 0 ( 1 + r ) 2 . Year 3: A 3 = 102.1 + 102.1 ( 0.01 ) = 103.03 BD. Year 3: A 3 = A 2 + A 2 ( r ) = A 2 ( 1 + r ) = A 0 ( 1 + r ) 3 . Year n: A n = A n − 1 + A n − 1 ( r ) = A n − 1 ( 1 + r ) = A 0 ( 1 + r ) n . So in any year n , we have A n = P ( 1 + r ) n Dr. Abdulla Eid (University of Bahrain) Exponential 10 / 18 where P is the principal which is the initial money in the account.

  11. Example d i E Suppose you saved 3000 BD at 5% for 3 years. Find the compound a amount and the compound interest. l l u d Solution: n = 3, P = 3000, and r = 5% = 0.05. We have that b A 3 = P ( 1 + r ) n =3000 ( 1 + 0.05 ) 3 = 3472.875 BD. A The total interest is I = A 3 − P = 3472.875 − 3000 = 472.875 BD. . r D Dr. Abdulla Eid (University of Bahrain) Exponential 11 / 18

  12. The compound interest formula d i E In general, if the interest are given periodically (say m times a year), the a l formula is l u 1 + r � mn � A n = P d m b A . r D Dr. Abdulla Eid (University of Bahrain) Exponential 12 / 18

  13. Example Find the compund interest and the compound interest of (a) 500 BD for 7 years at 11% semi–annually d i Solution: n = 7, P = 500, r = 11% = 0.11, and m = 2. We have that E m ) nm =500 ( 1 + 0.11 2 ) 3 · 2 = 1658.04 BD. A 7 = P ( 1 + r a l The total interest is I = A 7 − P = 1658.04 − 500 = 1158.04 BD. l u d Example b A Find the compund interest and the compound interest of (b) 4000 BD for . 15 years at 8.5% quarterly r D Solution: n = 15, P = 4000, r = 8.5% = 0.085, and m = 4. We have m ) nm =4000 ( 1 + 0.085 4 ) 15 · 4 = 14124.86 BD. that A 15 = P ( 1 + r The total interest is I = A 15 − P = 14124.86 − 4000 = 10124.86 BD. Dr. Abdulla Eid (University of Bahrain) Exponential 13 / 18

  14. Exercise Find the compound amount and the compound interest of investing 1 300 BD at 7% for 9 years compounded yearly. 2 200 BD at 5% for 6 years compounded monthly. 3 1000 BD at 9% for 2 years compounded semi–annually. d i E 4 200 BD at 1% for 2 years compounded daily (365 days in one year). a 5 (Old exam question) 11000 BD at 3% for 9 years compounded l l u monthly. d b 6 (Old exam question) 1020 BD at 6% for 8 years compounded A monthly. . r D Dr. Abdulla Eid (University of Bahrain) Exponential 14 / 18

  15. 3 - The Euler Number d Example i E (Motivational Example) Suppose you invest 1 BD in an account that pays a 100%. Find the compound amount for one year in every possible period. l l u What happen if the interest are paid continuously in every single moment? d b Solution: P = 1, r = 100% = 1, n = 1, m = m . So the compound A amount is . m ) nm = 1 ( 1 + 1 A 1 = P ( 1 + r r D m ) m Dr. Abdulla Eid (University of Bahrain) Exponential 15 / 18

  16. Continue... A 1 = ( 1 + 1 m ) m Period m A m d i yearly 1 E semi–annually 2 a l quarterly l 4 u d monthly 12 b daily 365 A hourly 365(24) . r Minutely 365(24)(60) D secondly 365(24)(60)(60) 365 ( 24 )( 60 )( 60 )( 10 ) 3 mini–secondly 365 ( 24 )( 60 )( 60 )( 10 ) 6 micro–secondly 365 ( 24 )( 60 )( 60 )( 10 ) 9 nano–secondly Continuously ∞ Dr. Abdulla Eid (University of Bahrain) Exponential 16 / 18

  17. As you can see from the example above that as m → ∞ , A 1 → 2.718281828 . . . . We define m → ∞ ( 1 + 1 m ) m = e 2.718281828 · · · = lim d i E a e is called the Euler number . l l u e is not a rational number,i.e., the decimal expansion of e never ends d nor repeat in a pattern. b A e = 1 + 1 1 ! + 1 2 ! + 1 3 ! + 1 4 ! + 1 . 5 ! + . . . r D Exercise 1 Find the value of (a) e 2.5 (b) e − 1 3 . (c) e Dr. Abdulla Eid (University of Bahrain) Exponential 17 / 18

  18. Graphing Exponential Function with a > 1 d Example i E Graph the function a f ( x ) = − e − x + 3 l l u d Solution:Using the calculator, we fill the following table b A x − 2 − 1 0 1 2 3 . r y D Dr. Abdulla Eid (University of Bahrain) Exponential 18 / 18

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