- 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 :
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.
Practice:
For the function below, which is greater: f'(0) or f'(1)?
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For the function shown below, which is greater: f'(-1) or f'(1)?
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For the function f and value a, determine if f'(a) is positive, negative, or zero.
For the function f and value a, determine if f'(a) is positive, negative, or zero.
For the function f and value a, determine if f'(a) is positive, negative, or zero.
For the function f and value a, determine if f'(a) is positive, negative, or zero.
For the function f and value a, determine if f'(a) is positive, negative, or zero.
- For the function shown below, which is greater: f'(x1) or f'(x2)?
- For the function shown below, which is greater: f'(x1) or f'(x2)?
- For the function shown below, list in order from least to greatest: f'(x1), f'(x2), f'(x3).
- For the function shown below, list in order from least to greatest: f'(x1), f'(x2), f'(0).
