Die Heuning Pot Literature Guide
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Introduction to Derivatives - At A Glance:

The following phrases all mean the same thing:

  • the slope of f at a
  • the derivative of f at a
  •  f ' (a)
  • the slope of the tangent line to f at a
  • the instantaneous rate of change of f at a

Since f ' (a) (the derivative of f at a) is equal to the slope of the tangent line to f at a, we can determine the sign of f ' (a) by looking at the tangent line to f at a.

Sample Problem

Consider this function:

The tangent line to f at a is sloping upwards. This means the slope of the tangent line to f at a is positive. Since the slope of the tangent line to f at a equals f ' (a), we know f ' (a) is also positive.

We can also tell the sign of f ' (a) by looking at the function f and not thinking about tangent lines.

  • If f is increasing at a then f ' (a) is positive.
      
  • If f is decreasing at a then f ' (a) is negative.
      
  • If f "flattens out" at a then f ' (a) is zero.

We can also use tangent lines to compare values of f' at different points.

Remember these pictures. We'll use them again when we discuss concavity and second derivatives.

Example 1

For the function below, which is greater: f'(0) or f'(1)?


Example 2

For the function shown below, which is greater: f'(-1) or f'(1)?


Exercise 1

For the function f and value a, determine if f'(a) is positive, negative, or zero.

Exercise 2

For the function f and value a, determine if f'(a) is positive, negative, or zero. 

Exercise 3

For the function f and value a, determine if f'(a) is positive, negative, or zero.

Exercise 4

For the function f and value a, determine if f'(a) is positive, negative, or zero.

Exercise 5

For the function f and value a, determine if f'(a) is positive, negative, or zero.

  •  

Exercise 6

  • For the function shown below, which is greater: f'(x1) or f'(x2)?

Exercise 7

  • For the function shown below, which is greater: f'(x1) or f'(x2)?

Exercise 8

  • For the function shown below, list in order from least to greatest: f'(x1), f'(x2), f'(x3).

Exercise 9

  • For the function shown below, list in order from least to greatest: f'(x1), f'(x2), f'(0).

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