(MTH 250)

Lecture 11

Calculus

Previous Lecture’s Summary

•Summary of differentiation rules: Recall

•Chain rules

•Implicit differentiation

•Derivatives of logrithemic functions

•Derivatives of hyperbolic functions.

•Derivatives of inverse trigonometric functions

•Derivatives of inverse hyperbolic functions

•Summary of results

Today’s Lecture

•Recalls

•Differentials

•Local linear approximations

•Indetermined forms

•L’Hopitâl rule

Theorem: (Chain rule) If is differentiable at an arbitrary point and is

differentiable at, then the composition is differentiable at Moreover,

Corollary: If and thenand

Recalls

Rule of Thumb: The derivative of is the derivative of the outsidefunctionevaluatedat the insidefunction times the derivative of the insidefunction.

Recalls

Recalls

Implicit differentiation: Differentiatebothsides of the equationwith respect to treating as (temporarilyunspecified) differentiablefunction of .

Example: Find when .

Solution:

DifferentialsWe have been interpreting as one quantityrepresenting the derivative of with respect to

We will now give the quantities and separate meanings that will allow us to treat as a ratio.

and terminology and the concept of differentials will also be used to approximate functions by simpler linear functions.

Differentials• The ratio canbeinterpreted as the slope of the secant line

joining the points and .

• is the vertical change in the secant line (rise) given by .

• If the derivativeisconsidered as the ratio of dy and dx then

is the vertical chage in the tangent line.

• The quantities and are called the differentials.

Differentials

The variable x is an indendent variable and sodoes dx. It canbeassignedanyarbitrary value and

However, and

• Let

• Thus

• But whenverysmall

Differentials

Example: Let Find and atwith.

Solution.

Differentials

Differentials

• The quantity

• If erroris positive thenestimateislessthanactual value.

• If errorisnegativethenestimateisgreaterthanactual value.

Local linear approximations• If the graph of a function is magnified at a point P that is

differentiable, the function is said to be locally linear at P.

• The tangent line through P closely approximates the graph.

• A technique called local linear approximation is used to evaluate function at a particular value.

• When measurements of independent variables have small errors then the computed functions will also be affected. This is known as error propagation.

• Our goal is to estimate errors in the function using local linear appraoximation and differentials.

Local linear approximations

• Let be the exact value of the quatitiybeingmeasured.

• is the exact value of the quantitybeingcomputed.

• is the measured value of

• is the computed value of y.

• Wedefine to be the measurementerror of

• to be the propagatederror of y.

• It followsthat the propagateerrorcanbeapproximated by

• As is not known, we use instead.

Local linear approximationsExample: Suppose that the side of a square ismeasuredwith a ruler to be 10 incheswith a measurementerror of atmostEstimate the error in the computed area of the square.

Solution: Let .

With, if , the canbeapproximated as

But to saythat the measurementerrorisatmostmeansthat

Local linear approximationsExample: The diameter of a polyurethanesphereismeasuredwithprercentageerrorwith. Estimate the percentageerror in the calculated volume of the sphere.

Solution:

Local linear approximationsExample: The diameter of a polyurethanesphereismeasuredwithprercentageerrorwith. Estimate the percentageerror in the calculated volume of the sphere.

Solution:

Local linear approximationsExample: Use the differential to approximate and estimate the relative error percent

Solution: • Let and (since is the perfect square of and is near to ), then .

• Substituting and in the approximating formula we get

• So

• Since then

Indeterminate From

• There are times when we need to evaluate functions which are rational

• We end up with the indeterminate form

• At a specific point it may evaluate to an indeterminate form

• Note why this is indeterminate

3

2

27( )

9

xf x

x

001 0

0

0

0

00 0 ?

0n n n

L’Hôpital Rule

L’Hôpital Rule

We can confirm L’Hôpital’s rule by working backwards, and using the definition of derivative:

f a

g a

lim

lim

x a

x a

f x f a

x ag x g a

x a

limx a

f x f a

x ag x g a

x a

limx a

f x f a

g x g a

0lim

0x a

f x

g x

limx a

f x

g x

L’Hôpital Rule

• Suppose gives an indeterminate form (and the limit exists)

• It is possible to find a limit by

• Note: this only works when the original limit gives an indeterminate form.

( )lim

( )x c

f x

g x

'( )lim

'( )x c

f x

g x

L’Hôpital Rule

Example: Find. • By direct substitution, we get: .

• By canceling out the commonfactorsweget

• By l’Hôpital rulewe have

L’Hôpital Rule

Applying l’Hôpital’s Rule:

• Check that the limit of is an indeterminateform of type .

• Differentiate and separetely.

• Find the limit of . If this limit is finite, or , then it is equal to the limit

of .

L’Hôpital Rule

Example: Find

Solution:

Example: Find

Solution:

L’Hôpital RuleUse apply L’Hôpital’s rule as many times as necessary as long as the fraction is still indeterminate. For example.

1

1lim

1 1 lnx x

L’Hôpital again.

1

2

1

1 1lim

ln 1x x x

1

1 lnlim

1 lnx

x x

x x

Now it is in the form0

0

This is indeterminate form

1

11

lim1

ln ( 1)x

x

x xx

L’Hôpital’s rule applied once.

0

0Fractions cleared. Still

1

1lim

ln 1x

x

x x x

L’Hôpital Rule

L’Hôpital Rule

Example: Find

Solution:

L’Hôpital Rule

Example: Find

Solution: Remark that now we have form. However we can make it or form.

L’Hôpital Rule

Other indeterminateforms: :

Limits of the form can give rise to indeterminate forms of the types . For example is of the form .

• Introduce

• Take the ln :

• Use the alreadystudiedruleto evaluatelimitor

L’Hôpital RuleExample:

Solution. Let , then

Thus

Then

Since, we have exponentialfunctioniscontinuous and as This impliesthatTherefore,

L’Hôpital RuleExample: Find

Solution:

0e1

1/lim x

xx

1/lim ln x

xx

e 1

lim lnx

xxe ln

limx

x

xe 1

lim

1x

x

e

L’Hôpitalapplied

1/lim x

xx

0

Lecture Summary

•Recalls

•Differentials

•Local linear approximations

•Indetermined forms

•L’Hopitâl rule