- 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

There are three steps to solving a math problem.

- Figure out what the problem is asking.

- Solve the problem.

- If you get stuck, figure out why you're stuck.

- Check the answer.

What does the Mean Value Theorem say about the function *f*(*x*) = 4*x* + *x*^{2} on the interval (4, 6)?

**Figure out what the problem is asking.**

We need to understand what all the words in the problem mean, and what any theorems mentioned say. The MVT says if

*f* is continuous on [*a*, *b*], and

*f* is differentiable on (*a*, *b*),

then there is some *c* in (*a*,* b*) with

.

The problem is actually two problems. Are we allowed to apply the MVT to the given function on the given interval? If so, what does the conclusion of the MVT say for this particular function on this particular interval?

**Solve the problem.**

To solve the problem we need to do two smaller problems.

Are we allowed to apply the MVT to the given function on the given interval?

First, we need to understand what the problem is asking. This problem is asking "do the hypotheses/assumptions of the MVT hold in this case?"

Next, we solve the problem. The answer is yes. *f* is a polynomial, since it's continuous and differentiable everywhere. In particular, *f* is continuous on [4, 6] and differentiable on (4, 6). There's not anything to check for this answer. If so, what does the conclusion of the MVT say for this particular function on this particular interval? First we need to understand what the problem is asking. This problem is asking "if we plug in the right values of *a* and *b* to the conclusion of the MVT, what does it say?" To solve the problem, we plug in *a* = 4 and *b* = 6. The MVT says there is some *c* in (4, 6) with

Again, there's not anything to check for this answer.

**Check the answer.**

Problems like this that don't involve much calculation, don't have answers to check. A good thing to do, though, would be to read through the answer again and make sure it's believable!