×

### Let's log you in.

or

Don't have a StudySoup account? Create one here!

×

or

by: Rosa Farrell

16

0

3

# SI Calculus II MATH 1220

Rosa Farrell
Weber State University
GPA 3.53

Staff

These notes were just uploaded, and will be ready to view shortly.

Either way, we'll remind you when they're ready :)

Get a free preview of these Notes, just enter your email below.

×
Unlock Preview

COURSE
PROF.
Staff
TYPE
Class Notes
PAGES
3
WORDS
KARMA
25 ?

## Popular in Math

This 3 page Class Notes was uploaded by Rosa Farrell on Wednesday October 28, 2015. The Class Notes belongs to MATH 1220 at Weber State University taught by Staff in Fall. Since its upload, it has received 16 views. For similar materials see /class/230807/math-1220-weber-state-university in Math at Weber State University.

×

## Reviews for SI Calculus II

×

×

### What is Karma?

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 10/28/15
ADDITIONAL NOTES AND EXERCISES FOR 71 These notes probably belong towards the end of chapter 6 rather than at the start of chapter 7 Unfortunately I was unable to write them in time for that In this discussion we will concern ourselves with linear approximations Some of this we have already discussed in class De nition 1 Let f z I 7gt R be a function on the open interval I Let a E I If f is differentiable at a then the linear approximation of f at a is given by 1 1 m 111 WWI a 1 Notice that L is simply the equation of the tangent line to f at z a The point is that if f is differentiable at a then L is a good approximation of f for 1 near a Put another way7 if f is differentiable at a then under a microscope f will look very much like a straight line Exercise 2 Sketch a graph of a nonlinear function f that illustrates this point Example 3 Let 13 At 1 2 the linear approximation of f is given by 2 fx mLz 322z722312z728 since 312 Example 4 Let xz 4 Then 2 The linear approximation to f at z 5 is z 7 5 z 7 5 lt3 fltzgtLltzgt M 6 3 As an immediate application we can approximate square roots of numbers near 10 by hand Computing m is difficult to compute directly since m is irrational The linear approximation gives us a simple way of estimating Notice that 6 7 5 19 7 4 W6 f6T3g316 It turns out that m w 316227766 correct to 8 decimal places so our linear approximation is only accurate to one decimal p ace Exercise 5 In the preceding example our approximation of was an overes timate Show that we always get an overestimate that is7 show that Lz 2 for z gt 74 Hint What does the second derivative tell you about With advanced calculators and computing software it may not appear necessary to use linear approximations Keep in mind7 however that when computing with irrational numbers both calculators and software use approximations As such7 it can be helpful to have some idea of how the approximations happen Although these approximations may not be linear7 an advantage of studying linear approximations initially is that they are relatively simp e 1 2 ADDITIONAL NOTES AND EXERCISES FOR 71 It should also be pointed out that one can estimate how good our linear approxi mation is without actually knowing the precise value ofwhat we are estimatingi We discuss this more when we consider approximate integration methods like Simpson s ruler As a practical example7 consider the trigonometric functions sinz and cos zi These functions come up frequently in any science where angles play a roleli Exercise 6 Show that the linear approximation of sinz at z 0 is z and the linear approximation of cos z at z 0 is 1 Do not use the subsequent discussion We will see in math 1220 that 00 7 71n12n1 7 Is 15 5 s1nzigmizigmiiu and 00 707L127 12 I4 6 717 i 7 H CO 7 2n 2 24 Notice that if z is close to zero then sinz m z and cosz m 1 because the higher order terms are much smaller than z This linearization simpli es many calculations without serious loss of accuracy and indeed makes otherwise intractable calculations feasible De nition 7 Let y be a differentiable function We de ne a new indepen dent variable dzi The domain of dz is any real number We de ne dy f zdz and note that dy is a dependent variable We say that dz and dy are differentials Notice that dy is a function both of z since is a function of z and of dzi Here is the reason for introducing differentials Fix a point Pa7 Let 7 Az z 7 d Note that if z is near a then Az is small Set dz Az and let Ay 7 fai If Az is small then dy m Ay and dy is the linear approximation of for z near We illustrate the point in gure 1 Exercise 8 Let z4i If a 1 and dz Az what are Ay and dy Exercise 9 Let If a l and dz Az i what are Ay and dy Exercise 10 Let sin2zi If a 7r and dz Az W72 what are Ay and dy Exercise 11 Using differentials estimate the amount of paint needed to apply a coat of paint 002 cm think to a sphere with diameter 40 meters Recall that the volume of a sphere of radius 7 is given by the formula V 47 Notice that you are given that d7quot 0 02 1For example optics and mechanics ADDITIONAL NOTES AND EXERCISES FOR 71 y fx FIGURE 1 Differentials

×

×

### BOOM! Enjoy Your Free Notes!

×

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

Bentley McCaw University of Florida

#### "I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

Kyle Maynard Purdue

#### "When you're taking detailed notes and trying to help everyone else out in the class, it really helps you learn and understand the material...plus I made \$280 on my first study guide!"

Jim McGreen Ohio University

#### "Knowing I can count on the Elite Notetaker in my class allows me to focus on what the professor is saying instead of just scribbling notes the whole time and falling behind."

Parker Thompson 500 Startups

#### "It's a great way for students to improve their educational experience and it seemed like a product that everybody wants, so all the people participating are winning."

Become an Elite Notetaker and start selling your notes online!
×

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com