A department store issues its own credit card, with an interest rate of 2% per month. Explain why this is not the same as an annual rate of 24%. What is the effective annual rate?
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Appendix A
Problems for Appendix A
Appendix B
Problems for Appendix B
1
REVIEW PROBLEMS FOR CHAPTER ONE
1.1
WHAT IS A FUNCTION?
1.10
PERIODIC FUNCTIONS
1.2
LINEAR FUNCTIONS
1.3
AVERAGE RATE OF CHANGE AND RELATIVE CHANGE
1.4
APPLICATIONS OF FUNCTIONS TO ECONOMICS
1.5
EXPONENTIAL FUNCTIONS
1.6
THE NATURAL LOGARITHM
1.7
EXPONENTIAL GROWTH AND DECAY
1.8
NEW FUNCTIONS FROM OLD
1.9
PROPORTIONALITY AND POWER FUNCTIONS
2
REVIEW PROBLEMS FOR CHAPTER TWO
2.1
INSTANTANEOUS RATE OF CHANGE
2.2
THE DERIVATIVE FUNCTION
2.3
INTERPRETATIONS OF THE DERIVATIVE
2.4
THE SECOND DERIVATIVE
2.5
MARGINAL COST AND REVENUE
3
REVIEW PROBLEMS FOR CHAPTER THREE
3.1
DERIVATIVE FORMULAS FOR POWERS AND POLYNOMIALS
3.2
EXPONENTIAL AND LOGARITHMIC FUNCTIONS
3.3
THE CHAIN RULE
3.4
THE PRODUCT AND QUOTIENT RULES
3.5
DERIVATIVES OF PERIODIC FUNCTIONS
4
REVIEW PROBLEMS FOR CHAPTER FOUR
4.1
LOCAL MAXIMA AND MINIMA
4.2
INFLECTION POINTS
4.3
GLOBAL MAXIMA AND MINIMA
4.4
PROFIT, COST, AND REVENUE
4.5
AVERAGE COST
4.6
ELASTICITY OF DEMAND
4.7
LOGISTIC GROWTH
4.8
THE SURGE FUNCTION AND DRUG CONCENTRATION
5
REVIEW PROBLEMS FOR CHAPTER FIVE
5.1
DISTANCE AND ACCUMULATED CHANGE
5.2
THE DEFINITE INTEGRAL
5.3
THE DEFINITE INTEGRAL AS AREA
5.4
INTERPRETATIONS OF THE DEFINITE INTEGRAL
5.5
TOTAL CHANGE AND THE FUNDAMENTAL THEOREM OF CALCULUS
5.6
AVERAGE VALUE
6
REVIEW PROBLEMS FOR CHAPTER SIX
6.1
ANALYZING ANTIDERIVATIVES GRAPHICALLY AND NUMERICALLY
6.2
ANTIDERIVATIVES AND THE INDEFINITE INTEGRAL
6.3
USING THE FUNDAMENTAL THEOREM TO FIND DEFINITE INTEGRALS
6.4
APPLICATION: CONSUMER AND PRODUCER SURPLUS
6.5
APPLICATION: PRESENT AND FUTURE VALUE
6.6
INTEGRATION BY SUBSTITUTION
6.7
INTEGRATION BY PARTS
7
REVIEW PROBLEMS FOR CHAPTER SEVEN
7.1
DENSITY FUNCTIONS
7.2
CUMULATIVE DISTRIBUTION FUNCTIONS AND PROBABILITY
7.3
THE MEDIAN AND THE MEAN
8
REVIEW PROBLEMS FOR CHAPTER EIGHT
8.1
UNDERSTANDING FUNCTIONS OF TWO VARIABLES
8.2
CONTOUR DIAGRAMS
8.3
PARTIAL DERIVATIVES
8.4
COMPUTING PARTIAL DERIVATIVES ALGEBRAICALLY
8.5
CRITICAL POINTS AND OPTIMIZATION
8.6
CONSTRAINED OPTIMIZATION
9
REVIEW PROBLEMS FOR CHAPTER NINE
9.1
MATHEMATICAL MODELING: SETTING UP A DIFFERENTIAL EQUATION
9.2
SOLUTIONS OF DIFFERENTIAL EQUATIONS
9.3
SLOPE FIELDS
9.4
EXPONENTIAL GROWTH AND DECAY
9.5
APPLICATIONS AND MODELING
9.6
MODELING THE INTERACTION OF TWO POPULATIONS
9.7
MODELING THE SPREAD OF A DISEASE
10
REVIEW PROBLEMS FOR CHAPTER TEN
10.1
GEOMETRIC SERIES
10.2
APPLICATIONS TO BUSINESS AND ECONOMICS
10.3
APPLICATIONS TO THE NATURAL SCIENCES
Textbook Solutions for Applied Calculus
Chapter Appendix B Problem 10
Question
A bank account is earning interest at 6% per year compounded continuously. (a) By what percentage has the bank balance in the account increased over one year? (This is the effective annual rate.) (b) How long does it take the balance to double? (c) For a continuous interest rate r, find a formula for the doubling time in terms of r.
Solution
The first step in solving Appendix B problem number 10 trying to solve the problem we have to refer to the textbook question: A bank account is earning interest at 6% per year compounded continuously. (a) By what percentage has the bank balance in the account increased over one year? (This is the effective annual rate.) (b) How long does it take the balance to double? (c) For a continuous interest rate r, find a formula for the doubling time in terms of r.
From the textbook chapter Problems for Appendix B you will find a few key concepts needed to solve this.
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Title
Applied Calculus 5
Author
Deborah Hughes-Hallett Patti Frazer Lock Andrew M. Gleason Daniel E. Flath, Sheldon P. Gordon, David O. Lomen, David Lovelock, & 7 more
ISBN
9781118174920
A bank account is earning interest at 6% per year compounded continuously. (a) By what
Chapter Appendix B textbook questions
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Chapter 0: Problem 1 Applied Calculus 5
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Chapter 0: Problem 2 Applied Calculus 5
A deposit of $10,000 is made into an account paying a nominal yearly interest rate of 8%. Determine the amount in the account in 10 years if the interest is compounded: (a) Annually (b) Monthly (c) Weekly (d) Daily (e) Continuously
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Chapter 0: Problem 3 Applied Calculus 5
A deposit of $50,000 is made into an account paying a nominal yearly interest rate of 6%. Determine the amount in the account in 20 years if the interest is compounded: (a) Annually (b) Monthly (c) Weekly (d) Daily (e) Continuously
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Chapter 0: Problem 4 Applied Calculus 5
Use a graph of y = (1+0.07/x)x to estimate the number that (1 + 0.07/x)x approaches as x. Confirm that the value you get is approximately e0.07.
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Chapter 0: Problem 5 Applied Calculus 5
(a) Find (1 + 0.04/n)n for n = 10,000, and 100,000, and 1,000,000. Use the results to predict the effective annual rate of a 4% annual rate compounded continuously. (b) Confirm your answer by computing e0.04.
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Chapter 0: Problem 6 Applied Calculus 5
Find the effective annual rate of a 6% annual rate, compounded continuously.
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Chapter 0: Problem 7 Applied Calculus 5
What nominal annual interest rate has an effective annual rate of 5% under continuous compounding?
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Chapter 0: Problem 8 Applied Calculus 5
What is the effective annual rate, under continuous compounding, for a nominal annual interest rate of 8%?
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Chapter 0: Problem 9 Applied Calculus 5
(a) Find the effective annual rate for a 5% annual interest rate compounded n times/year if (i) n = 1000 (ii) n = 10,000 (iii) n = 100,000 (b) Look at the sequence of answers in part (a), and predict the effective annual rate for a 5% annual rate compounded continuously. (c) Compute e0.05. How does this confirm your answer to part (b)?
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Chapter 0: Problem 10 Applied Calculus 5
A bank account is earning interest at 6% per year compounded continuously. (a) By what percentage has the bank balance in the account increased over one year? (This is the effective annual rate.) (b) How long does it take the balance to double? (c) For a continuous interest rate r, find a formula for the doubling time in terms of r.
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Chapter 0: Problem 11 Applied Calculus 5
Explain how you can match the interest rates (a)(e) with the effective annual rates IV without calculation. (a) 5.5% annual rate, compounded continuously. (b) 5.5% annual rate, compounded quarterly. (c) 5.5% annual rate, compounded weekly. (d) 5% annual rate, compounded yearly. (e) 5% annual rate, compounded twice a year. I. 5% II. 5.06% III. 5.61% IV. 5.651% V. 5.654%
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Chapter 0: Problem 12 Applied Calculus 5
In 1989, US inflation was 4.6% a year. In 1989 Argentina had an inflation rate of about 33% a month. (a) What is the yearly equivalent of Argentinas 33% monthly rate? (b) What is the monthly equivalent of the US 4.6% yearly rate?
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Chapter 0: Problem 13 Applied Calculus 5
Between December 1988 and December 1989, Brazils inflation rate was 1290% a year. (This means that between 1988 and 1989, prices increased by a factor of 1 + 12.90 = 13.90.) (a) What would an article which cost 1000 cruzados (the Brazilian currency unit) in 1988 cost in 1989? (b) What was Brazils monthly inflation rate during this period?
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