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Derivatives

Derivatives

At a Glance - Graphs of f ( x ) and f ' ( x )

From a graph of a function f(x) we can sketch graph of its derivative f ' (x). To do this, we use some things we talked about earlier.

  • If f is decreasing, its slope (and hence its derivative) is negative. If f is increasing, its slope (and hence its derivative) is positive.
      
  • From drawing tangent lines to f, we can compare relative values of the derivative and tell where the derivative is greatest.
      
  • If f is a line, its slope is constant.
      

Be Careful: The word "it" is dangerous.

Look at these two sentences: 

  • It's increasing, so it's positive. 
      
  • f is increasing, so ' is positive.
       

The first sentence is unclear. What does "it" mean? There's no way to know. Whenever we use the words "increasing, decreasing, positive, negative," we need to be clear about what (f, f ', or something else?) is increasing, decreasing, positive, or negative.

Example 1

From the graph of f(x), draw a graph of its derivative f ' (x).



Example 2

From the graph of f(x), draw a graph of f ' (x).


Example 3

From the graph of f(x), draw a graph of f ' (x).


Example 4

From the graph of f ' (x), draw a graph of f(x).


Example 5

From the graph of f ' (x), draw a graph of f(x).


Example 6

From the graph of f ' (x), draw a graph of f(x).


Exercise 1

For the function f(x) below, draw a graph of f ' (x). Don't worry too much about whether f ' is straight or curvy—focus on getting it to cross the x axis in the right places.


Exercise 2

For the function f(x) below, draw a graph of f ' (x). Don't worry too much about whether f ' is straight or curvy—focus on getting it to cross the x axis in the right places.


Exercise 3

For the function f(x) below, what would the graph of f ' (x) look like?


Exercise 4

For the function f(x) below, what would a rough sketch of f ' (x) look like?


Exercise 5

From the graph of f ' (x), what might the graph of f(x) look like? Make sure to label each graph.


Exercise 6

From the graph of f ' (x), draw a graph of f(x). Make sure to label each graph.


Exercise 7

From the graph of f ' (x), draw a graph of f(x). Make sure to label each graph.


Exercise 8

From the graph of f ' (x), what will the graph of f(x) look like? Make sure to label each graph.


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