Calculus (Math 1A) Lecture 5 - PDF Free Download

Calculus (Math 1A) Lecture 5 Vivek Shende September 5, 2017

Hello and welcome to class!

Hello and welcome to class! Last time

Hello and welcome to class! Last time We discussed composition, inverses, exponentials, and logarithms.

Hello and welcome to class! Last time We discussed composition, inverses, exponentials, and logarithms. Today

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

Zeno s paradox of the arrow At each instant, the arrow occupies a single position in space.

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 change position in any instant, it is not moving at any instant.

Lesson from Zeno

Lesson from Zeno We should be careful discussing anything at an instant.

Lesson from Zeno We should be careful discussing anything at an instant. Answer to Zeno (?):

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,

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 sense to discuss the rate of motion at an instant, and this is not zero.

Rate

Rate rate of change = amount of change time it took

Rate rate of change = amount of change time it took If what is changing is described by a number a varying with time as a(t),

Rate rate of change = amount of change time it took If what is changing is described by a number a varying with time as a(t), then the rate of change over some interval [t 0, t 1 ] is

Rate rate of change = amount of change time it took If what is changing is described by a number a varying with time as a(t), then the rate of change over some interval [t 0, t 1 ] is a(t 1 ) a(t 0 ) t 1 t 0

Secant line

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

Tangent line

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

Tangent line

Tangent from secants Nearby secant lines get closer and closer to the tangent line.

Tangent from secants

Tangent from secants Let s try with formulas.

Tangent from secants Let s try with formulas. Consider f (x) = x 3, and let s find the tangent line through (2, 8).

Tangent from secants Let s try with formulas. Consider f (x) = x 3, and let s find the tangent line through (2, 8). We ll do this by studying the secant lines passing through (2, 8) and (x, x 3 ), when x is close to 2. Since x is close to 2, we write it as 2 + h.

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

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 =

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 = (h 3 + 6h 2 + 12h + 8) 8 h = h 2 + 6h + 12

Tangent from secants As h gets close to zero, h 2 + 6h + 12 gets close to 12.

Tangents from secants The slope of the tangent to the graph of f (x) = x 3 at (2, 8) is 12.

Tangents from secants The slope of the tangent to the graph of f (x) = x 3 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) = x 3 and g(x) = 12(x 2) + 8.

Try it yourself!

Try it yourself! Using only these ideas (finding tangents from secants), find the tangent line to x 2 + 2x + 3 at (1, 6).

Limits

Limits In order to compute the slope of the tangent line, we ended up trying to understand how the quantities behave as x approaches 2. f (x) f (2) x 2

Limits In order to compute the slope of the tangent line, we ended up trying to understand how the quantities behave as x approaches 2. f (x) f (2) x 2 To do this kind of thing in an organized way, it helps to introduce the notion of limit.

Limits

Limits For a function f (x), the question lim f (x) =? x x 0 asks about the behavior of f (x) near but not at x 0.

Limits For a function f (x), the question lim f (x) =? x x 0 asks about the behavior of f (x) near but not at x 0. In terms of the previous discussion, the near but not at distinction is important because the secant through (x 0, f (x 0 )) and (x, f (x)) is not defined when x = x 0.

Limits More precisely, we say lim x x 0 f (x) = y 0

Limits More precisely, we say lim x x 0 f (x) = y 0 if, by restricting our attention to x very close to x 0

Limits More precisely, we say lim x x 0 f (x) = y 0 if, by restricting our attention to x very close to x 0 we can guarantee that f (x) is very close to y 0.

Limits Even more precisely, we say lim x x 0 f (x) = y 0

Limits Even more precisely, we say lim x x 0 f (x) = y 0 if, for every ɛ > 0, there is some δ > 0, such that in order to guarantee that f (x) is within ɛ of y 0, it suffices to take x 0 within δ over x.

Limits

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

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 lim x 2 =? x 1

Limits lim x 2 =? x 1 We are asking: as x gets close to 1, what happens to x 2?

Limits As x gets close to 1, what happens to x 2?

Limits As x gets close to 1, what happens to x 2? Looks like it gets close to 1.

Limits More quantitatively:

Limits More quantitatively: writing x = 1 + h, we have x 2 = (1 + h) 2 = 1 + 2h + h 2

Limits More quantitatively: writing x = 1 + h, we have x 2 = (1 + h) 2 = 1 + 2h + h 2 x 2 1 = (1 + 2h + h 2 ) 1 = 2h + h 2

Limits More quantitatively: writing x = 1 + h, we have x 2 = (1 + h) 2 = 1 + 2h + h 2 x 2 1 = (1 + 2h + h 2 ) 1 = 2h + h 2 That is, the difference between x 2 and 1 becomes small as h 0, i.e., as x 1.

Limits More quantitatively: writing x = 1 + h, we have x 2 = (1 + h) 2 = 1 + 2h + h 2 x 2 1 = (1 + 2h + h 2 ) 1 = 2h + h 2 That is, the difference between x 2 and 1 becomes small as h 0, i.e., as x 1. The above formula in fact tells us how small, which we will use when we return to the precise ɛ δ definition.

Try it yourself lim (x 2 + 3x + 2) =? x 4

Limits

Limits We will see (next week) that for any polynomial, rational, algebraic, trigonometric, exponential function, f, or for functions that are combinations of these, and for any point x 0 in the domain of f, one has lim f (x) = f (x 0 ) x x 0

Limits Consider the function with the following graph:

Limits In symbols we might write 1 x > 0 h(x) = 1/2 x = 0 0 x < 0

Limits In symbols we might write 1 x > 0 h(x) = 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.

Limits There is no limit as x 0. Indeed, a limit would have to be equal to both 0

Limits There is no limit as x 0. Indeed, a limit would have to be equal to both 0 because numbers arbitrarily close to zero but smaller have h(x) = 0,

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

Limits There is no limit as x 0. Indeed, a limit would have to be equal to both 0 because numbers arbitrarily close to zero but smaller have h(x) = 0, and 1 because numbers arbitrarily close to zero but larger have h(x) = 1.

Limits At which x 0 does lim x x0 f (x) exist? What is it?

Limits At which x 0 does lim x x0 f (x) exist? What is it? The limit exists except when x = 1, 1, 2. When defined, the limit is given by the value of the function.

Limits At which x 0 does lim x x0 f (x) exist? What is it?

Limits At which x 0 does lim x x0 f (x) exist? What is it? The limit always exists, and is given by the value of the function.

Limits At which x 0 does lim x x0 f (x) exist? What is it?

Limits At which x 0 does lim x x0 f (x) exist? What is it? The limit exists except at x = 2, and is given by the value of the function.

Limits At which x 0 does lim x x0 f (x) exist? What is it?

Limits At which x 0 does lim x x0 f (x) exist? What is it? The limit always. It is given by the value of the function except at x = 1, where it is 2.

Limits at infinity

Limits at infinity We say lim f (x) = x x 0 when, (roughly), as x gets close to x 0, the quantity f (x) becomes increasingly large and positive.

Limits at infinity We say lim f (x) = x x 0 when, (roughly), as x gets close to x 0, the quantity f (x) becomes increasingly large and positive. Similarly, we say lim x x 0 f (x) = when f (x) becomes very negative near x 0.

Limits at infinity Example:

Limits at infinity Example: For small x near 0, the quantity 1 x 2 is large and positive:

Limits at infinity Example: For small x near 0, the quantity 1 x 2 is large and positive: lim x 0 1 x 2 =

Limits at infinity Example:

Limits at infinity Example: For small x near 0, the quantity 1 x has large absolute value, but can be either positive or negative.

Limits at infinity Example: For small x near 0, the quantity 1 x has large absolute value, but can be either positive or negative. 1 lim x 0 x is undefined

One-sided limits

One-sided limits We say lim x x + 0 f (x) = y when, roughly, as x gets close to x 0 while remaining larger than it, the quantity f (x) approaches y.

One-sided limits We say lim x x + 0 f (x) = y when, roughly, as x gets close to x 0 while remaining larger than it, the quantity f (x) approaches y. We say lim x x 0 f (x) = y when, roughly, as x gets close to x 0 while remaining smaller than it, the quantity f (x) approaches y.

One-sided limits If lim x x0 f (x) exists,

One-sided limits If lim x x0 f (x) exists, then

One-sided limits If lim x x0 f (x) exists, then Conversely, if lim x x 0 f (x) and lim x x + 0 f (x) exist

One-sided limits If lim x x0 f (x) exists, then Conversely, if lim x x 0 f (x) and lim x x + 0 f (x) exist and are equal,

One-sided limits If lim x x0 f (x) exists, then Conversely, if lim x x f (x) and lim 0 x x + f (x) exist and are equal, 0 then also lim x x0 f (x) exists and is equal to both.

One-sided limits If lim x x0 f (x) exists, then Conversely, if lim x x f (x) and lim 0 x x + f (x) exist and are equal, 0 then also lim x x0 f (x) exists and is equal to both. However, it is possible that lim x x 0 but are not equal. f (x) and lim x x + 0 f (x) exist

One-sided limits If lim x x0 f (x) exists, then Conversely, if lim x x f (x) and lim 0 x x + f (x) exist and are equal, 0 then also lim x x0 f (x) exists and is equal to both. However, it is possible that lim x x f (x) and lim 0 x x + f (x) exist 0 but are not equal. In this case lim x x0 f (x) does not exist.

One-sided limits Which one-sided limits lim x x + f (x) and lim 0 x x 0 What are they? When are they equal? f (x) exist?

One-sided limits Which one-sided limits lim x x + f (x) and lim 0 x x 0 What are they? When are they equal? f (x) exist? Except at x = 0, the usual limit lim x x0 f (x) exists and is equal to f (x 0 ); hence the same for both one-sided limits.

One-sided limits At which x 0 does lim x x + 0 f (x) or lim x x 0 f (x) exist? What is it?

One-sided limits At which x 0 does lim x x + 0 f (x) or lim x x 0 f (x) exist? What is it? Already the ordinary limit existed except at x = 1, 1, 2. At these values, the one-sided limits exist, but are different.

One-sided limits at infinity

One-sided limits at infinity 1 lim x 0 + 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).

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)