calculus (math 1a) lecture 4vivek/1a/4.handout.pdf · calculus (math 1a) lecture 4 vivek shende...

Calculus (Math 1A)Lecture 4

Vivek Shende

August 31, 2017

Hello and welcome to class!

Last time

We discussed shifting, stretching, and composition.

Today

We finish discussing composition, then discuss inverses and theexponential function.

Picking up from last time

Let’s write

multa(x) = ax addb(x) = x + b

these functions compose as follows:

multa ◦multa′ = multaa′ addb ◦ addb′ = addb′ ◦ addb

Last time we called these functions stra and shb because we werethinking about how composition with them affects the graph.

Composing stretches and shifts

Exercise: Find some c, d such that

addb ◦multa = multc ◦ addd

Let’s just evaluate both sides.

(addb ◦multa)(x) = addb(ax) = ax + b

(multc ◦ addd)(x) = multc(x + d) = cx + cd

So we should take c = a and d = b/a.

addb ◦multa = multa ◦ addb/a

Associativity

A priori, f ◦ (g ◦ h) and (f ◦ g) ◦ h have different meanings:

the first means first do the function g ◦ h, then do the function fto the result

the second means first do the function h, then do the functiong ◦ h to the result

However, unpacking further, both amount to: first do the functionh, then do the function g to the result of that, then do the functionf to the result of that. We write f ◦ g ◦ h to mean either of these.

Associativity example

We saw doing a vertical shift then a vertical stretch is not thesame as doing these in the reverse order.

What about a vertical shift up by b, then a horizontal stretch by afactor of 1/a, versus the same in the opposite order?

Given our starting function f , the first of these would be(addb ◦ f ) ◦multa, and the second addb ◦ (f ◦multa).

Associativity tells us that these are the same.

The identity function

Consider the function I (x) = x . Composing with it does not affectother functions:

(f ◦ I )(x) = f (I (x)) = f (x) (I ◦ f )(x) = I (f (x)) = f (x)

I (x) is to composition as 1 is to multiplication, as 0 is to addition.

For this reason it is sometimes called the identity function. (Youdo not need to remember this name.)

The inverse of a function

We say g is the inverse of f if:

f (g(x)) = x g(f (x)) = x

Sometimes we write g = f −1.

DO NOT GET CONFUSED: f −1 DOES NOT MEAN 1/f .

What is being inverted is not the value of the function, but insteadthe operation that the function is doing.

Another perspective: f ◦ g = I = g ◦ f . I.e., the inverse function iswhat one must compose with to get the identity function.

The inverse of a linear function

What’s the inverse of f (x) = ax + b?

That is, what function g(x) has the properties that

g(f (x)) = g(ax + b) = x f (g(x)) = ag(x) + b = x

Solving the second equation for g gives g(x) = x−ba . This also

satisfies the first equation, so is the inverse function.

In particular, (addb)−1 = add−b and (multa)−1 = mult1/a.

The inverse of f (x) = x2

We want a function g such that g(x2) = x and g(x)2 = x .

Of course we want to take g(x) =√x .

Note however that√x has domain only [0,∞) and moreover√

x2 = |x |, which is equal to x only if x is non-negative.

Thus√· is the inverse, not of the function f (x) = x2 with domain

(−∞,∞), but instead of the function f (x) = x2 with domain andrange [0,∞).

Graph of the inverse:

For g(x) is the inverse of f (x), consider a point on the graph of g :

(x , g(x)) = (f (g(x)), g(x))

This looks like a point on the graph of f (x) with its coordinatesreversed. The graph of the inverse is the reflection of the originalgraph across the line y = x .

When is there an inverse?

The function f (x) has an inverse g(x), if and only if: for every y inthe range of f , there is only one x in the domain of f such thatf (x) = y .

This is a necessary condition since, if f (x) = f (x ′) and g is aninverse to f , then

x = g(f (x)) = g(f (x ′)) = x ′

Moreover, under this condition, we can simply define g(y) to bethe unique x such that f (x) = y .


In terms of the graph: if f has an inverse g , its graph, being thegraph of a function, must satisfy the vertical line test

or equivalently, the graph of f must satisfy a horizontal line test(i.e. every horizontal line meets its graph at most once)

If the graph of f satisfies the horizontal line test, intersecting withthis line can be used to define (and calculate) the inverse.


These functions, on their original domains do not have inverses:

However, an inverse can be define after restricting the domain, e.g.to [0,∞) for f (x) = x2 and to (−π, π] for sin(x).

Inverse functions you know

Often the inverse of a function is not a function whose name youalready know.

Many years ago, you learned addition. Then subtraction, asfollows: if you want to subtract 4 from 5, you ask for the numberso that, if you add 4 to it, you get 5.

Said differently, if a4(x) = x + 4 is the function which adds 4, thensubtracting 4 from y is (a4)−1(y).

Similarly, you learned division as the inverse to multiplication

Inverse functions you know

Likewise, the meaning of the expression n√x is: the quantitywhose n’th power is x .

In other words, the function f (x) = n√x is by definition theinverse function to g(x) = xn.

Note for n even, we have to restrict the domain of g(x) to [0,∞)before discussing its inverse.

Exponentials

For n an integer, an means: multiply a by itself n times.

We just discussed the meaning of a1/n in this case.

Putting these together, you can make sense of

am/n = (am)1/n = (a1/n)m

when m, n are integers.

That is, ar makes sense when r is a rational number (fraction).

Exponentials

Note that when a > 1, then for integers N < M one has aN < aM .

The same holds for rational numbers: if r < s, then ar < as .

You can see this by putting r , s over a common denominator asr = m/n and s = l/n. Since then m < l , we have:

ar = am/n = (a1/n)m < (a1/n)l = al/n = as

Exponentials

Thus we see that for any real number a > 1, the expression ax isan increasing function defined on rational numbers x .

Fact: there is an increasing function defined on the real numbers,also denoted ax , which agrees with the function we have defined sofar on the rational numbers.

To compute this for some arbitrary real number x , you could takebetter and better rational approximations of x . Saying thisprecisely requires a discussion of limits.

For a < 1, the same holds, except the function is decreasing.

Exponentials

Here are f (x) = 2x , g(x) = 3x , (1/2)x , and ex .

Exponential versus polynomial

Here are f (x) = x2, g(x) = x4, x6, and ex .

Some uses of exponential functions

Radioactive decay


Interest


Population growth (when not limited by resources)


A sample problem: You invest $100 at 3% interest, compoundedannually. How many years will it take to grow to $10, 000?

Here we need to solve:

100× (1.05)x = 10000

or simplifying(1.05)x = 100

To find x , we need an inverse to the exponential function.

Logarithms

The logarithm is by definition the inverse of the exponential.

More precisely, the function y 7→ logb(y) is the inverse of thefunction x 7→ bx .

One reads logb(y) as the logarithm in base b of y .

So if (1.05)x = 100 then x = log1.05 100.

Logarithm and exponential facts

The log and exp have the following complementary behaviors:

abac = ab+c loga(BC ) = loga(B) + loga(C )

ab/ac = ab−c loga(B/C ) = loga(B)− loga(C )

(ab)c = abc loga(BC ) = C loga B

The last property can be rewritten:

logm n =loga n

loga m

Note this means that to compute logarithm in any base, it isenough to know how to compute logarithm in one particular base.

e

When doing calculus, it is particularly convenient to use the base e.

The number e is a certain irrational number. Approximately it is

e ∼ 2.7182818284590452353602874713526624977572470936999595749669676277

What is e?

Being irrational means that any description of e must involve alimit since we only know how to write rational numbers explicitly.So the following won’t really make sense until we discuss limits.

e =∞∑n=0

1

n!

e = limN→∞

(1 +

1

N

)N

Also: the tangent line at (0, 1) to the graph of ex has slope 1.

calculus (math 1a) lecture 4vivek/1a/4.handout.pdf · calculus (math 1a) lecture 4 vivek shende...

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