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Liceo Scientifico Isaac Newton Roma School Year 2011-2012 Maths course Exponential function X Y O (0,1)

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12 M = M + M i = M ( 1 + i ) = C ( 1 + i )² 11 Sum of money M = C + C i = C ( 1 + i ) 1 M = C ( 1 + i ) x Its great! where the variable x can indicate also fractional values of time. ( 1 + i ) > 1 Compound interest

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General exponential function y = a x a > 0 a > 1 0 < a < 1 a = 1 + Its domain is the set of real numbers R, while its codomain is the set of real positive numbers R x є R

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X Y O (0,1) 0 xy 1 1 3 2 -2 -3 4 2 1 8 1 4 1 2 8 1 If a > 1 x y = 2 First condition

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X Y O (0,1) 1 y = 2 x y = 3 x Key:

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0 xy 1 1 3 2 -2 -3 4 2 1 8 1 4 1 2 8 y = 2 - xX Y O (0,1) 1 if 0 < a < 1 Second condition

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y = 2 - x y = 3 - x Key: X Y O (0,1) 1

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If a = 1 X Y O (0,1) 1 y = 1 Third condition

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y = 2 x - x Key: X Y O (0,1) 1 The symmetry about the y axis

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In fact if we apply the equations of symmetry about the y axis to the function y = a we obtain the following curve: Y = a - x y = a x X = - x Y = y This property is true for every pair of exponential functions of this type: y = a x - x x

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the injectivity the surjectivity the exponential function is bijective Properties of the exponential function so its invertible

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X Y O (0,1) (1,0) If a > 1 y = log x a y = a x y = x Logarithmic function

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If 0 < a < 1 y = log x a y = a xX Y (0,1) (1,0) y = x O Logarithmic function

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a > 1 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. DEFINITION the exponential function grows faster than any polynomial function

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X Y O (0,1) 1 y = 2 x y = x + 2 x + 1 2 Key: y = 2 x + 1 y = ln2 x + 1 P

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Eulers number e Values of a Exponential function Slope m a = 2y = 2m = 0.6931 a = 2.5 y = 2.5 m = 0.9163 a = 2.7y = 2.7m = 0.9933 a = 2.71y = 2.71m = 0.9969 a = 2.718y = 2.718m = 0.9999 a = 2.719y = 2.719m = 1.0003 x x x x x x this means that theres a value of a between these two values for which the slope of the tangent in P is equal to 1. a 2.718 then m < 1, If a grows, m also grows a 2.719 then m > 1;

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The exponential function having base equals e is called a natural exponential function and its equation is: y = e x It is an irrational number and a transcendental number because it isnt the solution of any polynomial equation with rational coefficients. e = 2.7 This number is called Eulers number e in honour of the mathematician who discovered it.

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y = e x y = x + 1 Key: X Y O (0,1) 1 y = e x y = x + 1 45 °

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Y = - a x y = a x X = x Y = - y or compressions.2) translations,3) dilations 1) symmetries, Graphs of non-elementary exponential functions: 1) if we apply the equations of the symmetry about the x axis to the function y = a we obtain the following curve: x

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2) if we apply the equations of a translation by a vector v having components (h,k) to the function y = a we obtain the following curve: x Y = a + k Y – k = a X - h y = a x X = x + h Y = y + k X - h 3) if we apply the equations of a dilation to the function y = a we obtain the following curve: x y = a x X = h x Y = k y = a Y k X h

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x x 1) the equation y = 2 + 1 represents the curve y = 2 shifted up by one: y = 2 x y = 2 + 1 x Key: Y (0,1) 1 y = 1 O X y = 2 x y = 2 + 1 x Examples of graphs:

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y = 2 x x + 1 Key: Y (0,1) 1 x + 1 x 2) the equation y = 2 represents the curve y = 2 shifted one point to the left: X O y = 2 x x + 1

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y = -2 Y (0,1) x - 3 3) the equation y = 2 - 2 represents the curve y = 2 shifted three points to the right and shifted down by two: x y = 2 x y = 2 - 2 x - 3 Key: X O 1 y = 2 x y = 2 - 2 x - 3

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4) the equation y = 3 2 represents the curve y = 2 in which the abscissas dont change while the ordinates are tripled: x x y = 2 x y = 3 2 x Key: Y (0,1) 1 X O y = 2 x y = 3 2 x

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X O x 5) the equation y = 2 represents the curve y = 2 in which the ordinates dont change while the abscissas are tripled: 3 x y = 2 x Key: y = 2 3 xY (0,1) 1 y = 2 x x 3

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x x - x Key: Y (0,1) 1X O 6) the equation y = 2 represents two curves: y = 2 x - x if x 0 if x < 0 y = 2 x - x

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X 7) the equation y = 2 represents two curves: x + 1 y = 2 - x - 1 if x -1 if x < -1 Y (0,1) O 1 y = 2 x +1 y = 2 - x -1 Key:

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X O 8) the equation y = 2 - 1 represents the graph of the previous curve number 7 shifted down by one: x + 1Y (0,1) 1 y = 2 - 1 x +1 y = 2 - 1 - x -1 Key:

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9) the equation y = - 2 + 1 can be rewritten as follows: x + 1 this equation represents the symmetrical curve of the function y = 2 - 1 about the x axis y = - ( 2 - 1 ) x + 1Y (0,1) 1 O (0,-1) X y = 2 - 1 x +1 Key: y = - (2 - 1 ) x +1

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X 10) the equation y = 2 - 2 x + 1Y O (0,1) 1 y = 2 -2 x + 1 the part of the negative ordinates is substituted by its mirror image about the x axis:

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. Copyright 2012 © eni S.p.A.

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