- 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

We've already done this in the case where the function is a line. What happens if the line isn't straight? What if it's snake-shaped or U-shaped? What if it's as curvy as a mudflap? Now we'll find the slope of a line between two points on any wonky function we like.

Find the slope of the line between the two points shown.

Since we're given a formula for the function *f*, we can say

Then we can find the rise and run from this picture:

A line between two points on a function is called a **secant line**.

Asking to find the slope of the "secant line" between two points on a function means the same thing as asking to find the slope of the "line" between those two points.

A secant line is a line between two points on a function. We've been thinking about a secant line as a line that starts at the point on *f* where *x* = *a*, and ends at the point on *f* where *x* = *b*. For what's coming, it will be helpful to think of starting at (*a*, *f*(*a*)) and ending at some "other point." The "other point" will be described by how far away its *x*-value is from *a*. Poor *b* is getting fired, to be replaced by *a* + *h*.

*h* is whatever we need to make *a* + *h* equal the number formerly known as *b*, not to be confused with the artist formally known as Prince.

Before, we had *a* = 1 and *b* = 1.5. *a* and *b* were equally important:

Now we have *a* = 1 and *h* = 0.5. *a* is important, and the distance of the other point from *a* is important:

Before, we had *a* = 1 and *b* = -2:

Now we have *a* = 1 and *h* = -3:

The moral of the story is that we can think of a secant line on the function *f* as a line that starts at the point on *f* where *x* = *a* and ends at the point on *f* where *x* = *a* + *h*:

Since *h* measures the distance (and direction) from our starting point to our ending point, *h* is conveniently equal to the "run" we need for the slope formula:

We find a lovely formula for the slope of the secant line on the function *f* from *x* = *a* to *x* =* a *+ *h*:

If *h* is negative the formula will be the same, and the picture will look slightly different:

Example 1

Find the slope of the line between the two points shown. |

Example 2

If |

Example 3

Let |

Exercise 1

For the given function *f* and values of *a* and *b*, find the slope of the secant line between the points (*a*, *f*(*a*)) and (*b*, *f*(*b*)):

*f*(*x*) = *x*^{2}, *a* = 1, *b* = 2.

Exercise 2

For the given function *f* and values of *a* and *b*, find the slope of the secant line between the points (*a, f*(*a*)) and (*b*, *f*(*b*)):

*f*(*x*) = *x*^{2}, *a* = 1, *b* = 1.5.

Exercise 3

For the given function *f* and values of *a* and *b*, find the slope of the secant line between the points (*a, f*(*a*)) and (*b*, *f*(*b*)):

*f*(*x*) = sin(*x*), *a* = 0, *b* = π/2.

Exercise 4

For the given function *f* and values of *a* and *b*, find the slope of the secant line between the points (*a, f*(*a*)) and (*b*, *f*(*b*)):

*f*(*x*) = *x*^{3} - 2*x* + 3, *a* = 1, *b* = 4.

Exercise 5

*f* and values of *a* and *b*, find the slope of the secant line between the points (*a, f*(*a*)) and (*b*, *f*(*b*)):

*f*(*x*) = *x*^{4}* - 2*, *a* = -2, *b* = 2.

Exercise 6

Given the values of *a* and *b*, find *h* so that *a* + *h* = *b*: *a* = 4, *b* = 4.25.

Exercise 7

Given the values of *a* and *b*, find *h* so that *a* + *h* = *b*: *a* = -1, *b* = -1.5.

Exercise 8

For the given function *f*, value of *a*, and value of *h*, find the slope of the secant line between (*a*, *f*(*a*)) and (*a* + *h*, *f*(*a* + *h*)):

*f*(*x*) = *x*^{2}, *a* = 1, *h* = 0.1.

Exercise 9

For the given function *f*, value of *a*, and value of *h*, find the slope of the secant line between (*a*, *f*(*a*)) and (*a* + *h*, *f*(*a* + *h*)):

*f*(*x*) = 1 - *x*^{2}, *a* = 0, *h* = 0.1.

Exercise 10

For the given function *f*, value of *a*, and value of *h*, find the slope of the secant line between (*a*, *f*(*a*)) and (*a* + *h*, *f*(*a* + *h*)):

*f*(*x*) = cos(*x*), *a* = 0, *h* = -π/2.

Exercise 11

*f*, value of *a*, and value of *h*, find the slope of the secant line between (*a*, *f*(*a*)) and (*a* + *h*, *f*(*a* + *h*)):

*f*(*x*) = *x*^{3}- *x*, *a* = 1, *h* = 4.

Exercise 12

*f*, value of *a*, and value of *h*, find the slope of the secant line between (*a*, *f*(*a*)) and (*a* + *h*, *f*(*a* + *h*)):

*f*(*x*) = 3*x*, *a* =* -*2, *h* = -0.2.