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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...
Riemann sums are a way to estimate the area under a curve. Check out the video for all the deets.
Derivatives as Slope of a Curve 5239 Views
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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.
Transcript
- 00:04
Derivatives as the Slope of a Curve, a la Shmoop,
- 00:07
Mr. Snickerdoodle always wanted to be a racecar driver, but unfortunately, there were never
- 00:10
any racecar dealerships established around his area. So he decided to buy the next best
- 00:12
thing, a brand new leather furnished golf cart with extra-large cupholders.
- 00:18
To fulfill his ambitions of being a race-car driver, he decides to see how fast he can
Full 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.
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