- Topics At a Glance
- Derivative as a Limit of Slopes
- Slope of a Line Between Two Points
- Slope of a Line Between Two Points on a Function
- Slope at One Point?
- Estimating Derivatives Given the Formula
- Estimating Derivatives from Tables
- Finding Derivatives Using Formulas
- Derivatives as an Instantaneous Rate of Change
- Average Rate of Change
- Instantaneous Rate of Change
- Units, Words, and Notation
- Tangent Lines
- How Tangent Lines Look
- When Tangent Lines Don't Exist
- Tangent Lines and Derivatives
- Tangent Line Approximation
- Finding Tangent Lines
- Using Tangent Lines to Approximate Function Values
- Differentiability and Continuity
- The Derivative Function
- Graphs of f ( x ) and f ' ( x )
- Theorems
- Rolle's Theorem
- The Mean Value Theorem
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem
- Appendix: Speed v. Velocity
Introduction to :
From a graph of a function f(x) we can make a sketched 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 f ' 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," that we are clear about what (f, f ', or something else?) is increasing, decreasing, positive, or negative.
Practice:
From the graph of f(x), draw a graph of its derivative f'(x). |
From the graph of f(x), draw a graph of f'(x).
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From the graph of f(x), draw a graph of f'(x).
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From the graph of f'(x), draw a graph of f(x).
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From the graph of f'(x), draw a graph of f(x).
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From the graph of f'(x), draw a graph of f(x).
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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.
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.
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.
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.
From the graph of f'(x), draw a graph of f(x). Make sure to label each graph.
This means f will look something like one of these: 
From the graph of f'(x), draw a graph of f(x). Make sure to label each graph.
From the graph of f'(x), draw a graph of f(x). Make sure to label each graph.
From the graph of f'(x), draw a graph of f(x). Make sure to label each graph.
