Mathematics Dictionary Dr. K. G. Shih

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Exponent e^x

Symbol Defintion Example : Sqr(x) = square root of x

Q01 | - Properties of y = e^x Q02 | - Find derivative of y = e^x Q03 | - Series of y = e^x Q04 | - Prove that e^(i*pi) = -1 Q05 | - Formula

Q01. Properties of y = e^x Diagram of y = e^x y = e^x and inverse Deifintion of y = e^x If y = e^x then x = ln(y) If y = ln(x) then x = e^(ln(y)) Composite function ln(e^x) = x e^(ln(x)) = x. Value of y is between 0 and +infinite Properties of Y = e^x 1. Domain : real values of x between -infinite and infinite 2. Range : +0.0000 to +infinite 3. The curve is always increasing 4. The curve is concave upward 5. It is the inverse of y = ln(x) Special values e^0 = 1 e^1 = 1 + 1 + 1/(2!) + 1/(3!) + .... Go to Begin Q02. Find derivative of y = e^x Find y' Calculus Find d/dx(e^x) y = e^x, 1st serivative = e^x y = e^x, 2nd serivative = e^x y = e^x, 3rd serivative = e^x Go to Begin Q03. Series of e^x Taylor's expansion F(x) = F(0) + F1(0)*x + F2(0)*(x^2)/(2!) + F3(0)*(x^3)/(3!) + .... Where F1(0) = 1 is 1st derivative of e^x for x = 0 F2(0) = 1 is 2nd derivative of e^x for x = 0 F3(0) = 1 is 3rd derivative of e^x for x = 0 F4(0) = 1 is 4th derivative of e^x for x = 0 Etc. Hence e^x = 1 + x + (x^2)/(2!) + (x^3)/(3!) + ..... Value of e e = 1 + 1 + 1/(2!) + 1/(3!) + .... Go to Begin Q04. Prove that e^(i*pi) = -1 e^(i*x) = cos(x) + i*sin(x) Euler's defintion Prove that e^(i*x) = cos(x) + i*sin(x) e^(i*x) = 1 + i*x + ((i*x)^2)/(2!) + ((i*x)^3)/(3!) + ..... = (1 - (x^2)/(2!) + (x^4)/(4!) + ...) + i*(x - (x^3)/(3!) + .....) = cos(x) + i*sin(x) = cis(x) If x = pi cos(pi) = -1 sin(pi) = 0 Hence e^(i*pi) = -1 Go to Begin Q05. Formula e^x = 1 + x + x^2)/(2!) + ... e^0 = 1 e^1 = 1 + 1 + 1/(2!) + ... e^(ln(x)) = x If y = e^x, y' = e^x If y = e^x, y" = e^x ∫[e^x]dx = e^x Go to Begin

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