unit #6 : families of functions, taylor polynomials, l...

Unit #6 : Families of Functions, Taylor Polynomials, l’Hopital’sRule

Goals:

• To use first and second derivative information to describe functions.

• To be able to find general properties of families of functions.

• To extend our tangent line formula to higher-degree polynomial approximations(Taylor polynomials)

• To explore more advanced ways to evaluate limits.

Interpreting First and Second Derivatives - 1

Interpreting First and Second Derivatives

The information about the graph of a function f provided by the sign of f ′(x) andf ′′(x) on an interval (a, b) is expressed in the following table. (a and b are assumedto be finite.)

f ′(x) > 0 on (a, b) f increasing on [a, b]

f ′(x) < 0 on (a, b) f decreasing on [a, b]

f ′′(x) > 0 on (a, b) f concave up on [a, b]

f ′′(x) < 0 on (a, b) f concave down on [a, b]


All the indicators above deal with non-zero values of f ′(x) and f ′′(x). Whatis distinctive about the zero values of these derivatives?

f ′(x) = 0 :

f ′′(x) = 0 :


The intervals of increasing and decreasing described in the table earlier can lead toa surprising or counter-intuitive technical point.Example: For the function f (x) = x2, find the derivative.

Find the intervals where f ′(x) > 0 and f ′(x) < 0.


Based on this analysis, on which interval(s) is f (x) = x2 increasing, andwhich interval(s) is it decreasing?

(a) Increasing on (0,∞), decreasing on (−∞, 0).

(b) Increasing on [0,∞), decreasing on (−∞, 0].

(c) Increasing on [0,∞], decreasing on [−∞, 0].

Critical Points - 1

Critical Points

If f (x) is defined on the interval (a, b), then we call a point c in the interval acritical point if:

• f ′(c) = 0, or

• f ′(c) does not exist.

We will also refer to the point (c, f (c)) on the graph of f (x) as a critical point.We call the function value f (c) at a critical point c a critical value.

Critical Points - 2

Technical Notes:

1. By this definition, f (c) must be defined for c to be a critical point.

a) Sketch f (x) = 1/x.

By the critical point definition, is x = 0 a critical point?

(a) Yes.

(b) No.

Critical Points - 3

b) Sketch g(x) = |x|.

Is x = 0 a critical point?

(a) Yes.

(b) No.

Critical Points - 4

2. By this definition, if a function is defined on a closed interval, the endpoints ofinterval cannot be critical points.

a) Sketch the graph of f (x) =√x and decide whether x = 0 is a critical

point.

Critical Points - 5

b) Sketch the graph of g(x) = 3√x and decide whether x = 0 is a critical

point.

Critical Points - 6

Example: Identify all the critical points on the graph below, and character-ize any other interesting points by continuity, limits, or other properties.

First and Second Derivative Sign Chart Example - 1

Example: Consider the function

f (x) =x

x2 + 1

Construct a sign chart for both f ′ and f ′′, and use this information to sketchf (x).

First and Second Derivative Sign Chart Example - 2

f (x) =x

x2 + 1

Families of Functions - Example 1 - 1

Families of Functions

Looking at the first and second derivative properties can be generalized to allowus to sketch families of functions, rather than a single function at a time.(A family of functions is a set of functions that share a common mathematicalform, but differ in the particular value they might have for one or more parameters.)


Example: Consider the family of functions

f (x) =ax

x2 + b

Let b = 1, then sketch several members of the family with different positivevalues of a.


f (x) =ax

x2 + b, with b = 1


f (x) =ax

x2 + bSuppose a = 1 now. Create a sign chart for f ′(x), given that b can change.

Find the (x, y) coordinates of the critical points of f (x), in terms of b.


f (x) =ax

x2 + bSketch several members of the family, for a = 1 and then using different

positive values of b.


f (x) =ax

x2 + bWould the family change substantially if a or b could be negative? If so, whatwould the change look like?


f (x) =ax

x2 + bFind the member of this family which has its maximum at (1, 10).


Example: Show that, for positive constants a and b, g(x) = a(1 − e−bx) isboth increasing and concave down for all x.


g(x) = a(1− e−bx), a and b positive.

Evaluate g(0).


g(x) = a(1− e−bx), a and b positive.

Evaluate the limit limx→∞

g(x).

(a) limx→∞

g(x) = 0

(b) limx→∞

g(x) = 1

(c) limx→∞

g(x) = a

(d) limx→∞

g(x) = a(1− e−bx)


Sketch what members of the family g(x) = a(1 − e−bx) might look like forx ≥ 0.

Taylor Polynomial Intro - 1

Taylor Polynomials

A more technical application of derivative information, but a very powerful one, isthe construction of polynomial approximations to more complicated functions.Previously we found a formula for linear approximations to functions f (x) arounda point x = a:

This linear approximation, or tangent line formula, can also be called the Taylorpolynomial of degree 1 approximating f (x) near x = a.


Sketch the graph of cos(x) around x = 0, and add its tangent line based atx = 0.

The linearization or tangent line is clearly a very limited approximation tothis function. What might be a slightly more complex form of function thatwould work better in this case?


Taylor Polynomial of Degree 2

f (x) ≈ f (a) + f ′(a)(x− a) +f ′′(a)

2(x− a)2

is a quadratic approximation to f (x) near x = a.

For values of x close to a do you think this quadratic approximation will be abetter or worse approximation than the tangent line, and why?

(a) The quadratic approximation will be better than the linear approximation.

(b) The quadratic approximation will be worse than the linear approximation.

Taylor Polynomials - Examples - 1

Example: Find the quadratic Taylor approximation to f (x) = cos(x) nearx = 0.


Sketch the graph of cos(x) around x = 0, and add both its 1st and 2nd degreeTaylor polynomial approximations for x near 0.


For reference, here is a computer generated version of y = cos(x) and its linear andquadratic Taylor approximations around x = 0.

0 0.5 1.0 1.5−0.5−1.0−1.5

x

y


There is a very good reason for the particular form of the Taylor polynomial.What mathematical features will f (x) and its 2nd degree Taylor approximationshare at x = a?


If we wanted a still-better approximation for f (x) near a specific point x = a,how could we generalize our earlier 1st and 2nd degree Taylor polynomials?

Taylor Polynomials - Inverting the Process - 1

Example: You are told that a function can be approximated around x = 2with the quadratic

y = −2 + 3(x− 2)− 3(x− 2)2.

Sketch the function near x = 2, and indicate any specific values you candetermine from the information provided.

Taylor Polynomials - Inverting the Process - 2

Example: You are told that f (x) ≈ 7− (x−5)+(x−5)3 for x near 5. Whatcan you say about the value and derivatives of f (x) at x = 5?

Applications of Taylor Polynomials - 1

Applications of Taylor Polynomials

It is not immediately obvious to most students why we would ever want to replacea perfectly good function like y = ex with its approximation y ≈ 1 + x. However,it can be argued that these Taylor approximations (and related ones like Fourierseries) are comparable in importance to the fundamental calculus ideas of thederivative and integral.Let us see how Taylor polynomials can help us answer previously unanswerablequestions.


Example: Evaluate the limit limx→0

e−2x − 1

x.



sin(5x)

x.


Have you seen an alternative method for evaluating limits like this? If so,what was it called?

L’Hopital’s Rule - 1

L’Hopital’s Rule

Use Taylor polynomials to show that if the ratio limx→a

f (x)

g(x)gives the form

0

0,

then

limx→a

f (x)

g(x)= lim

x→a

f ′(x)

g′(x)


L’Hopital’s rule can be applied in slightly more general circumstances as well.

L’Hopital’s Rule

When a limit limx→a

or limx→±∞

off (x)

g(x)yields an indeterminate ratio form of

0

0or±∞±∞

, then

limx→a

or limx→±∞

f (x)

g(x)= lim

x→aor lim

x→±∞

f ′(x)

g′(x)



1− e−2x

x.


Example: Evaluate the limit limx→∞

1− e−2x

x.



1− cos(4x)

x2.


Example: Evaluate the limit limx→0+

x ln(x).

unit #6 : families of functions, taylor polynomials, l...

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