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Unit #6 : Rule

Families of Functions, Taylor Polynomials, l’Hopital’s

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) and f ′′(x) on an interval (a, b) is expressed in the following table. (a and b are assumed to 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]

Interpreting First and Second Derivatives - 2

All the indicators above deal with non-zero values of f ′(x) and f ′′ (x). What is distinctive about the zero values of these derivatives? f ′(x) = 0 :

f ′′(x) = 0 :

Interpreting First and Second Derivatives - 3

The intervals of increasing and decreasing described in the table earlier can lead to a 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.

Interpreting First and Second Derivatives - 4

Based on this analysis, on which interval(s) is f (x) = x2 increasing, and which 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 a critical 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 of interval 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) = point.

√ 3

x and decide whether x = 0 is a critical

Critical Points - 6

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

First and Second Derivative Sign Chart Example - 1

Example: Consider the function x x2 + 1 Construct a sign chart for both f ′ and f ′′ , and use this information to sketch f (x). f (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 allow us 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 mathematical form, but differ in the particular value they might have for one or more parameters.)

Families of Functions - Example 1 - 2

Example: Consider the family of functions ax f (x) = 2 x +b Let b = 1, then sketch several members of the family with different positive values of a.

Families of Functions - Example 1 - 3

f (x) =

ax , with b = 1 +b

x2

Families of Functions - Example 1 - 4

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

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

Families of Functions - Example 1 - 5

ax x2 + b Sketch several members of the family, for a = 1 and then using different positive values of b. f (x) =

Families of Functions - Example 1 - 6

ax x2 + b Would the family change substantially if a or b could be negative? If so, what would the change look like? f (x) =

Families of Functions - Example 1 - 7

ax x2 + b Find the member of this family which has its maximum at (1, 10). f (x) =

Families of Functions - Example 2 - 1

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

Families of Functions - Example 2 - 2

Evaluate g(0).

g(x) = a(1 − e−bx ), a and b positive.

Families of Functions - Example 2 - 3

g(x) = a(1 − e−bx ), a and b positive.

Evaluate the limit lim g(x). x→∞

(a) lim g(x) = 0 x→∞

(b) lim g(x) = 1 x→∞

(c) lim g(x) = a x→∞

(d) lim g(x) = a(1 − e−bx ) x→∞

Families of Functions - Example 2 - 4

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

Taylor Polynomial Intro - 1

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

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

Taylor Polynomial Intro - 2

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

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

Taylor Polynomial Intro - 3

Taylor Polynomial of Degree 2

f (x) ≈

f (a)

′

+ f (a)(x − a)

f ′′(a) (x − a)2 + 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 a better 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: x = 0.

Find the quadratic Taylor approximation to f (x) = cos(x) near

Taylor Polynomials - Examples - 2

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

Taylor Polynomials - Examples - 3

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

y

−1.5 −1.0 −0.5

0

0.5

1.0

x 1.5

Taylor Polynomials - Examples - 4

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 approximation share at x = a?

Taylor Polynomials - Examples - 5

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 = 2 with the quadratic y = −2 + 3(x − 2) − 3(x − 2)2. Sketch the function near x = 2, and indicate any specific values you can determine 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. What can 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 replace a 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 Fourier series) are comparable in importance to the fundamental calculus ideas of the derivative and integral. Let us see how Taylor polynomials can help us answer previously unanswerable questions.

Applications of Taylor Polynomials - 2

e−2x − 1 Example: Evaluate the limit lim . x→0 x

Applications of Taylor Polynomials - 3

sin(5x) . x→0 x

Example: Evaluate the limit lim

Applications of Taylor Polynomials - 4

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

then

f (x)

0, gives the form x→a g(x) 0

Use Taylor polynomials to show that if the ratio lim f (x) f ′(x) lim = lim ′ x→a g(x) x→a g (x)

L’Hopital’s Rule - 2

L’Hopital’s rule can be applied in slightly more general circumstances as well. L’Hopital’s Rule When a limit lim or lim of x→a

±∞ 0 , then or ±∞ 0

x→±∞

f (x) yields an indeterminate ratio form of g(x)

f (x) f ′(x) lim or lim = lim or lim ′ x→±∞ g(x) x→±∞ g (x) x→a x→a

L’Hopital’s Rule - 3

1 − e−2x Example: Evaluate the limit lim . x→0 x

L’Hopital’s Rule - 4

1 − e−2x Example: Evaluate the limit lim . x→∞ x

L’Hopital’s Rule - 5

1 − cos(4x) . x→0 x2

Example: Evaluate the limit lim

L’Hopital’s Rule - 6

Example: Evaluate the limit lim x ln(x). x→0+...