- 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

If we remember two things, we can write the equation for the tangent line to *f* at *a* given the formula for *f* and the value *a* where we want the tangent line to go.

- We can calculate
*f*(*a*).

- We can calculate
*f '*(*a*).

We can write the equation of a line given a point and a slope, so once we have a point and a slope for the tangent line we're all set to go.

We've left out one thing in this discussion of tangent lines: the magic formula. Most textbooks have a magic formula that produces the equation for the tangent line.

We have lots of reasons for leaving this formula out. We don't need it. It takes extra memory that could be better spent remembering the limit definition of the derivative. It can be confusing.

However, for the sake of completeness, we'll show the magic formula. But we're doing it our way.

There's one special case that neither finding the equation of a line nor knowing the magic formula will help with. If *f ' *(*a*) is undefined and infinite, then we have a vertical tangent line.

The equation of such a tangent line is x = a like any vertical line. Knowing this will probably be more important for finding derivatives of parametric functions than it will be right now.

We can also work backwards to figure out information about the function given information about its tangent line.

Example 1

Find the tangent line to |

Example 2

Find the equation for the tangent line to |

Example 3

Let |

Example 4

The picture below shows a function and its tangent line at What is |

Exercise 1

For the given function *f *andvalue *a, *find the tangent line to *f *at* a. *

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

Exercise 2

For the given function *f* and value *a*, find the tangent line to *f* at *a*.

*f*(*x*) =*x*^{2}+ 1,*a*= 0

Exercise 3

For the given function *f* and value *a*, find the tangent line to *f* at *a*.

*f*(*x*) =*x*^{3}, a = 1

Exercise 4

For the given function *f* and value *a*, find the tangent line to *f* at *a*.

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

Exercise 5

For the given function *f* and value *a*, find the tangent line to *f* at *a*.

Exercise 6

For the function *f* and value of *a*, use the magic formula to find the tangent line to *f* at *a*. We will need to calculate derivatives from scratch.

*f*(*x*) = 2*x*+ 3*x*^{2},*a*= 4

Exercise 7

For the function *f* and value of *a*, use the magic formula to find the tangent line to *f* at *a*. We will need to calculate derivatives from scratch.

*f*(*x*) = 2*x*^{3},*a*= -2

Exercise 8

For the function *f* and value of *a*, use the magic formula to find the tangent line to *f* at *a*. We will need to calculate derivatives from scratch.

Exercise 9

The graph shows a function *f* and a line that is tangent to *f* at *a*. For the graph determine *a*, *f*(*a*), and *f'*(*a*) (*a* refers to the *x*-value at which the line is tangent to *f*).