I.  Graphs of "Simple" Exponential Functions   

II. What is e? What is exp(x)

• Tools for Winplot used in this lab:

• Equation menu
• explicit
• point
• Animation menu
  
• Families on Inventory.
• Two
• Intersections...

I.Graphs of Exponential Functions

• Set the parameters: A= 1 and B=2.
• Plot the point (h, B^h)
• Plot the point (h, A*B^h)
• Use animator A and  h.
• Plot graphs with explicit for
  
• y = f(x) = B^x
• y = g(x) = A*B^x
• Use animator A; family A.
• Set A = 1, use animator for B; family B.
• Notes:
  
• The base of the function g is B.
• Since B^(r+s) = B^r * B^s,  B^(x+s) = B^x * A where A = B^s.
• Since B^(k*r) = (B^k)^r,  B^(kx) = B'^x where B' =B^k.
  

• Finding Parameters and Solving equations [With solutions.]
  
• Find A and B where y=A*B^x with y(0)=5, y(1)=15.
  
• y(0)=A*B^0=A*1=A, so given that y(0)=5 it must be that A = 5
• y(1)=A*B^1 = A*B = 5*B [since A = 5], so given that y(1) = 15, 5*B = 15 and B =3.
• Find A and B where y=A*B^x with y(1)=30, y(2)=120.
• y(1)=A*B^1=A*B=30.
  
• y(2)=A*B^2 = (A*B)*B = 30*B [since A*B = 30], so given that y(2) = 120, 30*B = 120 and B =4.
• But A*B = 30 so  A *4 = 30 and A = 30/4 = 15/2.
  
• Find x where  2*5^x =250.
• 5^x = 125, so x = 3.
  
• Find an estimate for x where  2* 5^x = 100.
• 5^x = 50. Use winplot to find x by finding the intersection of y=5^x and y = 50.
  

Record your answers for the next work and submit them on Moodle by Wednesday, April 14.

1. Find A and B where y=A*B^x with y(0)=7, y(1)=21.
   
2. Find A and B where y=A*B^x with y(1)=20, y(2)=100.
3. Find x where  5*3^x =45.
4. Find an estimate for x where  3* 7^x = 210. [Use Winplot]
   
5. Find an estimate for any and all x where 2^x  +  (1/2)^x  = 10. [Use Winplot]
   

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II.  What is e? What is exp(x)? Using Excel. Make a spead sheet showing the values for (1 + 1/n)^n  where n = 1, n=10, n=100, n=1000, ... , n = 1000000. Continue your spread sheet until n = 10^20. Record your result (approximately) for n= 1000000 and n = 10 ^20. Using winplot  Plot ( 1, (1+1/n)^n Use animator N; family N Graph y = ((1+1/n)^n)^x.   Use animator N; family N Graph y = exp(x). Compare the graphs.