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# Exam 2 Study Guide Math 240

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This 10 page Study Guide was uploaded by AnnMarie on Friday October 16, 2015. The Study Guide belongs to Math 240 at Louisiana Tech University taught by Jonathan B Walters in Fall 2015. Since its upload, it has received 109 views. For similar materials see Precalculus in Mathematics (M) at Louisiana Tech University.

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Date Created: 10/16/15

When reviewing over Chapter 4 the following concepts should be studied for Exam 2 Exponen alFunc on fxaquotWhere agt0andal Natural Exponential Function fx ex Properties of Exponential Functions 1 a0 Cll Cl 3 axay am 4 gdW 5 any aw 6 czbe ab 7 a ax Logarithmic Function logax y ltgt ay x Properties of Logarithmic Functions 1 ogal 0 2 ogaa 1 3 logaaquot x 4 6110ng x Common Logarithmic Function logx logmx Natural Logarithmic Function nx logex Properties of Natural Logarithms 1 In 1 O 2 me 1 3 nexx 4 6 x Laws of Logarithms 1 0gaAB ogaA ogaB 2 loga logaA logaB 3 logaAc C logaA Compound Interest At P1 where P is the initial principle r is the interest rate per year n is the number of times compounded per year and t is time Change of Base logax Oar toga Guidelines for Solving Exponential Equations 1 lsolate the exponential expression on one side of the equation 2 Take the logarithm of each side then use the laws of Logarithms to bring down the exponent 3 Solve for the variable Guidelines for Solving Logarithmic Equations 1 lsolate the logarithmic term on one side of the equation you may need to combine the logarithmic terms first 2 Write the equation in exponential form or raise the base to each side of the equation 3 Solve for the variable Exponential GrowthDecay At A0e where A0 is the initial amount r is the rate of growthdecay and t is time Radioactive Decay Mt Moe rt where r lihz Newton s Law of Cooling Tt TS T0 my where T S is the surrounding temperature T 0is the initial temperature k is constant depending on object and t is time pH Scale pH logC cgt C PH Since there are no calculators allowed on the exam you can leave your answers in logarithmic notation Solving Systems of Linear Equations with Substitution 1 Solve for One Variable a Choose one equation and solve for one variable in terms of the other variable 2 Substitute a Substitute the expression you found in Step 1 into the other equation to get an equation in one variable then solve for that variable 3 BackSubstitute a Substitute the value you found in Step 2 back into the expression found in Step 1 to solve for the remaining variable Solving Systems of Linear Equations with Elimination 1 Adjust the Coefficients a Multiply one or more of the equations by approprate numbers so that the coefficients of one variable in one equation is the negative of its coefficient in the other equation 2 Add the Equations a Add the two equations to eliminate one variable then solve for the remaining variable 3 BackSubstitution a Substitute the value you found in Step 2 back into one of the original equations and solve for the remaining variable Solving Systems of Linear Equations with Graphing 1 Graph each equation a Express each equation in a form suitable for the graphing calculator by solving for y as a function of x Graph the equations on the same screen 2 Find the Intersection Points a The solutions are the x and ycoordinates of the points of intersection Solving Systems of Linear Equations with Several Variables 1 If the equation is in Triangular Form A Solve the last equation B BackSubstitute the value you found in Step A back into the second equation and solve for the remaining variable C BackSubstitue for the value you found in Step A and Step B back into the first equation and solve for the remaining variable 2 If the equation is not in Triangular Form A Use the following to arrange the equations into Triangular Form without modifying the solution i Add a nonzero multiple of one equation to another ii Multiply an equation by a nonzero constant iii Interchange the positions of two equations B Perform Step 1 to solve the systems of equations Solving Matrixs with Elementary Row Operations 1 Write an Augmented Matrix of the system of linear equations Add a multiple of one row to another Multiply a row by a nonzero constant Interchange two rows Write the resulting Matric into linear equation system and solve with backsubstitution 912590 RowEchelon Form A matrix is in rowechelon form if it satisfies the following conditions 1 The first nonzero number of each row is 1 2 The leading entry in each row is to the right of the leading entry in the row immediately above it 3 All rows consisting entirely of zeros are at the bottom of the matrix 4 Every number above and below each leading entry is a zero Equality of Matrices The matrices A 2 an and B bij are equal if and only if they have the same dimensions m x n and corresponding entries are equal that is fori 1 2 m andj 1 2 n Sum Difference and Scalar Properties of Matrices LetA 2 an and B bij be matrices of the same dimension m x n and let c be any real number 1 The sum ofA B is the m x n matrix obtained by adding corresponding entries ofA and B A B aijbij 2 The difference ofA B is the m x n matrix obtained by subtracting corresponding entries ofA and B A 39 B aij39bij 3 The scalar product cA is the m x n matrix obtained by multiplying each entry ofA by c CA 2 can Properties of Addition and Scalar Multiplication of Matrices Let A B and C be m X n matrices and let c and d be scalars 1 Commutative Property of Matrix Addition A B B A 2 Associative Property of Scalar Addition ABCABC 3 Associative Property of Scalar Multiplication cdA ch 4 Distributive Property of Scalar Multiplication cd A cA dA cAB cA cB Matrix Multiplication lfA 2 an is an m x n and B bu is an n x k matrix then their product is m x k matrix C Cu where cij is the inner product of the ith row of A and the jth column of B We write the product as CAB Identitv Matrices 2 X 2 m X n with 1 on diagnal and O on everything else 3X3 Definition Let A be an m X n matrix If there is an matrix A391 with A39lA ln X m 2 AA391 that we say A391 is the inverse of A Findind Inverses 1 Augment matrix A with the identity of same size 2 Use row operations to reduce LHS left hand side to identity 3 RHS right hand side will be A39l Determinates of a 2 X 2 matrix A la b cd detAAabadbc IC d Determinates of a 3 X 3 matrix I 01 a2 03 I A 04 05 06 detA A 0105 19 0608 020409 0507 030408 0507 I 07 a8 G9 I Cramer s Rule xl detAi The solutions to Ax b is given by xi dam where x x2 and A is the matrix owahose x3 ith folumn has been replace by E 1 IX n After exam 1 we found that there is at least one question for each section we go over in class I will be using some of the Homework Problems and Suggested Assignments listed on the Syllabus was no joke With this being said I took the difficult looking Homework Problems and Suggested Assignment Problems to create this Review Questions Study Guide Please look over the Chapter 4 and Chapter 10 Study Material also Hopefully this will assist with your review for Exam 2 1 For the function fx 1 10x calculate the following values a f 3 f 1 f0 1 0 1 6 919 2 The graph of gx 2x is shown below On the same axis sketch the graph of where gx 2x 3 State the domain and range of the graph 3 Hannah would like to make an investment that will turn 7000 dollars into 33000 dollars in 5 years What quarterly rate of interest compounded four times per year must she receive to reach her goal 4 Graph of gx exiS shown below Graph the function gx 2 ex on the same graph and state the domain range and asymptote 5 Express the equation in exponential form a l0g381 4 b l0g31 O 6 Express the equation is logarithmic form a ex 3 b e4 x 7 Sketch the graph of the function fx 2Zogx 8 Rewrite the expression 4Zogx 2l0gx2 1 510gx 1 as a single logarithm log A 9 Find the solution of the exponential equation 22x15 3H6 10 The population of California was 2976 million in 1990 and 338 million in 2000 Assume that the population grows exponentially a Find a functionn that models the population 1 years after 1990 b Find the time required for the population to double 0 Use the function from part a to project the population of California in the year 2010 11 The halflife of Strontium90 is 28 years How long will it take a 72 mg sample to decay to a mass of 18 mg 12 The pH reading of a sample of each sustance is givem Calculate the hydrogen ion concentration of the substance a Vinegar pH 30 b Milk pH 65 13 Solve the system x 2y 7 5x y 2 14 Solve the system 6x4y52 4 3x 4y 22 19 6x 5y4z 23

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