ShmoopTube

Where Monty Python meets your 10th grade teacher.

Search Thousands of Shmoop Videos


Calculus Videos 5 videos

Graphing and Visualizing Limits
1041 Views

Breathe in deeply through the nose... Now slowly exhale... Breathe in... And out... Now visualize the graph of the limit of f(x) as x approaches 2....

Derivatives as Slope of a Curve
5239 Views

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...

Introduction to Integrals with Riemann Sums
620 Views

Riemann sums are a way to estimate the area under a curve. Check out the video for all the deets.

See All

Derivatives as Slope of a Curve 5239 Views


Share It!


Description:

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.

Language:
English Language

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

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.

Related Videos

Compound Interest
45304 Views

Who wants to be a millionaire? In this video, learn about compound interest, interest rates, and the compound interest formula. You'll be buying th...

ACT Math 5.2 Pre-Algebra
1939 Views

ACT Math: Pre-Algebra Drill 5, Problem 2. If a and b are prime numbers, how many factors does ab have?

Simplifying Radicals
9741 Views

We don't like knocking people down to size, but we do like simplifying radicals. Join us?

Arithmetic Math
2251 Views

If fleeing criminals always fled the scene of the crime at perfect right angles, it would be much easier to determine their whereabouts. Fortunatel...