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Liceo Scientifico “Isaac Newton” Roma Y School Year 2011-2012 Maths course (0,1) Exponential function O X Compound interest M1= C + C i = C ( 1 + i ) M2 = M 1 + M1 i = M 1 ( 1 + i ) = C ( 1 + i )² Sum of money M=C(1+i) x where the variable x can indicate also fractional values of time. (1+i)>1 It’s great! General exponential function y=a x a>0 xєR Its domain is the set of real numbers R, while its codomain is the set of real positive numbers R + a>1 0<a<1 a=1 First condition If x -3 -2 a>1 x y=2 Y y 1 8 1 4 -1 1 2 0 1 1 2 (0,1) 2 4 3 8 O 1 X Y Key: y=2 y=3 x x (0,1) O 1 X Second condition if 0 < a < 1 Y x y -3 8 -2 4 -1 2 0 1 1 1 2 2 1 4 3 1 8 -x y=2 (0,1) O 1 X Y Key: y=2 y=3 -x -x (0,1) O 1 X Third condition If a = 1 Y (0,1) O y=1 1 X The symmetry about the y axis Y Key: y=2 y=2 x -x (0,1) O 1 X This property is true for every pair of exponential functions of this type: y=a x y=a -x In fact if we apply the equations of symmetry about the y axis to x the function y = a we obtain the following curve: y=a x X=-x Y= y -x Y=a Properties of the exponential function the injectivity the surjectivity the exponential function is bijective so it’s invertible Logarithmic function y=a Y x y=x If a > 1 y = log x a (0,1) O (1,0) X Logarithmic function Y y=a x y=x If 0 < a < 1 (0,1) O (1,0) X y = log x a a>1 the exponential function grows faster than any polynomial function DEFINITION Let C and C’ be two curves both passing through a point P; we say that C is steeper than C’ in P if the slope of the tangent in P to C is greater than the one of the tangent in P to C’. Key: 2 y=x+2x+1 y = ln2 ∙ x + 1 Y y=2x+1 y=2 x (0,1) O P 1 X Euler‘s number e Values of a Exponential function x a=2 y=2 a = 2.5 y = 2.5 a = 2.7 y = 2.7 a = 2.71 y = 2.71 Slope m m = 0.6931 x m = 0.9933 x y = 2.718 a = 2.719 y = 2.719 a ≤ 2.718 then m < 1, m = 0.9163 x a = 2.718 If a grows, m also grows m = 0.9969 x x m = 0.9999 m = 1.0003 a ≥ 2.719 then m > 1; this means that there’s a value of a between these two values for which the slope of the tangent in P is equal to 1. This number is called Euler’s number e in honour of the mathematician who discovered it. It is an irrational number and a transcendental number because it isn’t the solution of any polynomial equation with rational coefficients. e = 2.7 The exponential function having base equals e is called a natural exponential function and its equation is: y=e x Y y=e x Key: y=e x y=x+1 y=x+1 (0,1) 45 ° O 1 X Graphs of non-elementary exponential functions: 1) symmetries, 2) translations, 3) dilations or compressions. 1) if we apply the equations of the symmetry about the x axis to the function y = a x we obtain the following curve: y=a x X= x Y=-y Y=-a x 2) if we apply the equations of a translation by a vector v having x components (h,k) to the function y = a we obtain the following curve: y=a X=x+h x Y=y+k Y–k=a X-h X-h Y=a + k 3) if we apply the equations of a dilation to the function y = a obtain the following curve: x X y=a x X=hx Y Y=ky k = ah we Examples of graphs: x y=2 +1 x 1) the equation y = 2 + 1 x represents the curve y = 2 shifted up by one: Y x y=2 Key: y=2 y=2 x x +1 (0,1) O y=1 1 X x+1 2) the equation y = 2 x represents the curve y = 2 shifted one point to the left: x+1 y=2 Y x y=2 Key: y=2 y=2 x x+1 (0,1) O 1 X x-3 3) the equation y = 2 - 2 x represents the curve y = 2 shifted three points to the right and shifted down by two: Y Key: y=2 y=2 x x-3 - 2 x y=2 (0,1) x-3 y=2 O 1 - 2 X y = -2 x 4) the equation y = 3 ∙ 2 x represents the curve y = 2 in which the abscissas don’t change while the ordinates are tripled: Y y=3∙2 x x y=2 Key: y=2 x y=3∙2 x (0,1) O 1 X x 5) the equation y = 2 3 x represents the curve y = 2 in which the ordinates don’t change while the abscissas are tripled: Y x y=2 Key: y=2 x x y=2 y=2 3 (0,1) O 1 X x 3 x 6) the equation y = 2 represents two curves: y=2 x Y if x ≥ 0 -x y=2 -x x y=2 y=2 if x < 0 Key: y=2 y=2 x -x (0,1) O 1 X x+1 7) the equation y = 2 represents two curves: Y y=2 x+1 -x-1 y=2 if x ≥ -1 if x < -1 Key: y=2 x +1 - x -1 y=2 (0,1) -1 O 1 X x+1 8) the equation y = 2 -1 represents the graph of the previous curve number 7 shifted down by one: Y Key: y=2 y=2 x +1 -1 - x -1 -1 (0,1) -1 O 1 X x+1 9) the equation y = - 2 + 1 can be rewritten as follows: y=-(2 Y x+1 -1) this equation represents the symmetrical curve of the function y = 2 x + 1 - 1 about the x axis Key: y=2 (0,1) O x +1 -1 x +1 y = - (2 -1) (0,-1) 1 X 10) the equation y=2 x+1 y=2 x+1 -2 -2 Y the part of the negative ordinates is substituted by its mirror image about the x axis: (0,1) O 1 X . Copyright 2012 © eni S.p.A.