Calculus / Unit 3: The Derivative
3.2 The Tangent Line and Its Slope
Notes
A tangent line touches a curve at a point and has the same direction as the curve there.
- Its slope is the limit of secant slopes: $\displaystyle m = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$.
- That slope is called the derivative of $f$ at $a$, written $f'(a)$.
- Example: for $f(x) = x^2$ at $x = 1$, the secant slope from 1 to 1.01 is $\displaystyle \frac{1.0201 - 1}{0.01} = 2.01$, close to the tangent slope 2.
- A horizontal tangent has slope 0.
Practice
For $f(x) = x^2$, the secant slope between $x = 1$ and $x = 1.01$ is 2.01. What is the best estimate of the tangent slope at $x = 1$?
As the second point approaches 1, the slopes approach 2.
Which expression gives the slope of the tangent line to $f$ at $x = a$?
It is the limit of the difference quotient.
The tangent line to a curve at a high point is horizontal. What is the derivative there?
A horizontal line has slope 0.
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