Chapter 1  Functions and Variables


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1 Business Clculus 1 Chpter 1  Functions nd Vribles This Acdemic Review is brought to you free of chrge by preptests4u.com. Any sle or trde of this review is strictly prohibited.
2 Business Clculus 1 Ch 1: Functions nd Vribles Function of one vrible hs only one independent vrible. For emple, y = f () is function of one vrible,, which might tke ny vlue in its domin nd produce only one vlue for dependent vrible, y. On the other hnd, the dependent vrible might depend on two or more independent vribles such s A = f(,y) or V = f(,y,z). However, Business Clculus 1 minly focus on the functions of one nd two vribles. The most common functions eplored in this chpter re: Polynomils Given: n n 1 n 1 p( ) = n n 1 n 1 0 where s re the rel numbers, clled coefficients, nd n's re nonnegtive integers (i,e. zero or positive integers), clled powers. The highest power is clled the degree of the polynomil nd the corresponding coefficient is clled leding coefficient. Emple 1: 4 f ( ) = is polynomil of degree 4 with leding coefficient ( 5). The term with power three is missing. This mens tht the coefficient for tht term is zero. So, it is perfectly lright to hve one or more terms missing in polynomil, but hving ll of them missing will be trgedy!!! One of the frequently used polynomil is Qudrtic Eqution. f ( ) = + b + c. Grph of this function is concve up if < 0, nd it is concve down if > 0. Coefficients, b, nd c re rel numbers with the condition tht 0. Rtionl Functions R( ) f ( ) = is clled rtionl function if both R() nd S() re polynomils; S( ) f () S( ) = 0 nd ( ) 0 A function otherwise it is clled n lgebric eqution. is not defined for ll the rel vlue(s) where R. In cse both R() nd S() re zero for one or more vlues of, then one should use the concept of limits to find the limiting vlue(s) of f(). The vlue(s) t which S( ) = 0 nd R( ) 0 re clled nonremovble singulrities (discontinuities). Pge of 7 This Acdemic Review is brought to you free of chrge by preptests4u.com. Any sle or trde of this review is strictly prohibited.
3 Business Clculus 1 Ch 1: Functions nd Vribles The verticl symptotes of this function re locted t the vlues where nd R( ) 0. S( ) = 0 Emple : f ( ) = Find:. All the vlues in which f () is not defined 4 = 0, =, = (mke sure you check these nswers in the numertor) b. All the verticl symptotes Lines = nd = re the verticl symptotes. c. Find ll nonremovble singulrities (discontinuity) = nd = re nonremovble singulrities. These vlues do not vnish the numertor; otherwise, one should pply the rules of limits to check for removble singulrities. Eponentil Functions ( ) is clled eponentil function. is the bse nd is the eponent. Depending on the f = vlue of constnt >0, f () could be either incresing or decresing for ll rel vlues of independent vrible. If > 1, then f () is n incresing function, or growth function; otherwise, it is decresing function, commonly clled decy function. Emple 3: Grph the function f ( ) = 3 nd lbel s incresing or decresing function. By following the eponentil function rule, this is n incresing function, growth function, since > 1. In order to grph this function, we need to find the intercept, verticl nd horizontl symptotes (if ny), domin, nd rnge. Pge 3 of 7 This Acdemic Review is brought to you free of chrge by preptests4u.com. Any sle or trde of this review is strictly prohibited.
4 Business Clculus 1 Ch 1: Functions nd Vribles let = 0, find y. y = 1, this is yintercept. intercept, let y = 0, find. As you observe there is no vlue of which mkes the vlue of y = 0; therefore, there is no  intercept, mening the grph does not cross is. In the section of grph of functions, we will elborte on other items needed for resonble grphing f( ) Emple 4: Grph the function 1 f ( ) = nd lbel s incresing or decresing function. 3 By following the eponentil function rule, this is n decresing function, decy function, since < 1. In order to grph this function, we need to find the intercept, verticl nd horizontl symptotes (if ny), domin, nd rnge. let = 0, find y. y = 1, this is yintercept. intercept, let y = 0, find. Pge 4 of 7 This Acdemic Review is brought to you free of chrge by preptests4u.com. Any sle or trde of this review is strictly prohibited.
5 Business Clculus 1 Ch 1: Functions nd Vribles As you observe there is no vlue of which mkes the vlue of y = 0; therefore, there is no  intercept, mening the grph does not cross is. In the section of grph of functions, we will elborte on other items needed for resonble grphing. f () Logrithmic Functions The function f ( ) = Log is logrithmic function where > 0 nd constnt > 0. In fct, the grphs of logrithmic nd eponentil functions re symmetric with respect to the line y = in the yplne. This function crosses is t = 1 but does not cross yis. For constnt vlue of > 1, f () is n incresing function. For 0 < < 1, the function f () is decresing. Emple 5 : Grph f ( ) = Log decresing. intercept, let y = 0, find. = 1, this is yintercept., find nd y intercepts. Indicte if this function is incresing or Since > 0 (by definition) there is no mening tht f () does not cross yis. This function is incresing ccording to the condition outlined bove. Pge 5 of 7 This Acdemic Review is brought to you free of chrge by preptests4u.com. Any sle or trde of this review is strictly prohibited.
6 Business Clculus 1 Ch 1: Functions nd Vribles f ( ) Emple 6: Grph f ( ) = Log decresing. intercept, let y = 0, find. = 1, this is yintercept., find nd y intercepts. Indicte if this function is incresing or Since > 0 (by definition) there is no mening tht f () does not cross yis. This function is decresing ccording to the condition outlined bove. f ( ) Pge 6 of 7 This Acdemic Review is brought to you free of chrge by preptests4u.com. Any sle or trde of this review is strictly prohibited.
7 Business Clculus 1 Ch 1: Functions nd Vribles Trigonometric Functions A complete ccount of trigonometry functions is provided in Trigonometry Review mterils vilble on the website. However, quick review nd some importnt properties of trigonometry functions re given below. Fundmentl Trigonometric Functions: 1 f ( ) = Sin( ) 1 Sin ( ) 1 Period = π f ( ) = Cos( ) 1 Cos ( ) 1 Period = π 3 f ( ) = Tn( ) Period = π 4 f ( ) = Cot( ) Period = π 5 f ( ) = Sec( ) Period = π 6 f ( ) = Csc( ) Period = π Grphs of these equtions re given in the following sections. Pge 7 of 7 This Acdemic Review is brought to you free of chrge by preptests4u.com. Any sle or trde of this review is strictly prohibited.
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