Exponential and Logarithmic Functions: Identifying Growth and Decay Rates, Exams of Nursing

Download Exponential and Logarithmic Functions: Identifying Growth and Decay Rates and more Exams Nursing in PDF only on Docsity! MATH 225N Week 8 Finl exam (Top prediction) 2023-2024 summer QTR GRADED A+ In an exponential function there is 1 _____ but both ends ________ it - ANSWER>>Horizontal Asymptote don't Exponetial function formula and conditions - ANSWER>>f(x) = a * b^x a cannot = 0 b > 1 Look of an exponential function - ANSWER>> If a > 0 and b > 1 the function is an ________________ function add limits - ANSWER>>exponential growth lim x-->-inf f(x) = 0 lim x --> inf f(x) = inf If a >0 and 0<b<1 the function is an ______________ function add limits - ANSWER>>exponential decay lim x-->-inf f(x) = inf lim x--> inf f(x) = 0 Is the following function exponential? If so, state the initial value and base. If not explain why y = z^9 - ANSWER>>Not an exponential function because the base is a variable Is the following function exponential? If so, state the initial value and base. If not explain why y = 5^z - ANSWER>>Is an exponential function inital value is 1 (a) base is 5 Compute the exact value of the function for the given x-value f(x) = 5*2^x for x = 0 - ANSWER>>5 Compute the exact value of the function for the given x-value f(x) = -5*2^x for x = 1/3 - ANSWER>>-5*cbrt(2) when given an exponent of 1/x it is the same as - ANSWER>>x root of base Determine the formula for the exponential whose values are given (0,5/4) and (1,5/16) - ANSWER>>1. Write formula y = a * b^x 2. Plug in the point of the y-intercept and solve for a 5/4 = a *b^0 (b^0 is 1) 5/4 = a 3. Solve for b using the other point and a plugged in 5/16 = 5/4 * b^1 1/4 = b^1 b = 1/4 4.Write as the function f(x) = 5/4 * 1/4^x Determine the formula for the exponential whose values are given (0,7) and (2,14) - ANSWER>>1. Write formula y = a * b^x 2. Plug in the point of the y-intercept and solve for a 7 = a *b^0 (b^0 is 1) a = 7 If being halfed every ___ # of days divide the t variable by it Determine a formula for the exponential function with points (0, 2.5) and (1,4.7) - ANSWER>>1. Write formula: P(t) = P(1 + r)^t 2. Plug in (0, 2.5) 2.5 = P(1+r)^0 P = 2.5 3. Plug in (1, 4.7) 4.7 = 2.5(1+r)^1 r = .88 4. Write as function P(t) = 2.5(1.88)^t Key to logartithms - ANSWER>>y = a *b^x x = log<b> y Properties of Log, Natural Logs, Common Logs - ANSWER>>log<b> 1 = 0 ; ln 1 = 0; log 1 = 0 log<b> b = 1; ln e = 1; log 10 = 1 log<b> b^n = n; ln e^n = n; log 10^n = n b^log<b> m = m; e^ ln m = m; 10 ^ log m = m Evaluate the logarithimic expression: log<11> 11 - ANSWER>>Using log<b> b = 1 log<11> 11 = 1 Evaluate the logarithimic expression: log<6> 1296 - ANSWER>>1. write as exponential 1296 = 6^x 6^4 = 1296 log<6> 1296 = 4 Evaluate the logarithimic expression: log 1/10 - ANSWER>>1. Write as exponential 1/10 = 10^x 10^-1 = 1/10 log 1/10 = -1 Evaluate the logarithimic expression: ln (e^-5) - ANSWER>>Use ln e^n = n ln (e^-5) = -5 1/b^# is the same as - ANSWER>>b^-# Evaluate the logarithimic expressionln ln 9throot(e^7) - ANSWER>>ln (e^7)^1/9 ln (e^7/9) ln 9throot(e^7) = 7/9 Evaluate the logarithimic expression: 6^log<6> 2 - ANSWER>>b^log<b> m = m 6^log<6> 2 = 2 logarithm arguments cannot be - ANSWER>>negative Solve for x log x = 2 - ANSWER>>x = 10^2 x = 100 Graph the function below f(x) = log(-3 - x) - ANSWER>>1. Rewrite -(3 + x) --> -(x + 3) --> log(-x - 3) 2.Identify transformations Right 3, Reflect across y-axis 2.5 Identify VA -x-3 = 0 -x = 3 x = -3 3. Apply to base points of logartihms (1,0) and (b,1) (1,0) --> (-4,1) (10,1) --> (-13, 1) 4. Graph Graph the function below f(x) = -ln(x-5) - ANSWER>>1. Identify Transformations Right 5 units, reflect across the x-axis 2. Identify VA x-5 = 0 x = 5 3. Apply to Base points of logarithms (1,0) --> (6,0) (e, 1) --> (e+5 , -1) 4. Graph Graph the function below f(x) = 4 + log(x) - ANSWER>>1. Rewrite log(x) + 4 2. Identify transformations Up 4 units 3. Identify VA x = 0 3. Apply transformation to base points (1, 0) --> (1, 4) (10, 1) --> (10, 8) 4. Graph -1/ln(4) * (ln x) 2. Identify Transformations Vertical shrink by a factor of 1/ln(4), and reflected across the x-axis 3. Identify VA: x = 0 4. Apply Transformations (1,0) --> (1 , 0) (e, 1) --> (e , -0.721) 5. Graph If b^u = b^v then - ANSWER>>u = v If log<b>U = log<b>V then - ANSWER>>U = V Find the exact solution algebriacally 2 * 2^x/3 = 32 - ANSWER>>1. Divide 2^x/3 = 16 2. Find out what 2 raised to = 16 2^4 = 16 2^x/3 = 2^4 x/3 = 4 x = 12 {12} Find the exact solution algebriacally log<6>(2x-7) = 4 - ANSWER>>1. If solving a logarithm with a argument = to something 6^4 = 2x-7 1296 = 2x-7 2x = 1303 x = 651.5 {651.5} 1.23^x = 5.7 - ANSWER>>1. If solving and exponent is a variable log 1.23^x = log 5.7 x * log 1.23 = log 5.7 x = log 5.7 / log 1.23 x = 8.408 {8.408} State the domain of the function of f(x) = log[x(x+7)] - ANSWER>>1. Find the Vertical asymptote x= 0 x = -7 2. Place into a sign chart + -7 - 0 + 3. Find the x intercepts: x(x+7) = 1 x^2 + 7x -1 = 0 quadratic formula x = 0.140 x = -7.140 4. State the domain (-inf , -7) U (0, inf) 5. Choose two random points to the left and right, and then plot them and draw curves Solve for x e^x - e^-x ------------ = 5 2 - ANSWER>>1. Multiply by 2 e^x - e^-x = 10 2. Multiply each term by [e]^x [e]^2x - 1 = 10[e]^x 3. Rearrange to a Quadratic [e]^2x - 10[e]^x - 1 4. Quadtratic formula [e]^x = Y Y^2 - 10Y - 1 Y= 5 +- sqrt(26) [e]^x = 5 + sqr(26) [e]^x = 5 - sqrt(26) ---> x can't be negative e^x = 5 + sqrt(26) ln 5 + sqrt(26) = x x = 2.312 Solve log x^2 = 2 - ANSWER>>When dealing with solving, don't use exponential logarithm properties 10^2 = x^2 100 = x^2 x = +- 10 Solve ln(3x-2) + ln(x-1) = 2 ln x - ANSWER>>1. Simplify ln (3x-2 * x-1) = ln x^2 2. Same base logarithms = each other (3x-2 * x-1) = x^2 3x^2 - 5x + 2 = x^2 2x^2 - 5x + 2 = 0 (2x - 1) (x -2) x = 2, x = 1/2 (extraneous) {2} Graph Find the zeros of the function algebraically: f(x) = x^2 + x - 30 - ANSWER>>1. See if you can factor out anything - no 1.5 Factor (x + 6) (x - 5 ) 2. Set factors equal to zeros x + 6 = 0 x = -6 x - 5 = 0 x = 5 Find the zeros of the function algebraically: f(x) = 6x^2 + 25x - 9 - ANSWER>>1. See if you can factor out anything - no 2. Factor (3x - 1) (2x + 9) 3. Set factors equal to zero 3x - 1 = 0 x = 1/3 2x+9 = 0 x = -9/2 Find the zeros of the function algebraically: f(x) = 7x^3 - 26x^2 - 8x - ANSWER>>1. See if you can factor out anything x (7x^2 - 26x - 8) 2. Factor x (7x + 2) (x -4) 3. Set factors equal to 0 x = 0 7x + 2 = 0 x = -2/7 x-4 = 0 x = 4 State the degree and list the zeros of the polynomial function. State the multiplicity of each zero, and whether the graph crosses the x-axis at the location Sketch a graph f(x) = x (x-3)^2 - ANSWER>>1. State the degree by adding the multiplicities of each factor 2 + 1 = 3 degree = 3 2. State the multiplicity of each zero and whether it crosses the x-axis Multilicity of x = 1; crosses x axis because it is odd Multiplicity of (x-3)^2 = 2; touches x axis because even 3. Reference sheet for graph Factored form format - ANSWER>>a(x-k)(x-k)(x-k) Using only algebra, find a cubic function with the given zeros: -1 , 2 , -6 - ANSWER>>1. Write as factors (x + 1) ( x - 2) ( x + 6) 2. Multiply x^3 + 5x^2 -8x - 12 Multiplying two of the same radical = - ANSWER>>the number underneath the radical Using only algebra, find a cubic function with the given zeros: sqrt(3) , -sqrt(3), 5 - ANSWER>>1. Write as factors (x - sqrt(3) ) (x + sqrt(3) ) (x - 5) 2. Multiply x^3 - 5x^2 - 3x + 15 Rules for Synthetic and Long division - ANSWER>>1. Must be written in standard form 2. All terms must be present long division of polynomials - ANSWER>>1polynomial / 2polynomial 1p first term / 2p first term = thing on top multiply thing on top by 2p subtract result from original polynomial repeat with left over polynomial polynomial form: Ex: 2x^4-x^3 - 2 ------------- 2x^2 + x + 1 - ANSWER>>top = divisor * quotient + remainder Ex: 2x^4-x^3 - 2 ------------- 2x^2 + x + 1 2x^4 - x^3 -2 = (2x^2 + x +1) * (x^2 - x) + (x -2) Fraction form: Ex: 2x^4-x^3 - 2 ------------- 2x^2 + x + 1 - ANSWER>>problem = quotient + remainder /divisor Ex: 2x^4-x^3 - 2 ------------- 2x^2 + x + 1 2x^4-x^3 - 2 ------------- = x^2 - x + ( x-2 / 2x^2 + x + 1) 2x^2 + x + 1 synthetic division of polynomials - ANSWER>>zero of divisor | term1 term 2 term 3 1. Bring down 1 2. Multiply by zero 3. Add to term 2 4. Repeat 5. Last number is the remainder all non-negative numbers Use synthetic division to check that the number k is an lower bound for the real zeros of the function: f(x) = 5x^3 - 7x^2 + x -5 k = -7 - ANSWER>>It is a lower bound Alternating non-positive numbers Find rational zeros by using - ANSWER>>The Rational Zero Therom gives you the possible rational zeros of a polynomial Irrational zeros are - ANSWER>>zeros not on the list Find all real zeros of the function, finding exact values whenever possible, identify each zero as rational or irrational: f(x) = 7x^3 - 2x^2-35x + 10 - ANSWER>>Zeros: 2/7 rational +/- sqrt(5) irrational Find the remainder when f(x) = 8x^26 - 3x is divided by x+1 - ANSWER>>1. Identify k -- > -1 2. Use the remainder therom to find the remainder by plugging k in for x 3. remainder is 11 i = - ANSWER>>sqrt(-1) i^2 = - ANSWER>>-1 The number of complex zeros a function has is based on the - ANSWER>>functions degree Complex zeros are always found in - ANSWER>>conjugate pairs 2 irrational 1 rational or 2 irrattional 2 rational etc. Write the polynomial in standard form, and identify the zeros of the function and the x- ints of the graph f(x) = (x - 9i)(x + 9i) - ANSWER>>x^2 + 81 --> standard form zeros: +/- 9i (no x intercepts because imiginary numbers) Write a polynomial function minimum degree in standard form with real coefficents whose zeros include those listed zeros: 3i and -3i - ANSWER>>1. Convert zeros to factors (x - 3i) (x + 3i) 2. Multiply and Simplify x^2 + 9 Find a polynomial with real coefficents that has the given zeros: -1, 5-4i - ANSWER>>1. Convert zeros to factors (x + 1) (x - (5 - 4i) ) (x - (5 + 4i) ) (X + 1) (x -5 +4i) (x -5 -4i) 2. Multiply and Simplify x^3-9x^2+31x+41 Write a polynomial function of minimum degree in standard form with real coefficents whose zeros and their multiplicities include those listed: 1 (multiplicity of 2) -5 (multiplicity of 3) - ANSWER>>1. Write out all factors (x-1)(x-1)(x+5)(x+5)(x+5) 2. Multiply and Simplify x^5 + 13x^4 + 46x^3 -10x^2 -175x +125 Sketch the polynomial function graph for the given zeros and multiplicities -4 multiplicity of 3 3 multiplicity of 2 - ANSWER>>1. Convert zero to factor form (x+4)^3 (x-3)^2 2. Odd multiplicities = cross x axis 3. Even multiplicites = touch x axis State how many complex and real zeros the function has: f(x) = x^2 -10x +41 - ANSWER>>1. # of complex zeros = degree = 2 2. # of real zeros = plug into graphing calc = 0 3. Sketch graph with viewing window Irredusible Quadratic - ANSWER>>A quadratic that results in a imaginary number Use sqrt( b^2 -4ac) to determine if it is a irredusible quadratic Find all of the zeros and write a linear factorization of the function: f(x) = x^3 - 11x^2 + 9x -99 - ANSWER>>Zeros: 11, +- 3i Factorization: (x-11)(x-3i)(x+3i) For the following polynomial, one zero is given. Find the remaining zeros: f(x) = x^4 + 7x^2 - 144 Zero: 4i - ANSWER>>Zeros: 4i, -4i, 3, -3 Write the function as a product of linear and irredusible quadratic factors with all real coefficents f(x) = x^3 -14x^2 -8x -105 - ANSWER>>Zero: 15 Factor form: (x - 15)(x^2 + x + 7) Find the unique polynomial with real coefficents that meets these conditions: Degree = 4 Zeros: 1, -4, 2 - i; f(0) = -60 - ANSWER>>3x^4 - 3x^3 -33x^2 + 93x -60 The recipricol function - ANSWER>>1/x lim x --> 6+ f(x) = inf lim x --> inf f(x) = 0 lim x --> -inf f(x) = 0 Refer to graphs of rational functions for sketch #6 Describe how the graph of f(x) = 3x-4/x + 2 can be obtained by transforming the graph of the recipricol function g(x) = 1/x. Identify the horizontal and vertical asymototes and use limits to describe the cooresponding behavior. Sketch the graph of the function - ANSWER>>1. Write as vertex form synthetically divide the numerator by denominator 3 + -10/x+ 2 --> -10(1/x+2) + 3 From 1/x left 2 units, followed by a vertical stretch by a factor of 10, then a reflection across the x-axis, and then up 3 units 2. Identify asymptotes and their limits VA: x = -2 HA: y = 3 lim x --> -2 - f(x) = inf lim x --> -2 + f(x) = -inf lim x --> -inf f(x) = 3 lim x --> inf f(x) = 3 3. Sketch Refer to graphs of rational functions #7 Describe how the graph of f(x) = 3x-1/x+1 can be obtained by transforming the graph of the recipricol function g(x) = 1/x. Identify the horizontal and vertical asymototes and use limits to describe the cooresponding behavior. Sketch the graph of the function - ANSWER>>1. Write as vertex form synthetically divide the numerator by denominator 3 - 4/x+ 1 --> -4(1/x+1) + 3 From 1/x left 1 units, followed by a vertical stretch by a factor of 4, then a reflection across the x-axis, and then up 3 units 2. Identify asymptotes and their limits VA: x = -1 HA: y = 3 lim x --> -1 - f(x) = inf lim x --> -1 + f(x) = -inf lim x --> -inf f(x) = 3 lim x --> inf f(x) = 3 3. Sketch Refer to graphs of rational functions #8 How to find x intercepts of rational functions and their format - ANSWER>>Find x-int by setting the numerator = to 0 x-ints: x = #, x = # How to find y intercepts of rational functions and their format - ANSWER>>Find y-int by plugging 0 in for x y int: y = # How to find vertical asymptotes and their format - ANSWER>>Find by setting the denominator = to 0 Once found make sure that the numerator is not 0 when plugging it in for x. If it does then it is not a VA VA: x = # How to find horizontal asymptotes and their format - ANSWER>>Find by comparing the degrees of the numerator and denominator If numerator < denominator then y = 0 is a HA If numerator = denominator then y = LC num / LC denom If numerator > denominator then No Horizontal Asymptote, but need to look for End Behavior Asymptote HA: y = # How to find end behavior asymptotes - ANSWER>>1. Sythetically divide the numerator by the denominator and find the remainder 2. Drop the remainder and the rest of the division is the end behavior asymptotes Find the horizontal and vertical asymptotes of f(x). Use limits to describe the cooresponding behavior. f(x) = 5x^2 + 2 / x^2 + 4 - ANSWER>>VA: None HA: y = 5 lim x --> -inf f(x) = 5 lim x --> inf f(x) = 5 Graph the function f(x) = x+8 / x^2 -4x -32 Find all asymptotes. List x and y intercepts - ANSWER>>x-int: x = -8 y-int: y = -1/4 VA: x = -4, x= 8 lim x --> -4 - f(x) = inf lim x --> -4 + f(x) = -inf lim x --> 8 - f(x) = -inf lim x --> 8 + f(x) = inf HA: y = 0 lim x --> -inf f(x) = 0 lim x --> inf f(x) = 0 Refer to graph on Asymptotes and Intercepts # 3 Graph the function f(x) = 11x^2 + x -11 / x^2 -1 Find all asymptotes. List x and y intercepts - ANSWER>>x-int: x = 0.96, x = -1.05 y-int: y = 11 2. Multiply each term by LCD q+10 + q-11 = 1 3. Get all terms on 1 side 2q-2 =0 4. Solve for the zeros q = 1 5. Check for extraneous solutions q = 1 is a solution {1} Solve the equation and check your answer q + 4 q-5 1 ------- + ------ = ----- 2 2 2 - ANSWER>>1. Identify LCD --> 2 2. Multiply each term by LCD q+4 + q-5 = 1 3. Get all terms on 1 side 2q-2 =0 4. Solve for the zeros q = 1 5. Check for extraneous solutions q = 1 is a solution {1} Solve the equation algebraically r+3 = 28/r - ANSWER>>1. Identify LCD --> r 2. Multiply each term by LCD r^2 + 3r = 28 3. Get all terms on 1 side r^2 + 3r -28 =0 4. Solve for the zeros (r +7) (r -4) r = -7, r= 4 5. Check for extraneous solutions Both r=-7, and r=4 are a solutions {4,-7} Solve the equation algebraically x + 9x/x-8 = 72/x-8 - ANSWER>>1. Identify LCD --> x-8 2. Multiply each term by LCD x^2 -8x + 9x = 72 3. Get all terms on 1 side x^2 + x -72 =0 4. Solve for the zeros (x+9)(x-8) x = -9, x = 8 5. Check for extraneous solutions x = 8 is an extraneous solution x = -9 is a solution {-9} Solve algebraically 2 - (5/x-4) = (20/x^2 +4x) - ANSWER>>1. Identify LCD --> (x)(x+4) 2. Multiply each term by LCD 2x(x+4) - x = 4 3. Get all terms on 1 side 2x^2 + 7x - 4 =0 4. Solve for the zeros (2x-1)(x+4) x = 1/2, x = -4 5. Check for extraneous solutions x = -4 is an extraneous solution x = 1/2 is a solution {1/2} Solve the equation algebraically (7x / x+5) + (1/x-2) = (7 / x^2+3x-10) - ANSWER>>1. Identify LCD --> (x+5)(x-2) 2. Multiply each term by LCD 7x(x-2) + x+5 = 7 3. Get all terms on 1 side 7x^2 - 13x - 2 =0 4. Solve for the zeros (7x+1)(x-2) x = -1/7, x = 2 5. Check for extraneous solutions x = 2 is an extraneous solution x = -1/7 is a solution {-1/7} Solve: (2y/y+2) + (4/y) + 2 = (8/y^2 + 2y) - ANSWER>>1. Identify LCD --> y(y+2) 2. Multiply each term by LCD 2y(y) + 4(y+2) = 8 3. Get all terms on 1 side Determine the x value that causes the polynomial to a) be zero b) be positive c) be negative f(x) = (5x^2 + 9)(x-6)^2(x+2)^3 - ANSWER>>*Note: b/c 5x^2 +9 is squared inside the parentheses and squares can only result in non-negative numbers, there is no zero for this factor a) Zeros: x = 6, x = -2 b) f(x) > 0: (-2, 6) U (6, inf) c) f(x) < 0: (-inf, -2) 0 0 <------------|------------------------|-----------------> -2 6 Test conditions x = -3 (+)(+)(-) == - x = 0 (+)(+)(+) == + x = 7 (+)(+)(+) == + Factor the polynomial and solve the inequality using a sign chart f(x) = 4x^3 - 7x^2 -21x + 18 >= 0 - ANSWER>>1.Synthetically divide to find the zeros x = 3, x = -2, x = 3/4 2. Plot on sign chart and test given conditions x = 3 (-)(+) == - x = 0 (-)(-) == + x = 1 (-)(+) == - x = 4 (+)(+) == + 3. f(x) >= 0: [-2, 3/4] U [3, inf) When writing out a linear factorization with just factors ... - ANSWER>>make sure to include an a value that causes the y-intercepts to be comparable How to solve polynomial inequalities - ANSWER>>1. Find the Zeros and Undefined Values 2. Place Zeros on the sign chart 3. Test the "Regions" How to Find Undefined Values - ANSWER>>1. What makes the number underneath the radical negative If it makes a radical negative you use an in equality such as x < # to describe it, then proceed to cross out all the numbers behind it. 2. What makes the denominator zero Determine the real values of x that can cause the function to be a) Zero, b) Undefined, c) Positive, d) Negative f(x) = (x-1) / (5x+8)(x-6) - ANSWER>>1. Find the zeros (What makes the numerator 0) x-1 = 0 x = 1 2. Undefined (Negative roots, or 0 denominator) 5x+8 = 0 x = -8/5 x-6=0 x=6 3. Place values on sign chart and test regions 4. Determine what makes the function positive f(x) > 0: (-8/5, 1)U(6, inf) 5. Determine what makes the function negative f(x) < 0: (-inf, -8/5)U(1,6) Determine the real values of x that can cause the function to be a) Zero, b) Undefined, c) Positive, d) Negative f(x) = x sqrt(x+10) - ANSWER>>1. Find the zeros (What makes the numerator 0) x = 0 sqrt(x+10) = 0 x = -10 2. Undefined (Negative roots, or 0 denominator) x < -10 3. Place values on sign chart and test regions Cross out anything behind -10 4. Determine what makes the function positive f(x) > 0: (0, inf) 5. Determine what makes the function negative f(x) < 0: (-10, 0) Solve the inequality 1/x+4 + 1/x-8 <= 0 - ANSWER>>1. Identify LCD (x-8)(x+4) 2. Multiply each term by LCD 3. Zeros x = 2 4. Undefined x = 8, x = -4 5. Sign Chart 6. f(x) <= 0: (-inf, -4)U[2,8)