Chapter 1 Review

Chapter 1: Functions and change



• SECTION 1.1: How do we measure change?
  
  
  1. Suggested Problems: #2, 3, 5, 10, 14.
     
  2. If y is a function of t, then the change in y from t = a to t = b is denoted y, and is given by
     y = (the value of y at t = b) - (the value of y at t = a).
  3. The average rate of change of a quantity y between time t = a and time t = b is given by
     y / t = (the change in y) / (the change in t).
  4. The units of the change in y, y, are the same as the units of y.
     
  5. The units of the average rate of change of y with respect to t, y / t, are in (units of y) / (units of t).
     
  6. The quantity y / t is also known as a difference quotient.
     
  7. Average velocity is the average rate of change of distance with respect to time.
     
  8. Speed is the magnitude (the absolute value) of velocity.
     
  9. Velocity is speed with an associated direction.
     
  
  
  
• SECTION 1.2: What is a function?
  
  
  1. Suggested Problems: #5, 7, 15, 17.
     
  2. A function is a rule of association between two sets. For each input the function assigns a single output. The set of input values is the domain of the function, and the set of output values is the range of the function. The input is called the independent variable. The output is called the dependent variable.
     
  3. Functions can be represented in a variety of ways. They can be represented by means of tables, graphs, formulas, and words.
     
  4. Functional notation: y = f(t) means that the dependent variable y is a function of the independent variable t. This function has the name f.
     
  5. If y = f(t), then the change in y between t = a and t = b is given by
     y = f(b) - f(a).
  6. If y = f(t), then the average rate of change in y between t = a and t = b is given by
     y / t = (f(b) - f(a)) / (b - a).
  
  
  
• SECTION 1.3: Linear functions
  
  
  1. Suggested Problems: #4, 8, 17, 20, 21.
     
  2. A function is linear if its average rate of change is the same over every interval.
     
  3. A linear function is a function of the form
     y = f(x) = m x + b. The value m is referred to as the slope of the line. It is the same as the average rate of change of the function. The value b is referred to as the y-intercept or the vertical intercept of the function. It is the same as the value of y when x = 0.
     
  4. The slope m is a measure of the steepness of a line. The greater the magnitude of m, the steeper the line. If m > 0, then the line is increasing (its graph is from the lower-left to the upper-right). If m < 0, then the line is decreasing (its graph is from the upper-left to the lower-right).
     
  5. A horizontal line has slope m = 0, and so its equation is of the form y = b.
     
  6. A vertical line does not represent a function, and has no slope.
     
  7. The slope m of the line that passes through the points (x₁, y₁) and (x₂, y₂) is given by the average rate of change of the line over the interval from x = x₁ to x = x₂:
     m = rise / run = (y₂ - y₁) / (x₂ - x₁).
  8. The point-slope form of the equation of a line: The equation of the line with slope m that passes through the point (x₀, y₀) is given by
     y - y₀ = m (x - x₀)
  9. The change in the function y = f(x) from x = a to x = b is the same as the difference in height between f(x) at x = b and f(x) at x = a:
     y = f(b) - f(a)
  10. The average rate of change of any function y = f(x) between x = a and x = b is the same as the slope of the line that passes through the points (a, f(a)) and (b, f(b)):
      y / x = (f(b) - f(a)) / (b - a)
  11. A function y = f(x) is increasing if f(a) < f(b) whenever a < b.
      
  12. A function y = f(x) is decreasing if f(a) > f(b) whenever a < b.
      
  
  
  
• SECTION 1.4: Applications of functions to economics
  
  
  1. Suggested Problems: #2, 5, 6, 7, 8, 9, 30.
     
  2. The cost function, C(q), is the total cost of producing a quantity q of some good.
     
  3. We separate the cost function into two parts: the fixed costs, C(0), and the variable costs, C(q) - C(0).
     
  4. Fixed costs are the costs incurred even if nothing is produced.
     
  5. The revenue function, R(q), is the total revenue received from selling a quantity q of some good.
     
  6. The profit function, (q), is the revenue received from selling q units of some good minus the cost of producing q units of the good:
     (q) = R(q) - C(q)
  7. The break-even point of a company is the value of q for which the profit function is 0, or where the revenue functions equals the cost function.
     
  8. The supply curve, S, for a good is a representation of how the quantity, q, of the good that manufacturers are willing to make per unit time depends on the price, p, for which the item can be sold. Thus q = S(p).
     
  9. The demand curve, D, for a good is a representation of how the quantity, q, of the good that consumers demand per unit time depends on the price, p, of the item. Thus q = D(p).
     
  10. The supply curve S(p) usually increases with p, and the demand curve D(p) usually decreases with p.
      
  11. The point where the supply and demand curves, q = S(p) and q = D(p), cross is called the equilibrium point, and is denoted p = p^* and q = q^*. The value p^* is the equilibrium price and q^* is the equilibrium quantity. It is assumed that the market will naturally settle to the equilibrium point.
      
  12. A tax on producers will affect the supply curve S(p), decreasing the revenue on each item that producers receive by , and thus reforming the supply curve as S(p - ).
      
  13. A tax on consumers will affect the demand curve D(p), increasing the price of each item that consumers pay by , and thus reforming the demand curve as D(p + ).
      
  
  
  
• SECTION 1.5: Proportionality and power functions
  
  
  1. Suggested Problems: #26, 27, 28, 29.
     
  2. If y is (directly) proportional to x, then there is a constant k such that y = k x. k is known as the constant of proportionality.
     
  3. If y is inversely proportional to x, then there is a constant k such that y = k/x.
     
  4. If f(x) is a power function of x, then f(x) is proportional to a constant power of x. Thus f(x) = k x^p, where k is the constant of proportionality and p is the constant power.
     
  5. Be familiar with the general shape and comparitive steepness of the graphs of y = x^p for various values of p.
     
  6. A graph is said to be concave up if it bends upward with an increase in x.
     
  7. A graph is said to be concave up if it bends downward with an increase in x.
     
  8. If p is a real number with 0 < p < 1, then the graph of y = x^p is increasing and concave down for x > 0.
     
  9. If p is a real number with p > 1, then the graph of y = x^p is increasing and concave up for x > 0.
     
  
  
  
• SECTION 1.6: Exponential functions
  
  
  1. Suggested Problems: #1, 2, 13, 15, 18.
     
  2. The function P(t) is said to be an exponential function of t with base a (where a > 0 and a is not 1) if P(t) = P₀ a^t, where P₀ is the initial value of P(t), P(0) = P₀, and a is the factor by which P(t) changes when t increases by 1, a = P(t+1)/P(t).
     
  3. If P(t) = P₀ a^t with P₀ > 0 and a > 1, then P(t) is said to exhibit exponential growth.
     
  4. If P(t) = P₀ a^t with P₀ > 0 and 0 < a < 1, then P(t) is said to exhibit exponential decay.
     
  5. If P(t) is a positive, increasing exponential function, and r is its growth rate, then 1 + r is its growth factor, and P(t) = P₀ (1 + r)^t.
     
  6. If P(t) is a positive, decreasing exponential function, and r is its decay rate, then 1 - r is its decay factor, and P(t) = P₀ (1 - r)^t.
     
  7. If a and p are real numbers with a > 1 and p > 0, then a^t approaches infinity at a much greater rate than t^p, as t approaches infinity.
     
  
  
  
• SECTION 1.7: Continuous growth and the number e
  
  
  1. Suggested Problems: #2, 3, 7, 11.
     
  2. The greater the rate of interest, the faster an investment grows.
     
  3. The more often that interest is compounded, the faster an investment grows.
     
  4. If an initial amount P₀ is invested at an annual rate of r, and is compounded annually, then the investment at the end of t years is given by P = P₀ (1 + r)^[t], where [t] is the largest integer less than or equal to t.
     
  5. If an initial amount P₀ is invested at an annual rate of r, and is compounded continuously, then the investment at the end of t years is given by P = P₀ e^rt.
     
  6. If interest is compounded continuously, with growth equation P = P₀ e^rt, then r is the continuous growth rate. The annual growth rate is given by e^r - 1.
     
  
  
  
• SECTION 1.8: The natural logarithm
  
  
  1. Suggested Problems: #7, 8, 16, 17, 18, 22, 26, 32, 33, 40.
     
  2. The natural logarithm function, ln(x), is the value of y such that e^y = x.
     
  3. We then have the following equivalent statements:
     e^y = x if and only if y = ln(x).
  4. The functions ln(x) and e^x are inverses of each other, so that ln(e^x) = x for all values of x, and e^ln(x) = x for all x > 0.
     
  5. The domain of the function f(x) = ln(x) consists of all positive real numbers.
     
  6. The properties of ln(x): Assume that x > 0 and y > 0. Then
     
     1. ln(x y) = ln(x) + ln(y).
        
     2. ln(x/y) = ln(x) - ln(y).
        
     3. ln(x^k) = k ln(x).
        
  7. We can convert a function of the form P(t) = P₀ a^t to a function of the form P(t) = P₀ e^kt, and vice versa, by means of the relationship k = ln(a) or a = e^k.
     
  8. If P₀ > 0, then P(t) = P₀ a^t is increasing when a > 1. It is decreasing when 0 < a < 1.
     
  9. If P₀ > 0, then P(t) = P₀ e^kt is increasing when k > 0. It is decreasing when k < 0.
     
  10. Be able to solve equations of the form a · b^cx = d for x.
      
  
  
  
• SECTION 1.9: Exponential growth and decay
  
  
  1. Suggested Problems: #1, 2, 6, 10, 15.
     
  2. The Doubling Time of an increasing exponential function of the form P(t) = P₀ e^kt is given by T = ln(2)/k.
     
  3. The Doubling Time of an increasing exponential function of the form P(t) = P₀ a^t is given by T = ln(2)/ln(a).
     
  4. The Half-Life of a decreasing exponential function of the form P(t) = P₀ e^kt is given by T = -ln(2)/k.
     
  5. The Half-Life of a decreasing exponential function of the form P(t) = P₀ a^t is given by T = -ln(2)/ln(a).
     
  6. The Present value P of a future payment B is the amount that would need to be deposited today, into an interest bearing account, in order to exactly produce the future payment B at a specific point of time in the future.
     
  7. The Future value B of a payment P is the amount to which the payment P will grow if it is deposited into an interest bearing account today.
     
  8. Be able to find the Present value P of a future payment B.
     
  9. Be able to find the Future value B of a payment P.
     
  10. Be aware of how future values and present values relate.
      
  
  
  
• SECTION 1.11: Polynomials
  
  
  1. Suggested Problems: #1, 3, 6.
     
  2. A function that can be written in the form
     f(x) = a_n xⁿ + a_n-1 x^n-1 + ... + a₁ x + a₀, where n is a nonnegative integer, and the a_n, a_n-1, ..., a₁, a₀ are real numbers, is called a polynomial in the variable x. The degree of the polynomial is the value n, the leading coefficient of the polynomial is the value a_n, and the leading term of the polynomial is a_n xⁿ.
     
  3. For values of x of large magnitude, a polynomial behaves in a manner similar to that of its leading term.
     
  4. A polynomial of degree n changes direction, from increasing to decreasing or from decreasing to increasing, at most n - 1 times.
     
  
  
  
• SECTION: Focus on Modeling. Fitting formulas to data
  
  
  1. Suggested Problems: #3, 6, 7.
     
  2. Be familiar with what it means to find a regression curve for a set of data.
     
  3. Be able to find a specified curve of best fit (a line, exponential curve, etc.) given a collection of data points.
     
  4. Be able to use a regression curve to predict additional values of data.
     
  5. Using a regression curve to predict the output values of input values that lie between the input values of points used to generate the curve is called interpolation.
     
  6. Using a regression curve to predict the output values of input values that lie outside of the input values of points used to generate the curve is called extrapolation.
     
  7. In general, we tend to have more confidence in the accuracy of predicting output values by means of interpolation than by means of extrapolation.
     
  
  
  
• SECTION: Focus on Modeling. Compound interest and the number e
  
  
  1. Suggested Problems: #2, 4, 5, 8.
     
  2. What is the annual percentage rate of an investment?
     
  3. If an initial amount P₀ is invested at an annual rate of r, and is compounded n times a year, then the investment at the end of t years is given by P = P₀ (1 + r/n)^[nt], where [nt] is the largest integer less than or equal to nt.
     
  4. If an initial amount P₀ is invested at an annual rate of r, and is compounded continuously, then the investment at the end of t years is given by P = P₀ e^rt.
     
  5. The effective annual yield of an investment that grows according to the formula P = P₀ (1 + r/n)^[nt] is given by EAY = (1 + r/n)ⁿ - 1.
     
  6. The effective annual yield of an investment that grows according to the formula P = P₀ e^rt is given by EAY = e^r - 1.
     
  
  
  

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Last modified: Fri Apr 12 2002