Mathematics Dictionary Dr. K. G. Shih

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Exponent

Subjects

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

AL 06 00 | - Outlines AL 06 01 | - Defintions of e^x AL 06 02 | - Why do we need exponent ? AL 06 03 | - Why do we use expression e^x ranther than 2^x or 3^x ? AL 06 04 | - Laws of Expenonts AL 06 05 | - Properties of y = e^x AL 06 06 | - The exponential family founctions AL 06 07 | - Euler function e^(ix) = cos(x) + i*sin(x) AL 06 08 | - Inverse of y = e^x AL 06 09 | - Example : e^x + e^(2*x) + e^y + e^(2*y) = 12 AL 06 10 | - Solve e^(2*x) + 3*e^x - 18 = 0 AL 06 11 | - Example : How sketch e^x + e^(2*x) + e^y + e^(2*y) = 12 ? AL 06 12 | - Example : e^x + e^(2*x) + e^y + e^(2*y) = 12, find y if x = ln(2) AL 06 13 | - Example : Prove that (5^(n+1) + 5^(2*n))/(5 + 5^n) = 5^n AL 06 14 | - Example : Solve 8^(2*x+1) = 4^(5-x) AL 06 15 | - Example : Solve e^x + e^(2*x) < 12 AL 06 16 | - The exponential family functions AL 06 17 | - Identities in the exponential family functions

Answers

AL 06 01. Defintions Definition Symbol : e^x means e to power x. Symbol : EXP(x) means e to power x. Symbol : e^1 = 1 + 1 + 1/2! + 1/3! + 1/4! + 1/5! + .... Symbol : e^x = 1 + x + x^2/2! + x^3/3! + x^4/4! + x^5/5! + .... Note : Change expressions into mathematical forms used in classroom. Go to Begin AL 06 02. Why do we need exponent ? Answer Exponent makes mathematical expression simple to read. Example : 2 multiply 2 one hundred times can be expressed as 2^100. 2 multiply 2 one hundred times which need write hundred 2 and hundred x. 2^100 is clearly shown hundred 2 and we do not to count them. It is easier to read and simple to write. This is an example to show that mathematics will make method simple and easy. Example : Who did first use exponent ? Exponential notation was introduced into math field by French scientis Rene Descartes in 17th centrury. Go to Begin AL 06 03. Why do we use expression e^x ranther than 2^x or 3^x ? The expression e^x has the following properties. Expression y = e^x has slope = 1 when x = 0. Derivative of e^x is remaining as e^x. Diferentiation of e^x is also equal to e^x. y = e^(-x) is symmetrical to y = e^x about the y-axis. Go to Begin AL 06 04. Laws of exponet Laws 1. Multiplication : (a^x)*(a^y) = a^(x+y). 2. Division : (a^x)/(a^y) = a^(x-y). 3. Poser of power : (a^x)^m = a^(m*x). 4. Power of product : (a*b)^m = (a^m)*(b^m). 5. Power of quotient : (a/b)^m = (a^m)/(b^m). Special exponent 1. (a^1) = a. 2. Zero exponent : (a^0) = 1. 3. Negative exponent : a^(-n) = 1/(a^n). 4. Rational exponent : a^(1/n) = nth radical of a. Go to Begin AL 06 05. Properties of y = e^x Properties 1. y-intercept is at y = 1. 2. The slope is always positive and the curve is always increasing. 3. It has asymptote y = 0 and it has no negative value. 4. The curve is always concave upward. 5. y' = e^x and y" = e^x. 6. The series of e^x = Sum[(x^n)/n!] for n from 0 to infinite. Sketch diagram : quick fee hand sketch x = -1, y = e^x = 0.368 x = +0, y = e^x = 1 x = +1, y = e^x = 2.718 Go to Begin AL 06 06. The exponential family founctions Hyperbolic functions Study subject Hyperbolic functions. Functions. sinh(x) = (e^x - e^(-x)/2. cosh(x) = (e^x + e^(-x)/2. tanh(x) = sinh(x)/cosh(x). csch(x) = 1/sinh(x). coth(x) = 1/tanh(x). sech(x) = 1/cosh(x). Compare with trigonometric functions Study subject PM 14. Go to Begin AL 06 07. Euler function e^(ix) = cos(x) + i*sin(x) Reference Study subject Complex numbers. Properties Series e^x = 1 + x + x^2/2! + x^3/3! + .... cos(x) = 1 - x^2/2! + x^4/4! - ..... sin(x) = x - x^3/3! + x^5/5! - ..... e^(ix) = 1 + (ix)^2/2! + (ix)^4/4! + .... + (ix) + (ix)^3/3! + (ix)^5/5! .... e^(ix) = (1 - x^2/2! + x^4/4! + ....) + i*(x - x^3/3! + x^5/5! ....) Hence e^(ix) = cos(x) + i*sin(x). Example : e(i*pi) = cos(pi) + i*sin(pi) = -1. Example : e^(i*pi) = -1 includes most significant symbols -, 1, e, i and pi Go to Begin AL 06 08. Inverse function of y = e^x Definition Inverse of y = e^x is y = ln(x). Inverse of y = ln(x) is y = e^x. Properties Hence e^(ln(x)) = x. Hence ln(e^x) = x. Diagram Diagrams of y=e^x and y=ln(x) Program 06 05 Go to Begin AL 06 09. Example : e^x + e^(2*x) + e^y + e^(2*y) = 12 Question Find the equation of tangent to the curve e^x + e^(2*x) + e^y + e^(2*y) = 12 when x = ln(3). Solution Find dy/dx. e^x + 2*e^(2*x) + (e^y)*y' + 2*e^(2*y)*y' = 0. Hence y' = -(e^x + 2*e^(2*y))/(e^y + 2*e^(2*y)). Find e^(2*x) + e^x at x = ln(3) e^x + e^(2*x) = e^x + (e^x)^2. Since e^x = e^ln(3) = 3 (Formula). e^x + e^(2*x) = 3 + 3^2 = 12. Find e^(2*y) + e^y at x = ln(3) e^x + e^(2*x) + e^y + e^(2*y) = 12. Hence 12 + e^y + e^(2*y) = 12 Hence e^y + e^(2*y) = 0 Find slope at x = ln(3). y' = -(12)/(0) = - infinite Hence equation of tangent at x = ln(3) is a vertical line. Hence equation of tangent is x = ln(3). When x = ln(3), e^y + e^(2*y) = 12 - e^x -e^(2*x) = 12 - 3 - 9 = 0 Hence e^y + (e^y)^2 = 0. Or (e^y)*(1 + e^y) = 0. Or e^y = 0 and e^y = -1. Hence e^y = 0 and y = -infinite. Hence x = ln(3) is the asymptote. Exercise Express the expression in the form of y = F(x) Go to Begin AL 06 10. Solve e^(2*x) + 3*e^x - 18 = 0 Solution e^(2*x) = (e^x)^2. Let e^x = u and we have u^2 + 3*x - 18 = 0. Solve quadratic equation : (u - 3)*(u + 6) = 0. Hence u = 3 or u = -6. Since u = e^x is greater than zero, hence the solution is u = 3. Hence e^x = 3 or x = ln(3). Go to Begin AL 06 11. Example : How to sketch e^x + e^(2*x) + e^y + e^(2*y) = 12 ? Change it as y = F(x) e^(2*y) = (e^y)^2 Hence (e^2)^2 + e^y + e^x + e^(2*x)- 12 = 0. Use quadratic formula. Hence e^y = (-1 + Sqr(1^2 - 4*1*(e^x + e^(2*x) - 12)))/(2*1). Take logarithm on both sides Hence y = ln(-1 + Sqr(1^2 - 4*(e^x + e^(2*x) - 12))/2) Find y Since for ln(u) and u > 0. Hence -1 + Sqr(1 - 4*(e^x + e^(2*x) - 12) > 0. Or Sqr(1 - 4*(e^x + e^(2*x) - 12) > 1. For Sqr(t) and t > 0. Hence (1 - 4*(e^x + e^(2*x) - 12)) > 0. Or (e^x + 2^(2*x) - 12) < 0. Or (e^x)^2 + e^x - 12 < 0. Or (e^x)^2 + e^x - 12 < 0 Hence e^x <= (-1 + Sqr(1 + 4*1*12))/2 and e^x > 0. Hence x is between 0 and ln((-1 + Sqr(49))/2). Or x is between 0 and ln(3) Special points : use e^(ln(t)) = t x = ln(1) = 0 y = ln((-1 + Sqr(1 - 4*(1 + 1 - 12))/2) = ln((-1 + Sqr(41))/2). x = ln(2) = 0.6931 y = ln((-1 + Sqr(1 - 4*(2 + 4 - 12))/2) = ln((-1 + Sqr(25))/2) = ln(2). x = ln(3) = 1.0986 y = ln((-1 + Sqr(1 - 4*(3 + 9 - 12))/2) = ln(0) = -infinite. diagram Go to Begin AL 06 12. e^x + e^(2*x) + e^y + e^(2*y) = 12. Find y if x = ln(2). Solution e^(2*y) = (e^y)^2. Hence (e^2)^2 + e^y + e^(ln(2) + (e^(ln(2))^2 = 12. Use formula e^(ln(t)) = t. Hence (e^y)^2 + e^y + 2 + 2^2 = 12. Hence (e^y)^2 + e^y - 6 = 0. Hence (e^y - 2)*(e^y + 3) = 0 since e^y > 0, hence e^y = 2. Hence y = ln(2). Go to Begin AL 06 13. Prove that (5^(n+1) + 5^(2*n))/(5 + 5^n) = 5^n Proof 5^(n+1) = 5*5^n and 5^(2*n) = (5^n)^2 (5^(n+1) + 5^(2*n)) = 5*5^n + (5^n)^2 = (5^n)*(5 + 5^n). Hence (5^(n+1) + 5^(2*n))/(5 + 5^n) = 5^n. Go to Begin AL 06 14. Solve 8^(2*x+1) = 4^(5-x) Method 1 Since 8 = 2^3, hence 8^(2*x+1) = 2^(3*(2*x+1) Since 4 = 2^2, hence 4^(5-x) = 2^(2*(5-x) 2^(3*(2*x+1) = 2^(2*(5-x)) Hence 3*(2*x+1) = 2*(5-x) Hence Hence x = 7/8 Method 2 Take logarithm on both sides and use log(m^n) = n*log(m) Hence (2*x+1)*log(8) = (5-x)*log(4) Log(8) = log(2^3) = 3*log(2) Log(4) = log(2^2) = 2*log(2) Hence 3*(2*x+1) = 2*(5-x) Hence Hence x = 7/8 Verify x = 7/8 and 8^(2*x+1) = 2^(3*(2*7/8+1) = 2^(66/8) = 2^(33/4) x = 7/8 and 4^(5-x) = 2^(2*(5-7/8)) = 2^(33/4) Go to Begin AL 06 15. Solve e^(x) + e^(2*x) LT 12 Solution Since e^(2*x) = (e^x)^2. Hence (e^x)^2 + e^x - 12 < 0. Or (e^x - 3)*(e^x + 4) < 0. Since e^x is positvie for real x . Hence e^x + 4 greater than 0, hence e^x - 3 is negative. Hence e^x is less than 3. Hence x is greater than ln(1) and less than ln(3) Go to Begin AL 06 16. The exponential family functions Hyperbolic functions sinh(x) = (e^x - e^x)/2. cosh(x) = (e^x + e^x)/2. tanh(x) = sinh(x)/cosh(x) csch(x) = 1/sinh(x) sech(x) = 1/cosh(x) coth(x) = 1/tanh(x) Reference Study subject Diagrams of hyperbolic functions Study subject Properties of hyperbolic functions Study subject Properties of logarithm Compare with trigonometrical functions : See PM Section 14 Go to Begin AL 06 17. Examples of the exponential family functions Exponent Family Programs Study subject Properties of hyperbolic functions Questions in above programs 1. Prove that cosh(x)^2 - sinh(x)^2 = 1 2. Prove that sinh(2*x) = 2*sinh(x)*cosh(x) 3. Prove that sinh(x+y) = sinh(x)*cosh(x) + cosh(x)*sinh(x) 4. Prove that d/dx(sinh(x)) = cosh(x) Go to Begin AL 06 00. The exponential family functions Laws of Exponent 1. Multiplication : (a^x)*(a^y) = a^(x+y). 2. Division : (a^x)/(a^y) = a^(x-y). 3. Poser of power : (a^x)^m = a^(m*x). 4. Power of product : (a*b)^m = (a^m)*(b^m). 5. Power of quotient : (a/b)^m = (a^m)/(b^m). Special exponent 1. (a^1) = a. 2. Zero exponent : (a^0) = 1. 3. Negative exponent : a^(-n) = 1/(a^n). 4. Rational exponent : a^(1/n) = nth radical of a. Series of e^x e^x = 1 + x + (x^2)/2! + (x^3)/3! + ..... Value of e = 1 + 1 + 1/2! + 1/3! + ..... The value of e Lim[(1 + 1/x)^x] = e if x goes to infinite Lim[(1 + x)^(1/x)] = e if x goes to zero The family of exponent sinh(x) = (e^x - e^x)/2. cosh(x) = (e^x + e^x)/2. tanh(x) = sinh(x)/cosh(x) csch(x) = 1/sinh(x) sech(x) = 1/cosh(x) coth(x) = 1/tanh(x) Go to Begin

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