Mathematics for Life Sciences
Exam Materials
Course Introduction
Mathematics for Life Sciences introduces students to the fundamental mathematical concepts and quantitative methods essential for understanding and analyzing biological phenomena. Topics covered include basic algebra, calculus, probability, and statistics, all applied in the context of life sciences. Students learn to model biological systems, interpret experimental data, and use mathematical reasoning to solve problems in areas such as population dynamics, genetics, epidemiology, and ecology. This course equips students with the analytical tools needed to explore the quantitative aspects of modern biology and prepares them for advanced study and research in the life sciences.
Recommended Textbook
Introductory Mathematical Analysis for Business Economics and the Life and Social Sciences 14th
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Chapter 1: Review of Algebra
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Sample Questions
Q1) A boat with a speed of 12 mph travels 8 miles downstream in a current of c miles an hour in the same time a boat going the same speed travels 4 miles upstream.Write an equation describing this situation and find the speed of the current.
Answer: 11ea83f6_62b3_1baa_b5a5_3f1d3dffe696_TB6578_11_TB6578_11 = 11ea83f6_62b3_1bab_b5a5_5d39d51450bb_TB6578_00_TB6578_00 ; 4 mph
Q2) Solve 2q - 1 = 2(1 - q)
Answer: q = 11ea83f6_62af_72be_b5a5_b706d3df5f66_TB6578_11_TB6578_11
Q3) A number squared is four more than the opposite of three times the number.What is the number?
Answer: The number is -4 or 1.
Q4) Simplify: -4 - (-10)
Answer: 6
Q5) Simplify: (-4)(-2)
Answer: 8
Q6) Solve the equation: 1.4( 0.2x + 1.3)= 0.5(0.1x + 4.56)
Answer: x = 2
Q7) Solve: x = 2x - (6 - x)
Answer: x = 3
Page 3
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Chapter 2: Applications and More Algebra
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Sample Questions
Q1) A retired couple has $192,000 to invest.A safe investment yields 9.5% per year,but a riskier investment yields 12.5% per year.How much should they invest at each rate to earn $21,000 per year?
Answer: $100,000 in the safe investment and $92,000 in the riskier investment
Q2) The absolute value of the product of two numbers is equal to the product of the absolute values of the numbers.
Answer: 11ea83f6_62be_db96_b5a5_5d5729b6b584_TB6578_11_TB6578_11 = 11ea83f6_62be_db97_b5a5_259573ce2f13_TB6578_11_TB6578_11 11ea83f6_62be_db98_b5a5_015733293e00_TB6578_11_TB6578_11
Q3) A computer keyboard retails for $48.54,which includes a markup of 40% for the retailer and a markup of 20% for the wholesaler.What did the keyboard cost the wholesaler?
Answer: Approximately $28.89
Q4) Write the income range for line 5 as an inequality.Write the inequality that represents the tax owed.
Answer: I > 263,750; T > 84,020.50
Q5) Solve: Three times x plus four is a number -4 units from zero. Answer: A DISTANCE CANNOT BE NEGATIVE,SO THERE IS NO SOLUTION.
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Chapter 3: Functions and Graphs
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Sample Questions
Q1) List the graphical transformations,in the correct order,that must be done to y = f(x)to produce the graph of y = -f(x - 5)+ 4.
Answer: Order: 1.Move 5 right
2.Reflect about the x-axis
3.Move 4 up
Or any order in which 2.is before 3.
Q2) If f(x)= 7,find f(14).
Answer: 7
Q3) In June Gail decided to save $20.00 a week.She saved for 14 weeks and then for 14 weeks she spent $20.00 a week on gifts.Write an absolute-value function to represent the amount of money Gail had in savings.
Answer: f(x)= - 11ea83f6_62ca_74a3_b5a5_31d0951ecf0c_TB6578_11_TB6578_11 + 280
Q4) Bill has charged $2300 on his charge card.He plans to pay $60 a month on his charge cards.Write an equation to represent the amount he owes excluding any finance charges,and identify the intercepts.
Answer: y = -60x + 2300; x-intercept 11ea83f6_62d0_1999_b5a5_1fddfc5cad5f_TB6578_11_TB6578_11 ; y-intercept (0,2300)
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5
Chapter 4: Lines parabolas and Systems
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Sample Questions
Q1) Suppose f is a linear function with slope -3 and f(2)= 5.Find f(x).
A) f(x) = -3x + 2
B) f(x) = -3x + 5
C) f(x) = -3x + 7
D) f(x) = -3x + 11
E) f(x) = -3x + 13
Q2) A California homeowner with a 30-year fixed rate mortgage pays $170,000 after 5 years and $266,000 after 9 years.Use a graphing calculator to graph the resulting equation and determine how much the homeowner will have paid when the mortgage is paid off in 30 years.
Q3) A general linear equation of the line that has slope -2 and that passes through the point (1,3)is
A) 2x + y + 1 = 0.
B) 2x - y + 1 = 0.
C) 2x + y - 7 = 0
D) 3x + y + 5 = 0
E) 2x + y - 5 = 0
Q4) Determine the equation of the line which is perpendicular to 2x - y + 3 = 0 and has y-intercept 6.
Q5) Find the equation of a line having slope = 0.5 and x-intercept = 1.8.
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Chapter 5: Exponential and Logarithmic Functions
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Sample Questions
Q1) Of the following the best approximation of e is
A) 1.4.
B) 1.8.
C) 2.3.
D) 2.7.
E) 3.1.
Q2) The number of bacteria in a culture is doubling every hour.Currently the culture has 128 bacteria.Make a table of values for the number of bacteria present each hour for 0 to 4 hours.For each hour write an expression for the number of bacteria as a product of 128 and a power of 2.Use the expressions to make an entry in your table for the number of bacteria after t hours.Write a function N for the number of bacteria after t hours.
Q3) If $500 is invested for 3 years at 7% compounded semiannually,then the compound interest at the end of the period is
A) $112.52.
B) $114.63.
C) $120.37.
D) $122.56.
E) $150.36.
Q4) Find x: ln x = 1
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Chapter 6: Mathematics of Finance
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Sample Questions
Q1) Given a payment of $800 per quarter for five years,use a graphing calculator to graph the present value A as a function of the interest rate per quarter r.Determine the nominal interest rate if the present value of the annuity is $14,000.
Q2) A company repays a $50,000 loan by paying 10% of the outstanding loan each month.Find the amount the company pays in the fourth and twentieth months.
Q3) Suppose an initial investment grows from $12,000 to $30,000 over ten years.First find the nominal rate compounded quarterly and then find the equivalent effective rate.
Q4) A debt of $2000 due four years from now is to be repaid by a payment of $1000 now and a second payment at the end of two years.How much should the second payment be if the interest rate is 5% compounded annually?
Q5) If $1,000 is invested at a nominal rate of 4% compounded quarterly for 5 years,find the compound amount.
Q6) A man bought a stereo system for $3500 and agreed to pay off the loan by making monthly payments of $79.If the store charges an interest rate of 11% compounded monthly,how many months will it take to pay off the debt?
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Chapter 7: Matrix Algebra
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Sample Questions
Q1) EF
Q2) Use the matrix operations on a graphing calculator to find the total demand for the four types of energy from only the consumers.
Q3) What is the change in sales between 1994 and 1996?
Q4) Use the matrix operations on a graphing calculator to find her income if increasing it by 20% would cause an increase in profit of 50%.(Round to the nearest thousand dollars.)
Q5) Find the income earned by the gas industry and its cost for the other forms of energy it uses.Then calculate its profit.
Q6) Give an example of 2 matrices A and B where A × B B × A.
Q7) What would her income be if increasing it threefold would cause her profit to be five times as large? (Round to the nearest thousand dollars.)
Q8) Use the matrix operations on a graphing calculator to find her income if increasing it two times would cause her profit to be three times as large.(Round to the nearest thousand dollars.)
Q9) A boom in the economy causes an across the board increase of 10% in energy demands in the area.Calculate a new demand matrix which reflects this increase.
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Chapter 8: Linear Programming
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Sample Questions
Q1) Maximize
Z = 4x + y subject to -x + y 2
3x + y 18
x,y 0.
Q2) Maximize Z = 2x - 3y subject to
2x + y 1
x - y 1 x,y 0.
Also find the corner point where the value of z is attained.
Q3) To make some extra money,you make two types of picture frames,type A and type B,for sale.You have an initial start-up expense of $75.The production cost for type A is $3.60 per frame,and the production cost for type B is $5.20 per frame.The price for type A is $6.00 per frame and the price for type B is $10.00 per frame.Let x be the number of type A and y be the number of type B produced and sold.Write an inequality describing revenue less than cost.Solve the inequality and describe the region.Also,describe what this means in terms of frames.
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Chapter 9: Introduction to Probability and Statistics
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Sample Questions
Q1) Two cards are randomly drawn with replacement from a standard deck of 52 cards.Find the probability of drawing,in order,the queen of hearts and a diamond.
Q2) Ten tickets numbered from 1 to 10 are placed in a hat.If two tickets are randomly drawn with replacement,find the probability the sum is 15.
Q3) A United Nations diplomat can travel from Washington,D.C.to New York City in 6 ways.Depending on the urgency of the trip,she can fly by helicopter,fly by plane,drive the car to a suburban subway station and then take the subway the rest of the way,drive by limousine,take the train,or borrow a friend's yacht and sail most of the way.When she returns to Washington,D.C.,the same modes of transportation are available.How many round trip transportation combinations are available to her?
Q4) Find the probability of rolling the same number in four throws of a fair die.
Q5) An office worker spends each coffee break at the same deli.This deli has the same 8 varieties of coffee every day.To add variety and excitement to his or her day,the office worker decides to randomly choose a coffee variety in the morning and then randomly choose a variety in the afternoon.How many morning coffee - afternoon coffee choices are there?
Q6) In how many ways is it possible to answer a six-question true-false examination.
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Chapter 10: Additional Topics in Probability
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Sample Questions
Q1) P(X = 2); n = 5,p = 0.70
A) 0.132
B) 0.198
C) 0.700
D) 0.464
Q2) 25% of a video store's customers are female.Let X be the number of women among the next 5 customers,and find the distribution of X.
Q3) If a person watches a certain TV daily evening news program on one evening,then the probability that the person watches that program the next evening is 0.7.However,if the person does not watch the program one evening,then the probability that the person watches the program the next evening is 0.2.
(a)If the person watches the program on Monday,what is the probability that the person watches the program on Wednesday?
(b)If 20% of the population watches the program on Thursday,what percentage can be expected to watch on Friday?
Q4) A random variable X has a distribution given by f(0)= 0.3,f(1)= 0.6,f(2)= 0.1.(a)Find .(b)Find .
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Chapter 11: Limits and Continuity
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Sample Questions
Q1) Find: \(\lim_{x \rightarrow 5}\) \(\frac{x^{2}}{h}\)
Q2) Solve: (x + 6)(x - 1)(x - 4) 0
Q3) Find: lim 15 x 8
Q4) Suppose that the rent for an apartment in a city with rent control is $480 per month.Show that the rent function R(t)= 480 is continuous at t = 7.
Q5) Solve: (x - 1)(x + 2)(x - 3) 0
Q6) The solution of (x + 9)(x - 1)(x - 5) 0 is
A) x -9.
B) -9 x 1.
C) x 5.
D) x -9, 1 x 5.
E) -9 x 1, x 5.
Q7) Suppose that the rent for an apartment in a city with rent control is $740 per month.Show that the rent function R(t)= 740 is continuous at t = 3.
Q8) Suppose that the rent for an apartment in a city with rent control is $540 per month.Show that the rent function R(t)= 540 is continuous at t = 5.
Page 13
Q9) Use the definition of continuity to show f(x)= 2x + 3 is continuous at x = -1.
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Chapter 12: Differentiation
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Sample Questions
Q1) A stereo company sells 300 car stereos per month at a price of $280 per car stereo.Market research indicates that they can sell one additional stereo for each $1 they reduce the price.In this case the total revenue is given by: R(x)= (300 + x)(280 - x)where x is the number of additional stereos above 300.Find the marginal revenue function using the product rule.
Q2) If the revenue function is given by R(x)= 5x - 7,find the derivative of the revenue function (this is called the marginal revenue).
Q3) The concentration of acid in a buffer solution is given by A = 0.8.Graph this function on your graphing calculator and describe its slope.
Q4) A bus company will plan a group trip for groups of 10 or larger.If the group is exactly 10,the cost is $50 per person.The company offers to discount each person's ticket by $2 for each addition to the group above 10.The revenue for the bus company is given by: R(x)= (10 + x)(50 - 2x)where x is the number of additional people above 10.Find the marginal revenue function using the product rule.
Q5) Find the rate of change of the area of a square with respect to its one side.Evaluate it when side = 4 units.
Q6) Differentiate: g(x)= (x - 8)(2x - 7)(x + 5)
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Chapter 13: Additional Differentiation Topics
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Sample Questions
Q1) Find an equation of the tangent line to the curve y = ln(x + 3)when x = -2.
Q2) If the position of a particle moving along the x-axis at a given time is given by x(t)= 5t + 11,find all the higher-order derivatives of this function.
Q3) The total cost for a product is given by C(x)= 212 + 3 ln(2x + 1).Find the rate of change of C''(x).
Q4) The total cost for a product is given by C(x)= 1750 + 300 ln(x + 2).Find the rate of change of C''(x).
Q5) Determine the point elasticity of the demand equation p = -3q + 120,where p > 0 and q > 0.
Q6) The total cost for a product is given by C(x)= 36 + 5 ln(3x + 9).Find the rate of change of C''(x).
Q7) The total cost for a product is given by C(x)= 900 + 8 ln(3x + 7).Find the rate of change of C''(x).
Q8) Determine the point elasticity of the demand equation pq = 81,where p > 0 and q > 0.
Q9) Find the point elasticity of the demand equation p = 15 - 0.3q for the value q = 300,and determine whether demand is elastic,inelastic or has unit elasticity.
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Chapter 14: Curve Sketching
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Sample Questions
Q1) A cable TV company has 500 customers paying $20 each month.For each $1 reduction in price,the company attracts 50 more customers.Find the price that yields maximum revenue.
Q2) Find two positive numbers whose product is 25 and whose sum is a minimum.
Q3) A cable TV company has 200 customers paying $10 each month.For each $1 reduction in price,the company attracts 50 more customers.Find the price that yields maximum revenue.
Q4) A manufacturer has to produce annually 1000 units of a product that are sold at a uniform rate during the year.The production cost for each unit is $400,and carrying costs (insurance,interest,storage,and so on)are estimated to be 5% of the value of average inventory.Set-up costs per production run are $64.Find the economic lot size.
Q5) A company owns an apartment building containing 100 units.If the company charges $400 per month rent,then all units can be rented out,and for every increase of $20 per month in rent,the company will lose one customer.What rent should be charged to maximize revenue?
Q6) Find two numbers whose sum is 50 and such that the product of one of them and five more than the other is a maximum.
Q7) Find two numbers whose sum is 30 and whose product is a maximum.
Page 16
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Chapter 15: Integration
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Sample Questions
Q1) Use differentials to approximate ln(0.99).
Q2) Use differentials and C = 250 + 0.30x to approximate the change in the cost C (in dollars)to produce x pounds of candy if the number of pounds of candy increases from 10 pounds to 10.6 pounds.
Q3) Use differentials to approximate: ln(0.95).
Q4) If y'' = 6x + 2 and y'(1)= 2 and y(1)= 2,find y.
Q5) If y = x ln x,then dy =
A) 1 + ln x.
B) (1 + ln x) dx.
C) x + ln x.
D) (x + ln x) dx.
E) none of the above
Q6) If the marginal cost (in dollars)for a company is C'(x)= 200,and the cost for producing 8 units is $2250,that is C(8)= 2250,then find C(x).
Q7) Find y subject to the given conditions: y'' = 6x - 2; y'(1)= 2; y(1)= 2.
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17
Chapter 16: Methods and Applications of Integration
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Sample Questions
Q1) The demand equation for a certain product is p = 30 - 0.1q,where p is the price per unit (in dollars)for q units.If its supply equation is p = 0.15q + 5,find the consumers' surplus and the producers' surplus when market equilibrium has been established.
Q2) In a city whose population is 30,000,an outbreak of flu occurs.A researcher studying the spread of the flu determines from the surrounding hospitals and the health department that there where 600 infected persons one week ago.This week there are 1,000 infected persons.Assuming logistic growth,estimate the number of infected people 3 weeks from now.
Q3) The population of a certain city was 1,500,000 in the year 1580.The population grew by 5% each year of the next 15 years.The population declined by 10% each year for the rest of the century.What was the population of the city in the year 1600?
Q4) If 55% of the initial amount of a radioactive sample remains after 10 hours,find the decay constant and the half-life of the element.
Q5) Find the exact area of the region bounded by the graphs of y = -x,y = 2x,y = 1,and y = 2.Also sketch the region.
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Chapter 17: Continuous Random Variables
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Sample Questions
Q1) If X is normally distributed with = 10 and = 2,find P(4 X 12).
Q2) If X is normally distributed with = 25 and = 5,find P(20 X 35).
Q3) If Z has a standard normal distribution,find P(|Z| > 1.75).
Q4) If Z has a standard normal distribution,state the mean and standard deviation of Z.
Q5) If Z has a standard normal distribution,find P(2.5 Z 3).
Q6) If X is normally distributed with = 100 and = 10,find P(X < 118).
Q7) If X is normally distributed with mean and standard deviation ,then what is the approximate probability that - 2 X + 2 ?
A) 0.5
B) 0.68
C) 0.95
D) 0.997
E) 0.999
Q8) If X is a normal random variable with = 30 and = 5,determine the Z-value that corresponds to X = 28.
Q9) If X is normally distributed with = 18 and = 2,find P(11 X 16).
Q10) If Z has a standard normal distribution,find P(Z > -1.81).
Page 19
Q11) If Z has a standard normal distribution,find P(-0.65 < Z < 1.92).
Q12) If Z has a standard normal distribution,find P(0 < Z < 1.48).
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Chapter 18: Multivariable Calculus
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Q1) An open rectangular cardboard box is to have a volume of 4 cubic feet.Find the dimensions of the box so that the amount of cardboard is minimized.
Q2) Use the method of Lagrange multipliers to determine the critical points of f(x,y)= x + 2y subject to the constraint xy = 8.
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