The Transcendental Functions
The Natural Logarithmic and Exponential FunctionsFunctions: Differentiation and IntegrationObjective:Objective Find derivatives of functions involving the natural logarithmic function.The Derivative of the Natural Logarithmic FunctionLet u be a differentiable function of x.d1[ ln x ] , x 0dxx,d1 du u′ ,u 0[ ln u ] dxu dx udu′[ ln u ] , u 0dxu,Derivative of the Natural Exponential FunctionLet u be a differentiable function of x.d x e e xdxd udu e eudxdx,Ex: Differentiate :a. y ln(5 x 2)b. f ( x) ln2x(2 x 2 )3c. y 5x4 3e. y ln ( ln x )2d. y e5 x x 1( x 3) 2x3 2f. y x3e x,x 32Ln Rule for IntegrationLet u be a differentiable function of x1 x dx ln x C1 u du ln u C,Integration Rules for Exponential FunctionsLet u be a differentiable function of x. e dx exx C, e du euu CEx: Evaluatea.4 x3 1 x 4 x dxb.csc2 x cot x dxc.x 3 2x 2 x 1 x 2 1 dx
Ex: Find the area of the region bounded by the graph y 2x, the x-axis, the linex 32x 3.Ex: Evaluatea.e1/ x x 2 dxb.sin x cos xe dxc.e3 x 3e2 x e xdx e2 x1d.ex 0 1 e x dxIntegrals of the Six Basic Trig Functions1. sin udu cos u C2. cos udu sin u C3. tan udu ln cos u u C4. cot udu ln sin u u C5. sec udu ln sec u tan u C6. csc udu ln csc u cot u CBases Other than e and ApplicationsObjective:Objective Define exponential functions that have bases other than e. Differentiate and integrate exponential functionsthat have bases other than e. Use exponential functions to model compound interest and exponential growth.Definition of Exponential Function to Base a:a:If a is a positive real number (a 1) and x is any real number, then the exponentialfunction to the base a is denoted by a x and is defined bya x e (ln a ) xDefinition of Logarithmic Function to Base a:a:If a is a positive real number (a 1) and x is any positive real number, then thelogarithmic function to the base a is denoted by log a x and is defined aslog a x 1ln xln aDerivatives for Bases other than e:e:Let a be a positive real number (a 1) and let u be a differentiable function of x.d x a (ln a)a xdxd13. [log a x ] dx(ln a ) x1.d udu a (ln a)a udxdxd1 du4. [log a x ] dx(ln a ) x dx2.Ex: Find the derivative of each function
a. y 5 xb. y x 3 x2 1c. y 5log3 4 xEx:Ex Evaluate 5x dx .Previously the power rule required n to be a rational number. However, now the rulecan be extended to cover any real value number.The Power Rule for Real ExponentsLet n be any real number and let u be a differentiable function of x.d n x nx n 1dxd udu u nu n 1dxdxEx: Find the derivative of each function2a. y x 2b. y x 2 x 1Inverse Trigonometric Functions: DifferentiationObjective:Objective Develop properties of the six inverse trigonometric functions. Differentiate an inverse trigonometric function.Review the basic differentiation formulas for elementary functions.None of the six basic trigonometrictrigonometric functions has an inverse function!y arcsin x D: 1 x 1 R: π / 2 y π / 2y arctan x D: x R: π / 2 y π / 2y arcsec x D: x 1 R: 0 y π , y π / 2Ex: Evaluate:a. arcsin(- ½ )y arccos x Domain: 1 x 1 Range: 0 y πy arccot x D: x R: 0 y πy arccsc x D: x 1 R: π / 2 y π / 2, y 0b. arcos(0)c. arctan ( 3)
Properties of Inverse Trigonometric Functions1. If 1 x 1 and π 2 y π 2 , then2. If 1 -1-1-1-1-1-1sin(sin x) x and sin (sin y) y.yx 1 and 0 y π , thencos(cos x) x and cos (cos y) y.y3. If x is a real number and π 2 y π 2 , thentan(tan x) x and tan (tan y) y.ysimilar properties hold for other inverse trigonometric functionsEx: Solve arctan(2 x 3) π / 4Derivatives of Inverse Trigonometric FunctionsLet u be a differentiable function of x.du′[ arcsin u ] dx1 u2du′[ arctan u ] dx1 u2du′[ arcsec u ] dx u u2 1d u ′[arccos u ] dx1 u2d u ′[ arccot u ] dx1 u2d u ′[ arccsc u ] dx u u2 1To derive these formulas, you can use implicit differentiationEx: Differentiate:a. y arcsin( x 2 )b. f ( x) arctan(3x 2)c. y arccos x 1Ex:Ex Differentiate y arcsin x x 1 x 2d. y arcsec e x2
Inverse Trigonometric Functions: IntegrationObjective:Objective Integrate functions whose antiderivatives involve inverse trigonometric functions. Use completing the squareto integrate a function. Review the basic integration formulas involving elementary functions.Integrals Involving Inverse Trigonometric FunctionsLet u be a differentiable function of x, and let a 0. dua2 u 2 arcsinu Ca a,2du1u arc ta n C2 uaa, uduu2 a2 1 u arcsec CaaThese rules all come from the preceding derivative rules of inverse functions. Sincethe arcsin and arccos derivatives are the negative of each other you only need onederivative for the pair.Ex: Evaluatea. d. dx9 xb. 2dxe.e 12x dx5 16 x 2x 34 x2c. dxx 4x2 9dxRemember completing the square!?!?!? Well, you need it for Calculus too!2222 b b b b x bx c x bx c x c 2 2 2 2 dxEx: Evaluate 2x 4x 722
Ex:Ex Find the area of the region bounded by the graph of1f ( x) 3x x2the x-axis, and the lines x 3/2 and x 9/4 .Ex: Find as many of the following integrals as you can using the formulas andtechniques studied so far.a. d.dxx x 12dx x ln xb. e. xdxx 12ln xdxxc. dxx2 1f. ln xdx
The Natural Logarithmic and Exponential The Natural Logarithmic and Exponential and Exponential Function FunctionFunctions sss: . Differentiate and integrate exponential functions that have bases other than e. Use exponential functions to model compound interest and exponential
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transcendental . Cantor: Algebraic numbers are countable, so transcendental numbers exist, and are a measure 1 set in [0;1], but it is hard to prove transcendence for any particular number. Examples of (proported) transcendental numbers: e, ˇ, , eˇ, p 2 p 2, (3), (5)::: Know: e, ˇ, eˇ, p 2 p 2 are transcendental. We don't even know if and .
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Exponential, logarithmic, and trigonometric functions are transcendental. . the procedures can be generalized for the calculation of other transcendental functions as well. 1 Taylor Approximations One method for comp
