Calculus (Math 1A)Lecture 5

Vivek Shende

September 5, 2017

Hello and welcome to class!

Last time

We discussed composition, inverses, exponentials, and logarithms.

Today

We talk about motion, Zeno’s paradox, tangent lines, and limits.

Zeno of Elea (450 b.c.)

Zeno’s paradox of the arrow

At each instant, the arrow occupies a single position in space.

Motion means changing position; as the arrow does not changeposition in any instant, it is not moving at any instant.

If it is not moving at any instant, when is it moving?

Lesson from Zeno

We should be careful discussing anything “at an instant”.

Answer to Zeno (?):

While the amount of motion at an instant is zero, it makes senseto discuss the rate of motion at an instant, and this is not zero.

The correct notion of being in motion at an instant is having anonzero rate of motion.

Rate

rate of change =amount of change

time it took

If what is changing is described by a number a varying with timeas a(t), then the rate of change over some interval [t0, t1] is

a(t1)− a(t0)

t1 − t0

I.e., the slope of the line through (t0, a(t0)) and (t1, a(t1)).

Secant line

A line connecting two points on the graph of a function is called asecant line. It has slope ∆f /∆x .

Tangent line

The tangent line to through a point on a curve is the (?) linethrough that point meeting no other nearby points of the curve.

Tangent line

Tangent line

Tangent from secants

Nearby secant lines get closer and closer to the tangent line.

Tangent from secants

Nearby secant lines get closer and closer to the tangent line.

Tangent from secants

Nearby secant lines get closer and closer to the tangent line.

Tangent from secants

Nearby secant lines get closer and closer to the tangent line.

Tangent from secants

Nearby secant lines get closer and closer to the tangent line.

Tangent from secants

Let’s try with formulas. Consider f (x) = x3, and let’s find thetangent line through (2, 8).

We’ll do this by studying the secant lines passing through (2, 8)and (x , x3), when x is close to 2.

Since x is close to 2, we write it as 2 + h.

2 + h getting close to 2 is the same as h getting close to 0.

Tangent from secants

The slope of the secant through (2, f (2) = 8) and (2 +h, f (2 +h)):

∆f

∆x=

f (2 + h)− f (2)

(2 + h)− 2=

(2 + h)3 − 2

h=

(h3 + 6h2 + 12h + 8)− 8

h= h2 + 6h + 12

Tangent from secants

As h gets close to zero, h2 + 6h + 12 gets close to 12.

Tangents from secants

The slope of the tangent to the graph of f (x) = x3 at (2, 8) is 12.

The tangent line itself is given by (y − 8) = 12(x − 2)

Tangent from secants

Here we have f (x) = x3 and g(x) = 12(x − 2) + 8.

Try it yourself!

Using only these ideas (finding tangents from secants), find thetangent line to x2 + 2x + 3 at (1, 6).

Limits

In order to compute the slope of the tangent line, we ended uptrying to understand how the quantities

f (x)− f (2)

x − 2

behave as x approaches 2.

To do this kind of thing in an organized way, it helps to introducethe notion of limit.

Limits

For a function f (x), the question

limx→x0

f (x) = ?

asks about the behavior of f (x) near but not at x0.

In terms of the previous discussion, the near but not at distinctionis important because the secant through (x0, f (x0)) and (x , f (x))is not defined when x = x0. In terms of the formula, x = x0 is notin the domain of (f (x)− f (x0))/(x − x0).

Limits

More precisely, we say

limx→x0

f (x) = y0

if, by restricting our attention to x very close to x0 we canguarantee that f (x) is very close to y0.

Limits

Even more precisely, we say

limx→x0

f (x) = y0

if, for every ε > 0, there is some δ > 0, such that in order toguarantee that f (x) is within ε of y0, it suffices to take x0 within δover x .

Limits

Limits

We’ll return to these more precise definitions in a couple days.

For now, let’s look at examples to gain some intuition.

Limits

limx→1

x2 = ?

We are asking: as x gets close to 1, what happens to x2?

Limits

As x gets close to 1, what happens to x2?

Looks like it gets close to 1.

Limits

More quantitatively: writing x = 1 + h, we have

x2 = (1 + h)2 = 1 + 2h + h2

x2 − 1 = (1 + 2h + h2)− 1 = 2h + h2

That is, the difference between x2 and 1 becomes small as h→ 0,i.e., as x → 1.

The above formula in fact tells us how small, which we will usewhen we return to the precise ε− δ definition.

Try it yourself

limx→4

(x2 + 3x + 2) = ?

Limits

We will see (next week) that for any polynomial, rational,algebraic, trigonometric, exponential function, f , or for functionsthat are combinations of these, and for any point x0 in the domainof f , one has

limx→x0

f (x) = f (x0)

But this is not true for all functions!

Limits

Consider the function with the following graph:

Limits

In symbols we might write

h(x) =

1 x > 0

1/2 x = 0

0 x < 0

Sometimes this is called the “Heaviside function”

Heaviside

Limits

What’s the limit of this function as x → 0?

Limits

There is no limit as x → 0. Indeed, a limit would have to be equalto both 0 because numbers arbitrarily close to zero but smallerhave h(x) = 0, and 1 because numbers arbitrarily close to zero butlarger have h(x) = 1.

Limits

At which x0 does limx→x0 f (x) exist? What is it?

The limit exists except when x = −1, 1, 2. When defined, the limitis given by the value of the function.

Limits

At which x0 does limx→x0 f (x) exist? What is it?

The limit always exists, and is given by the value of the function.

Limits

At which x0 does limx→x0 f (x) exist? What is it?

The limit exists except at x = 2, and is given by the value of thefunction.

Limits

At which x0 does limx→x0 f (x) exist? What is it?

The limit always. It is given by the value of the function except atx = 1, where it is 2.

Limits at infinity

We saylimx→x0

f (x) =∞

when, (roughly), as x gets close to x0, the quantity f (x) becomesincreasingly large and positive.

Similarly, we saylimx→x0

f (x) = −∞

when f (x) becomes very negative near x0.

Limits at infinity

Example: For small x near 0, the quantity 1x2

is large and positive:

limx→0

1

x2=∞

Limits at infinity

Example: For small x near 0, the quantity 1x has large absolute

value, but can be either positive or negative.

limx→0

1

xis undefined

One-sided limits

We saylim

x→x+0

f (x) = y

when, roughly, as x gets close to x0 while remaining larger than it,the quantity f (x) approaches y .

We saylim

x→x−0

f (x) = y

when, roughly, as x gets close to x0 while remaining smaller thanit, the quantity f (x) approaches y .

One-sided limits

If limx→x0 f (x) exists, then

Conversely, if limx→x−0f (x) and limx→x+0

f (x) exist and are equal,

then also limx→x0 f (x) exists and is equal to both.

However, it is possible that limx→x−0f (x) and limx→x+0

f (x) exist

but are not equal. In this case limx→x0 f (x) does not exist.

One-sided limits

Which one-sided limits limx→x+0f (x) and limx→x−0

f (x) exist?

What are they? When are they equal?

Except at x = 0, the usual limit limx→x0 f (x) exists and is equal tof (x0); hence the same for both one-sided limits.

We also have limx→0+ = 1 and limx→0− = 0.

One-sided limits

At which x0 does limx→x+0f (x) or limx→x−0

f (x) exist? What is it?

Already the ordinary limit existed except at x = −1, 1, 2. At thesevalues, the one-sided limits exist, but are different. E.g.:

limx→1−

f (x) = 1 limx→1+

f (x) = 2

One-sided limits at infinity

limx→0+

1

x=∞ lim

x→0−

1

x= −∞

sin(1/x)

The function sin(1/x) can be made to assume any value between−1 and 1 for x arbitrarily close to zero (on either side).

Thus it has no limit as x → 0 (or one-sided limits).

(1/x) sin(1/x)