1. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §9.1a Exponential Fcns

2. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §8.5 → Rational InEqualities  Any QUESTIONS About HomeWork §8.5 → HW-41 8.5 MTH 55

3. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 3 Bruce Mayer, PE Chabot College Mathematics Exponential Function  A function, f(x), of the form  is called an EXPONENTIAL function with BASE a.  The domain of the exponential function is (−∞, ∞); i.e., ALL Real Numbers

4. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 4 Bruce Mayer, PE Chabot College Mathematics Recall Rules of Exponents  Let a, b, x, and y be real numbers with a > 0 and b > 0. Then

5. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 5 Bruce Mayer, PE Chabot College Mathematics Evaluate Exponential Functions  Example   Solution   Example   Solution 

6. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 6 Bruce Mayer, PE Chabot College Mathematics Evaluate Exponential Functions  Example   Solution 

7. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 7 Bruce Mayer, PE Chabot College Mathematics Example  Graph y = f(x) =3 x  Graph the exponential fcn:  Make T-Table, & Connect Dots xy 0 1 –1 2 –2 3 1 3 1/3 9 1/9 27

8. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 8 Bruce Mayer, PE Chabot College Mathematics Example  Graph Exponential  Graph the exponential fcn:  Make T-Table, & Connect Dots xy 0 1 –1 2 –2 –3 1 1/3 3 1/9 9 27 This fcn is a REFLECTION of y = 3 x

9. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 9 Bruce Mayer, PE Chabot College Mathematics Example  Graph Exponential  Graph the exponential fcn:  Construct SideWays T-Table x −3−3 −2−2 −1−1 0123 y = (1/2) x 84211/21/41/8  Plot Points and Connect Dots with Smooth Curve

10. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 10 Bruce Mayer, PE Chabot College Mathematics Example  Graph Exponential  As x increases in the positive direction, y decreases towards 0

11. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 11 Bruce Mayer, PE Chabot College Mathematics Exponential Fcn Properties  Let f(x) = a x, a > 0, a ≠ 1. Then A.The domain of f(x) = a x is (−∞, ∞). B.The range of f(x) = a x is (0, ∞); thus, the entire graph lies above the x-axis. C.For a > 1 (e.g., 7) i.f is an INcreasing function; thus, the graph is RISING as we move from left to right ii.As x→∞, y = a x increases indefinitely and VERY rapidly

12. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 12 Bruce Mayer, PE Chabot College Mathematics Exponential Fcn Properties  Let f(x) = a x, a > 1, a ≠ 1. Then iii.As x→−∞, the values of y = a x get closer and closer to 0. D.For 0 < a < 1 (e.g., 1/5) i.f is a DEcreasing function; thus, the graph is falling as we scan from left to right. ii.As x→−∞, y = a x increases indefinitely and VERY rapidly iii.As x→ ∞, the values of y = a x get closer and closer to 0

13. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 13 Bruce Mayer, PE Chabot College Mathematics Exponential Fcn Properties  Let f(x) = a x, a > 0, a ≠ 1. Then E.Each exponential function f is one-to-one; i.e., each value of x has exactly ONE target. Thus: i. ii.f has an inverse

14. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 14 Bruce Mayer, PE Chabot College Mathematics Exponential Fcn Properties  Let f(x) = a x, a > 0, a ≠ 1. Then F.The graph f(x) = a x has no x-intercepts In other words, the graph of f(x) = a x never crosses the x-axis. Put another way, there is no value of x that will cause f(x) = a x to equal 0 G.The x-axis is a horizontal asymptote for every exponential function of the form f(x) = a x.

15. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 15 Bruce Mayer, PE Chabot College Mathematics Translate Exponential Graphs TranslationEquationEffect on Equation Horizontal Shift y = a x+b = f (x + b) Shift the graph of y = a x, b units (i) Left if b > 0. (ii) Right if b < 0. Vertical Shift y = a x + b = f (x) + b Shift the graph of y = a x, b units (i) Up if b > 0. (ii) Down if b < 0.

16. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 16 Bruce Mayer, PE Chabot College Mathematics Example  Sketch Graph  By Translation Move DOWN y = 3 x by 3 Units  Note Domain: (−∞, ∞) Range: (−4, ∞) Horizontal Asymptote: y = −4

17. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 17 Bruce Mayer, PE Chabot College Mathematics Example  Sketch Graph  By Translation Move LEFT y = 3 x by 1 Unit  Note Domain: (−∞, ∞) Range: (0, ∞) Horizontal Asymptote: y = 0

18. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 18 Bruce Mayer, PE Chabot College Mathematics Alternative Graph: Swap x & y  It will be helpful in later work to be able to graph an equation in which the x and y in y = a x are interchanged

19. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 19 Bruce Mayer, PE Chabot College Mathematics Example  Graph x = 3 y  Graph the exponential fcn:  Make T-Table, & Connect Dots xy 1 3 1/3 9 1/9 27 0 1 –1 2 –2 3

20. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 20 Bruce Mayer, PE Chabot College Mathematics Example  Apply Exponential  Example  Bank Interest compounded annually.  The amount of money A that a principal P will be worth after t years at interest rate i, compounded annually, is given by the formula

21. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 21 Bruce Mayer, PE Chabot College Mathematics Example  Compound Interest  Suppose that $60,000 is invested at 5% interest, compounded annually a)Find a function for the amount in the account after t years  SOLUTION a) = $60000(1 + 0.05 ) t = $60000(1.05) t

22. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 22 Bruce Mayer, PE Chabot College Mathematics Example  Compound Interest  Suppose that $60,000 is invested at 5% interest, compounded annually b)Find the amount of money amount in the account at t = 6.  SOLUTION b) A(6) = $60000(1.05) 6

23. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 23 Bruce Mayer, PE Chabot College Mathematics Example  Bacterial Growth  A technician to the Great French microbiologist Louis Pasteur noticed that a certain culture of bacteria in milk doubled every hour.  Assume that the bacteria count B(t) is modeled by the equation Where t is time in hours

24. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 24 Bruce Mayer, PE Chabot College Mathematics Example  Bacterial Growth  Given Bacterial Growth Equation  Find: a)the initial number of bacteria, b)the number of bacteria after 10 hours; and c)the time when the number of bacteria will be 32,000.

25. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 25 Bruce Mayer, PE Chabot College Mathematics Example  Bacterial Growth a)INITIALLY time, t, is ZERO → Sub t = 0 into Growth Eqn: b)At Ten Hours Sub t = 10 into Eqn:

26. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 26 Bruce Mayer, PE Chabot College Mathematics Example  Bacterial Growth c)Find t when B(t) = 32,000  Thus 4 hours after the starting time, the number of bacteria will be 32k

27. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 27 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §9.1 Exercise Set 36, 40, 54  USA Personal Savings Rate

28. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 28 Bruce Mayer, PE Chabot College Mathematics All Done for Today Bacteria Grow FAST! Note: 37 °C = 98.6 °F (Body Temperature)

29. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 29 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

30. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 30 Bruce Mayer, PE Chabot College Mathematics Irrational Exponents  By The Properties of Exponents we Can Evaluate Bases Raised to Rational-Number Powers Such as  What about expressions with IRrational exponents such as:  To attach meaning to this expression consider a rational approximation, r, for the Square Root of 2

31. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 31 Bruce Mayer, PE Chabot College Mathematics Irrational Exponents  Approximate by ITERATION on: 1.4 < r < 1.5 1.41 < r < 1.42 1.414 < r < 1.415

32. BMayer@ChabotCollege.edu MTH55_Lec-54_sec_8-5a_PolyNom_InEqual.ppt 32 Bruce Mayer, PE Chabot College Mathematics Irrational Exponents  Thus by Iteration  Any positive irrational exponent can be interpreted in a similar way.  Negative irrational exponents are then defined using reciprocals.