- 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**

It's possible to mix the words "speed" and "velocity," especially since we almost never use the word "velocity" in everyday life.

**Speed is a number.** Some examples of speeds are 55 mph, 100 km/hr, and 33 ft/sec.

**Velocity is a vector.** For our current purposes, velocity is a speed with some kind of direction attached to it. Some examples of velocities are 55 mph due East, 100 km/hr away from home, and 33 ft/sec downwards.

The direction associated with velocity gives us additional information and lets us answer questions like this:

Jenna travels at 50 mph away from home for one hour, and then at 50 mph toward home for 30 minutes. How far away from home is she at the end of this time?

First, Jenna travels at 50 mph away from home for one hour, therefore she's 50 miles from home:

Then Jenna travels back towards home for half an hour, covering 50/2 = 25 miles:

She ends 25 miles away from home.

If we didn't have the information about direction, we couldn't answer the question.

Jenna travels at 50 mph for one hour, then at 30 mph for one hour. How far from home is she at the end of this time?

We can't answer this question. If Jenna went in the same direction for two hours, she would be 80 miles from home.

If she went out and then back, she'd be only 20 miles from home.

And if she went in funny directions, who knows where she would be?

Calculus classes often use problems in which velocity has only two possible directions: positive or negative. For example, a ball moving upwards would have positive velocity, and a ball falling downwards would have negative velocity. A driver driving away from home would have positive velocity, and driving toward home would have negative velocity. In such cases, speed is the absolute value of velocity. More generally, *speed is the magnitude of velocity*.

Find the speed associated to each velocity.

- 70 mph due North

- -16 ft/sec

- 15 meters per second at an angle of 45
^{°}to the ground and straight ahead.

For each of these, leave out the words that tell about direction and take absolute values of the numbers.

- Omit the words "due North." The number left over is 70 mph.

- Take the absolute value of -16 ft/sec to find 16 ft/sec.

- Ignore all the words except the unit.

The answer is 15 meters per second.