Derivatives as Slope of a Curve

What do snickerdoodles and velocity have in common? Derivatives! No, that wasn't a bad attempt at a joke. Get on our level by watching this speedy video.

AP CalculusDerivatives
LanguageEnglish Language
MPAC 1Reasoning with definitions and theorems
MPAC 3Implementing algebraic/computational processes
MathCalculus

Transcript

00:22

go in his new golf cart. He starts at his house and decides to have

00:27

the finish line at his friendÕs house. His friend, Mr. Macadamia Nut, Macnut for short,

00:32

decides to help him, by drawing a graph of his distance traveled in feet versus time

00:36

elapsed in seconds. Because Mr. SnickerdoodleÕs favorite number

00:40

is 10, he wants to figure out how fast the car can go at time equals 10 seconds.

00:46

How fast is Mr. Snickerdoodle going at 10 seconds?

00:47

One way to express how fast he is going is by using velocity, which equals the change

00:52

in position over change in time.

00:55

We know the function for position of the golf cart can be written as f of x equals x-squared.

01:02

If we can find the rate of change, or slope, of the position graph at 10 seconds, we will

01:07

find how fast he is going.

01:10

Typically, we can find the slope of a line by taking two points on the line, and taking

01:16

the quantity y2 minus y1 over x2 minus x1. However, our position graph is a curve, not

01:23

a line. This means that the slope of the graph varies at different points in time.

01:29

Even though the slope is changing, we can find the slope at a single x-value by finding

01:34

the slope of a line tangent to the curve at that point, when t equals 10.

01:41

To find the tangent line, we start by finding the line that goes through t equals 10 and

01:46

another point on the graph. This is called the secant line. To find the tangent line,

01:53

we can slide the other point closer and closer to t equals 10Éand eventually weÕll get

01:58

a line that is tangent to the graph at t equals 10.

02:02

A mathematical term we can use to describe getting closer and closer to something is

02:07

a limit. In other words, we want to find the limit as the second point approaches 10.

02:14

As you may recall, slope is y2 minus y1 over x2 minus x1. To find the slope at x equals

02:20

ten, we take the limit of the slope of the secant line as the second point approaches

02:25

the first. Another way of thinking about this is the

02:29

distance between the two points, which we can call the variable h, approaching zero.

02:34

Since we want to express everything in terms of variables, we can label the initial point

02:38

with the variable x, which we will later substitute 10 for. Our formula for slope now becomes

02:45

the limit as h approaches zero of f of x plus h minus f of x over x plus h minus x.

02:54

The xÕs on the bottom cancel out, which leaves us with the limit as h approaches zero of

02:59

f of x plus h minus f of x all over h.

03:04

This formula will show up again and again when you do derivatives, so MEMORIZE IT.

03:10

Plugging in Mr. SnickerdoodleÕs equation, we have f of x equals x squared. For the first

03:15

part, we plug in x plus h instead of x..to get the limit as h approaches zero of x plus

03:23

h squared minus x squared over h. If we FOIL x plus h squared, we get x squared plus 2xh

03:32

plus h-squared. We can cancel the x squaredÕs, leaving us with the limit as h approaches

03:38

zero of two x h plus h squared all over h. We can factor an h out of the numerator..to

03:44

get h times 2x plus h. The hÕs cancel on the top and bottom. WeÕre left with the limit

03:50

as h approaches 0 of 2x plus h.

03:53

Now we can plug h equals zero in, giving us that the slope equals two x.

03:59

What we just found is the rate of change of the position over time, which, if you think

04:04

about it, is just velocity. This is also called TAKING THE DERIVATIVE of a function. Which

04:10

is presumably why you came to watch this video.

04:13

So back to Mr. Snickerdoodle. To find the velocity at ten seconds, we just plug in ten

04:18

for x into 2x. to find that Mr. SnickerdoodleÕs speed is 2 times 10, or 20 feet per second.

04:25

For those of you not familiar with feet and seconds, heÕs going a blazing 13.6 miles

04:31

per hour. His race car driving dreams are finally fulfilled.