Chapter 2 Review

Chapter 2: Rate of change: the derivative



• SECTION 2.1: Instantaneous rate of change
  
  
  1. Suggested Problems: #7, 8, 9, 11.
     
  2. The instantaneous rate of change of a function y = f(x) at x = a is defined to be the limit of the average rate of change of f(x) over smaller and smaller intervals that contain the point x = a, provided this limit exists.
     
  3. The instantaneous rate of change of a function y = f(x) at x = a is the same as the slope of the line tangent to the graph of y = f(x) at the point (a, f(a)).
  4. Be able to approximate the instantaneous rate of change of a function at a point given a table of values.
     
  5. Be able to approximate the instantaneous rate of change of a function at a point given a graph.
     
  6. Be able to approximate the instantaneous rate of change of a function at a point given a formula.
     
  7. Given a function of the position of an object at time t, the instantaneous velocity of the object is given by the instantaneous rate of change in position with respect to time.
     
  
  
  
• SECTION 2.2: The derivative at a point
  
  
  1. Suggested Problems: #3, 7, 9, 11.
     
  2. The derivative of f(x) at x = a is given by the instantaneous rate of change in f(x) with respect to x at x = a, provided it exists. We denote this as f(a).
     
  3. The derivative of f(x) at x = a is the same as the slope of the line tangent to the graph of the curve y = f(x) at the point (a, f(a)).
     
  4. Be able to approximate the derivative of a function at a point given a table of values.
     
  5. Be able to approximate the derivative of a function at a point given a graph.
     
  6. Be able to approximate the derivative of a function at a point given a formula.
     
  7. Be able to approximate the equation of the line tangent to the graph of y = f(x) at x = a.
     
  8. Suppose that an object is moving along a straight line, and has position y = f(t) at time t. The instantaneous velocity of the object at time t = a is the same as the derivative of f(t) at t = a, f(a).
     
  9. Be able to use your calculator to approximate the derivative of a function at a point given a formula.
     
  
  
  
• SECTION 2.3: The derivative function
  
  
  1. Suggested Problems: #4, 5, 10, 15, 17.
     
  2. Given a function f(x), the derivative function f(x) is defined as the instantaneous rate of change of f(x) with respect to x at each point x, provided it exists.
     
  3. Be able to sketch a graph of f(x) if given the graph of f(x) and vice versa.
     
  4. The sign of f(x) tells us when f(x) is increasing and decreasing:
     
     1. If f(x) > 0 over an interval, then f(x) is increasing over that interval.
        
     2. If f(x) < 0 over an interval, then f(x) is decreasing over that interval.
        
     3. If f(x) = 0 over an interval, then f(x) is constant over that interval.
        
  5. The magnitude (absolute value) of f(x) tells us about the steepness of the slope of f(x):
     
     1. If |f(x)| is large, then f(x) has a steep slope at x.
        
     2. If |f(x)| is small (close to 0), then f(x) has a small (almost horizontal) slope at x.
        
  6. Be able to approximate values of f(x) given a table of values of f(x).
     
  
  
  
• SECTION 2.4: Interpretations of the derivative
  
  
  1. Suggested Problems: #1, 9, 17, 20.
     
  2. Leibniz's Notation for the derivative: (d/dx) f(x) is another way to write f(x).
     
  3. The units of f(x) are given in (units of f(x)) / (units of x).
     
  4. Be able to recognize when a word problem is making a reference to the derivative of a function.
     
  5. Interpretation of the derivative in terms of the problem.
     
  6. Local linear approximation: Given the values f(a) and f(a), we estimate the value of f(a+x) by means of the approximation
     f(a+x) f(a) + f(a) x
  7. Be able to recognize how values of the derivative of a function at a point indicate how the function is behaving locally.
     
  
  
  
• SECTION 2.5: The second derivative
  
  
  1. Suggested Problems: #5, 7, 11, 16, 17.
     
  2. Given a function f(x) having a derivative f(x), the second derivative of f(x), denoted f(x), is defined as the derivative of f(x), provided it exists.
     
  3. We denote the second derivative of the function f(x) as either f(x) or (d²/dx²) f(x).
     
  4. The units of f(x) are given in (units of f(x)) / (units of x)².
     
  5. The second derivative of a function f(x) tells us the following about the graph of f(x):
     
     1. If f(x) > 0 on an interval, then f(x) is increasing on that interval, and so f(x) is concave up on the interval.
        
     2. If f(x) < 0 on an interval, then f(x) is decreasing on that interval, and so f(x) is concave down on the interval.
        
  6. The second derivative of a function f(x) tells us about how the rate of change of f(x) is changing:
     
     1. If f(x) > 0 on an interval, then the rate of change of f(x) is increasing on the interval.
        
     2. If f(x) < 0 on an interval, then the rate of change of f(x) is decreasing on the interval.
        
  
  
  
• SECTION 2.6: Marginal cost and marginal revenue
  
  
  1. Suggested Problems: #1, 5, 6, 9, 12.
     
  2. Revenue is always the product of price and quantity.
     
  3. We take the definitions
     
     1. C(q) as Marginal Cost.
        
     2. R(q) as Marginal Revenue.
        
     3. (q) = R(q) - C(q) as Marginal Profit.
        
  4. Three observations about how the relationship between marginal cost and marginal revenue affects profit:
     
     1. The values of q that make (q) = 0 are the same values of q for which R(q) = C(q).
        
     2. If marginal cost is less than marginal revenue, then (q) > 0, and so profit is increased by increasing production.
        
     3. If marginal cost is greater than marginal revenue, then (q) < 0, and so profit is decreased by increasing production.
        
  5. Be able to determine whether or not one should add an additional item of production by considering the values of the Marginal Revenue and Marginal Cost functions at the current level of production.
     
  6. Provided that R(q) and C(q) exist for all values of q, the value of q for which profit is maximized must also satisfy R(q) = C(q). However, values of q for which R(q) = C(q) may not necessarily maximize profit.
     
  
  
  

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Last modified: Fri Apr 12 2002