Applied Calculus
Exam Materials
Course Introduction
Applied Calculus introduces the foundational concepts of calculus with a focus on real-world applications across various fields such as business, economics, life sciences, and social sciences. The course covers topics including functions, limits, derivatives, and integrals, emphasizing their practical use in modeling and solving problems involving rates of change and accumulation. Students develop problem-solving skills through contextual examples and learn to interpret mathematical results in applied settings. This course is designed for students seeking a practical understanding of calculus rather than a theoretical approach.
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) Simplify: 7(x - y)- 4(y - x)
Answer: 11x - 11y
Q2) Joe and Sue have agreed to pool their savings when they have saved the same amount of money.Joe can save $125 a week but he must use his first $375 to pay off his car.Sue has been saving $75 a week for 13 weeks already.When will they pool their savings? How much will they each have saved?
Answer: 27 weeks; $3000
Q3) The time it takes an airplane to travel a given distance with a headwind (against the wind)can be calculated by dividing the distance by the difference of the speed of the airplane and the speed of the wind.Write an equation which calculates the time t it takes an airplane traveling at a speed r against a wind w to cover a distance d.Solve your equation for w.
Answer: 11ea83f6_62b3_42be_b5a5_85f195f3cd45_TB6578_11_TB6578_11 ; 11ea83f6_62b3_42bf_b5a5_33dbed304f70_TB6578_11_TB6578_11
Q4) Completely factor: 3ax + 9ay
Answer: 3a(x + 3y)
Q5) Solve for x: a(2 -x)= 3b
Answer: 11ea83f6_62b0_0e14_b5a5_8f1010bbc185_TB6578_11_TB6578_11
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Chapter 2: Applications and More Algebra
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Sample Questions
Q1) A manufacturer has 4000 units of product x in stock and is now selling it at $10 per unit.Next month the unit price will increase by $2.The manufacturer wants the total revenue received from the sale of the 4000 units to be no less than $45,000.What is the maximum number of units that can be sold this month?
Answer: 1500
Q2) 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
Q3) The absolute value of the difference of two numbers is equal to the absolute value of the difference of the two numbers in reverse order.
Answer: 11ea83f6_62be_8d70_b5a5_c9935fa6db2a_TB6578_11_TB6578_11 = 11ea83f6_62be_b481_b5a5_4d70bcbb05c4_TB6578_11_TB6578_11
Q4) a = 12; d = -3
A) 12, 9, 6, 3, 0
B) 15, 12, 9, 6, 3
C) 9, 6, 3, 0, -3
D) 12, 9, 5, 3, 0
Answer: A
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Chapter 3: Functions and Graphs
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Sample Questions
Q1) A rectangular sheet of metal has a length that is 2 less than 4 times the width.
(a)Write the area of the rectangular sheet as a function of the width.
(b)Without simplifying,write the ratio of the length of the sheet to the area of the sheet as a function of the width.
(c)Simplify the function you wrote in b.
(d)What kind of function is this?
(e)What is its domain out of context?
(f)What is its domain in the given context?
Answer: (a)a(w)= w(4w - 2)
(b)f(w)= 11ea83f6_62c9_b140_b5a5_f5b0b8397011_TB6578_11_TB6578_11
(c)f(w)= 11ea83f6_62c9_b141_b5a5_03ce1afeeb3e_TB6578_11_TB6578_11
(d)rational function
(e)all real numbers except 0 and 11ea83f6_62c9_d852_b5a5_d71aca326aa4_TB6578_11_TB6578_11 (f)w > 11ea83f6_62c9_d853_b5a5_d5c842187a78_TB6578_11_TB6578_11
Q2) Find the domain of the function f(x)= 6.
Answer: all real numbers
Q3) Find an equation of the plane that is parallel to the x,z-plane and that passes through the point (-3,2,5).
Answer: y = 2
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Chapter 4: Lines parabolas and Systems
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Sample Questions
Q1) Suppose f is a linear function with slope 5 and such that f(1)= 4.Find f(x).
Q2) A hobby store packages mixtures of different beads for sale.From wood beads,glass beads,and metal beads,the owner wants to prepare a mixture which will sell for $4.20 for a 200 count bag.The cost per bead of these beads is $0.01,$0.03,and $0.05,respectively.The number of wooden beads is to be four times the number of metal beads.How many of each type of bead will be in the final package?
Q3) A chemist has two solutions that contain different concentrations of hydrochloric acid.One is a 20% concentration and the other is a 12% concentration.How many cubic centimeters of each should he mix to obtain 100 cc with a concentration of 15.2%?
Q4) The stock price of a company has risen at the rate of $5.00 per month over the last year.On January 1 it was $75.00.Write an equation that shows this relationship.
Q5) Suppose f is a linear function with slope 2 and such that f(-3)= 8.Find f(x).
Q6) A swimming pool owner has two solutions which contain different concentrations of chlorine.One is a 5% solution and the other is a 10% solution.How many liters of each should she mix to obtain 8 liters with a 6% concentration?
Q7) Determine an equation of the vertical line that passes through the point (3,-6).
Q8) Suppose f is a linear function such that f(-2)= 5 and f(5)= 2.Find f(x).
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Chapter 5: Exponential and Logarithmic Functions
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Sample Questions
Q1) What is the sum of the Richter Scale measurement of an earthquake which is 250,000 times the intensity of a zero-level earthquake and an earthquake with intensity twice the intensity of a zero-level earthquake? Write as an expression involving logarithms.Simplify by combining logarithms and then use a calculator to evaluate.
Q2) Find x: log 100,000 = x
Q3) Assume your $2000 investment,which is guaranteed to triple every decade,has a one time $10 service charge.
(a)Make a table of the value of your investment without the service charge at each decade from 0 to 3.
(b)Make a table of the value of your investment with the service charge deducted at each decade from 0 to 3.
(c)How could you use a graph of (a)to make a graph of (b)?
Q4) An investment increases by 10% every year.Write a function for the factor of increase in the investment as a function of years.Use a graphing calculator to graph your function.Use the graph to estimate when the investment will double.
Q5) What power of 18 is 36?
Q6) Find x: ln x = 1
Q7) Assume that log 5 = 0.6690.Determine the value of log 500. To view all questions and flashcards with answers, click on the resource link above.
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Chapter 6: Mathematics of Finance
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Sample Questions
Q1) Suppose you wish to purchase a mine that will yield an annual return of $32,000 for 12 years,after which the mine will have no value.You want to earn 8% annually on this investment and also set up a sinking fund to replace the purchase price.If money is placed in the fund at the end of each year and earns 6.2% compounded annually,how much should you pay for the mine?
Q2) At what nominal rate of interest,compounded quarterly,will money double in 10 years?
Q3) A trust fund for a 8-year-old child is being set up by a single payment so that when the child is 20 there will be $12,000.If the fund earns interest at the rate of 6.5% compounded semiannually,how much money should be paid into the fund initially?
Q4) Given that the effective rate is 9%,determine the interest rate r which is compounded continuously to give an effective rate of 9%.
Q5) Suppose that you can invest $5000 in a business that guarantees you the following cash flows: $3000 at the end of 2 years,$2000 at the end of 4 years,and $2000 at the end of 6 years.Assuming an interest rate of 6% compounded monthly,find the present value of the cash flows.Is the investment profitable?
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Chapter 7: Matrix Algebra
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Sample Questions
Q1) Find the income earned by the gas industry and its cost for the other forms of energy it uses.Then calculate its profit.
Q2) An electronics store has 35 televisions,15 VCRs,and 25 CD players in stock.If the value of each television is $400,each VCR is $200 and each CD player is $150,use the matrix operations on a graphing calculator to find the total value of the electronics store's inventory.
Q3) Find the income earned by the nuclear industry and its cost for the other forms of energy it uses.Then calculate its profit.
Q4) Show that A + O = A.
Q5) If the orders of the matrices Q,R,and S are 4 × 5,3 × 2,and 5 × 3,respectively,then the order of QRS is
A) 4 × 5.
B) 3 × 5.
C) 3 × 3.
D) 2 × 4.
E) none of the above
Q6) What is the change in sales between 1994 and 1995?
Q7) What is the change in sales between 1994 and 1996?
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Q8) Use the matrix operations on a graphing calculator to show that A + C = C + A.
Chapter 8: Linear Programming
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Sample Questions
Q1) Use the simplex method to solve the following problem: United Blimpo Co.produces two types of exercise devices,regular and heavy-duty,each of which requires in its manufacture the use of two machines,A and B.A regular model requires the use of machine A for 2 hours and machine B for 3 hours.A heavy-duty model requires the use of machine A for 3 hours and machine B for 3 hours.Machine A can be used at most 18 hours a day and machine B can be used at most 21 hours a day.If the profits on the regular and heavy-duty models are $20 and $15,respectively,and United Blimpo Co.can sell all it produces,how many of each model should be produced per day in order to realize maximum profit? What is the maximum profit per day?
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.
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Chapter 9: Introduction to Probability and Statistics
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Sample Questions
Q1) A grocery store displays 100 green peppers in 10 rows of 10 each.It is quite likely that customer A's choice of the 2 best peppers to buy would differ from customer B's choice,even if the peppers were more or less identical.In how many ways (sample points)could the first customer to see the green pepper array choose 2 peppers to buy?
Q2) Find the probability of rolling the same number in four throws of a fair die.
Q3) How many distinguishable horizontal arrangements of all the letters in BOOKS are possible?
Q4) Fourteen children from a day care center are the first to arrive at a movie theater that seats 600.They decide to pull a trick on their chaperons by hiding under the seats,one child per seat.If we consider the 14 seats selected to be one "hiding place",how many potential hiding places would the chaperons conceivably have to search? (Leave answer in factorial form.)
Q5) A quiz contains four multiple-choice problems.Each problem has five choices for the answer,but only one of them is correct.If a student randomly guesses the answer to each problem,find the probability that the student gets exactly three correct answers.
Q6) The plane arrives on time in L.A.even though it left Denver later than scheduled. To view all questions and flashcards with answers, click on the resource link above. Page 11
Chapter 10: Additional Topics in Probability
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Sample Questions
Q1) A manufacturer produces light bulbs,of which 3% are defective.From a production run of 5000 bulbs,three bulbs are randomly selected and each is tested.Find,to three decimal places,the probability that the sample contains exactly one defective bulb.Assume that the three trials are independent and that the number of defective bulbs in the sample has a binomial distribution.
Q2) A college dining hall has available two fruit juices for breakfast: orange and grapefruit.One hundred students drink juice on a daily basis.It is found that a student will not drink grapefruit juice on two successive days,and if a student drinks orange juice one day,then the following day the student is equally likely to drink orange juice as to drink grapefruit juice.If 40 of the regular juice drinkers drink orange juice on Monday,how many can be expected to drink orange juice on Wednesday?
A) 10
B) 30
C) 50
D) 60
E) 90
Q3) 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.
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Chapter 11: Limits and Continuity
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Sample Questions
Q1) Use the definition of continuity to show f(x)= 2 is continuous at x = 5.
Q2) Solve: (x - 1)(x + 2)(x - 3) 0
Q3) Consumers will purchase q units of a certain product when the price is 340 - 2q dollars per unit.How many units must be sold in order that sales revenue be no less than $14,000?
Q4) Solve: (x + 6)(x - 1)(x - 4) 0
Q5) 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.
Q6) Use the definition of continuity to show f(x)= 2x + 3 is continuous at x = -1.
Q7) 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 = 3.
Q8) Find the value(s)of x for which f(x)= 10 is discontinuous.
Q9) Find: \(\lim_{x \rightarrow 5}\) \(\frac{x^{2}}{h}\)
Page 13
Q10) 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.
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Chapter 12: Differentiation
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Sample Questions
Q1) If the revenue function is given by R(x)= 10x,find the derivative of the revenue function (this is called the marginal revenue).
Q2) If the revenue function is given by R(x)= 6x + 4,find the derivative of the revenue function (this is called the marginal revenue).
Q3) A cone has a height of 10 inches.It is being filled with ice cream.Find the rate of change of its volume with respect to the radius of the base of the cone.Evaluate this rate of change when the radius is 12 inches.
Q4) If the revenue function is given by R(x)= 5x - 7,find the derivative of the revenue function (this is called the marginal revenue).
Q5) If a function is differentiable at x = a,then it is continuous at x = a.
A)True
B)False
Q6) Differentiate: g(x)= (x - 8)(2x - 7)(x + 5)
Q7) If the revenue function is given by R(x)= 11x + 9,find the derivative of the revenue function (this is called the marginal revenue).
Q8) Find the rate of change of the area of a square with respect to its one side.Evaluate it when side = 4 units.
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Chapter 13: Additional Differentiation Topics
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Sample Questions
Q1) 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.
Q2) The total cost for a product is given by C(x)= 900 + 8 ln(3x + 7).Find the rate of change of C''(x).
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) The total cost for a product is given by C(x)= 36 + 5 ln(3x + 9).Find the rate of change of C''(x).
Q6) Determine the point elasticity of the demand equation pq = 81,where p > 0 and q > 0.
Q7) Determine the point elasticity of the demand equation p = -3q + 120,where p > 0 and q > 0.
Q8) 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.
Q9) Find an equation of the tangent line to the curve y = ln(x + 3)when x = -2.
Page 15
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Chapter 14: Curve Sketching
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Sample Questions
Q1) An open box is to be made by cutting equal squares from each corner of a 20-in.square piece of cardboard and then folding up the sides.Find the length of the side of the square that must be cut out if the volume of the box is to be maximized.
Q2) Find two numbers whose sum is 30 and whose product is a maximum.
Q3) A company owns an apartment building containing 80 units.If the company charges $300 per month rent,then all units can be rented out,and for every increase of $10 per month in rent,the company will lose one customer.What rent should be charged to maximize revenue?
Q4) A rectangular plot of ground is to be fenced in so that it has an area of 15,000 square feet.The plot is to be divided in half with a fence parallel to one pair of sides.Find the dimensions of the plot if the least amount of fencing is to be used.
Q5) Find all the critical values of f(x)= x ln x.
Q6) 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.
Q7) 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.
Page 16
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Chapter 15: Integration
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Sample Questions
Q1) Find y subject to the given conditions: y'' = 6x - 2; y'(1)= 2; y(1)= 2.
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) By using differentials,an approximation of ln(1.03)is
A) -0.01.
B) 0.01.
C) 0.02.
D) 0.03.
E) 0.04.
Q4) Use differentials to approximate ln(0.99).
Q5) 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).
Q6) Use differentials to approximate: ln(0.95).
Q7) If y'' = 6x + 2 and y'(1)= 2 and y(1)= 2,find y.
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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 = 25 - 0.005q,where p is the price per unit (in dollars)for q units.If its supply equation is p = 1 + 0.03q,then the consumers' surplus when market equilibrium is established is
A) $1176.49
B) $1250.87
C) $1500.
D) $1675.30
E) $1800.
Q2) If 65% of the initial amount of a radioactive sample has decayed after 5 hours,find the decay constant and the half life of the element.
Q3) Suppose that the population of a city in 1992 was 52,400 and in 1996 was 55,300.If the exponential law of growth is assumed,what is the expected population in 2007?
Q4) A company performed a marketing study and concluded that there is a potential of 100,000 customers for its product.At the beginning of the study there were 40,000 customers that used the product,and one year later there were 50,000 customers.Determine the number N of customers t years after the beginning of the study if N follows logistic growth.
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18
Chapter 17: Continuous Random Variables
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Sample Questions
Q1) Suppose X is normally distributed with mean 80 and standard deviation 15.Without using tables,determine the approximate value of P(65 < X < 95).
A) 0.997
B) 0.95
C) 0.68
D) 0.50
E) 0.33
Q2) If Z has a standard normal distribution,find P(0 < Z < 1.48).
Q3) The life expectancy (in years)of patients after they have contracted a certain disease is exponentially distributed with k = 0.4.Find the mean life expectancy and the standard deviation.
Q4) If X is normally distributed with = 100 and = 10,find P(X < 118).
Q5) If Z has a standard normal distribution,find P(-2 Z -1).
Q6) If X is a normal random variable with = 30 and = 5,determine the Z-value that corresponds to X = 28.
Q7) If Z has a standard normal distribution,find P(|Z| > 1.75).
Q8) If X is normally distributed with = 12 and = 2,find P(12 X 15).
Page 19
Q9) If Z has a standard normal distribution,find P(-0.65 < Z < 1.92).
Q10) If Z has a standard normal distribution,find P(Z < -1.23).
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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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