×

### Let's log you in.

or

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

×

or

## MIdterm 2 Study Guide

by: Saul Cervantes

388

1

16

# MIdterm 2 Study Guide Math 2419

Saul Cervantes
UTD

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

×
Unlock Preview

COURSE
Accelerated Calculus II
PROF.
Anotoly Ezlydon
TYPE
Study Guide
PAGES
16
WORDS
KARMA
50 ?

## Popular in Mathematics (M)

This 16 page Study Guide was uploaded by Saul Cervantes on Monday November 2, 2015. The Study Guide belongs to Math 2419 at University of Texas at Dallas taught by Anotoly Ezlydon in Fall 2015. Since its upload, it has received 388 views. For similar materials see Accelerated Calculus II in Mathematics (M) at University of Texas at Dallas.

×

## Reviews for MIdterm 2 Study Guide

×

×

### 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: 11/02/15
MIDTERM 2 STUDY GUIDE: CALCULUS II Equations ( )  Partial Derivative relative to x: fx∆????→∞im???? ????????+∆????,???????? −????(????????,????????) ∆????  Partial Derivative relative to y: fy = lim???? ????????,????????+∆???? −????(????????,????????) ∆????→∞ ∆????  Gradient: ∇???? = < ????????,????????,???????? >  Full differential: ∇???? ∙< ∆????,∆???? >  Implicit Differentiation: Given z = f(x, y), and F (x, y, z) = 0, then ???????? −???????? ???????? −???????? ???????? = ????????, and???????? = ???????? ???? ????+???????? −????(????)  Directional Derivative u f(X=lim ????→0 Where f is a function ???? of n variables, X =1,X2X ,… n ), and u is a unit vector1u 2 <u n u ,…u >. |????∙????|  To find angle of inclination: ????????????????| ???? |  Lagrange Multipliers: If f and g are differentiable, then: ∇???? = ????∇???? → ???? = ???? fx = ???????????? fy = ????????????  Volume under a surface: ????(????,????)???????? ∬ ????  Area of domain D: ∬ 1???????? ????  Mass: m = ∬???? ????(????,????)????????  First moment with respect to x:yM ∬ ????????????(????,????)???????? ( )  First moment with respect to y:xM ∬ ???????????? ????,???? ???????? ???????? ????????  Center of mass at (x, y) where x????= and y = ???? Concepts  If f is a function of two or more variables, then the gradient of f (???????????????????????????? ???????? ∇????) is the vector of the partial derivatives. Gradient is the derivative of a multivariable function.  Theorem: Sufficient Condition of Differentiability—If f is the function of x and y, and fx and fy are continuous in an open region, then f is differentiable in this region.  Theorem: If f is differentiable at (xo, yo), then it is also continuous at (xo, yo).  Chain Rule: Derivative of Composite Function = Derivative of Outer Function + Derivative of Inner Function.  Second Derivative Test: d = f fxx yy ) xy2 o If d > 0 and fxxa, b) > 0, then f has a relative minimum at the point (a, b) o If d > 0 and f (a, b) < 0, then f has a relative maximum at the point xx (a, b) o If d < 0, then point (a, b, f(a, b)) is a saddle point. o If d = 0, the test is inconclusive.  Theorem: Suppose f(x, y) is continuous on ????,???? × [????,????], then ???? ???? ???? ???? ∫ ∫ ???? ????,???? ???????????????? = ∫ ∫ ???? ????,???? ???????????????? = ∬ ????(????,????)???????? ???? ???? ???? ???? ????,???? × ????,???? o Note: This theorem only works on functions defined over a rectangle. Examples: MIDTERM 2 STUDY GUIDE: CALCULUS II Equations ( )  Partial Derivative relative to x: fx∆????→∞im???? ????????+∆????,???????? −????(????????,????????) ∆????  Partial Derivative relative to y: fy = lim???? ????????,????????+∆???? −????(????????,????????) ∆????→∞ ∆????  Gradient: ∇???? = < ????????,????????,???????? >  Full differential: ∇???? ∙< ∆????,∆???? >  Implicit Differentiation: Given z = f(x, y), and F (x, y, z) = 0, then ???????? −???????? ???????? −???????? ???????? = ????????, and???????? = ???????? ???? ????+???????? −????(????)  Directional Derivative u f(X=lim ????→0 Where f is a function ???? of n variables, X =1,X2X ,… n ), and u is a unit vector1u 2 <u n u ,…u >. |????∙????|  To find angle of inclination: ????????????????| ???? |  Lagrange Multipliers: If f and g are differentiable, then: ∇???? = ????∇???? → ???? = ???? fx = ???????????? fy = ????????????  Volume under a surface: ????(????,????)???????? ∬ ????  Area of domain D: ∬ 1???????? ????  Mass: m = ∬???? ????(????,????)????????  First moment with respect to x:yM ∬ ????????????(????,????)???????? ( )  First moment with respect to y:xM ∬ ???????????? ????,???? ???????? ???????? ????????  Center of mass at (x, y) where x????= and y = ???? Concepts  If f is a function of two or more variables, then the gradient of f (???????????????????????????? ???????? ∇????) is the vector of the partial derivatives. Gradient is the derivative of a multivariable function.  Theorem: Sufficient Condition of Differentiability—If f is the function of x and y, and fx and fy are continuous in an open region, then f is differentiable in this region.  Theorem: If f is differentiable at (xo, yo), then it is also continuous at (xo, yo).  Chain Rule: Derivative of Composite Function = Derivative of Outer Function + Derivative of Inner Function.  Second Derivative Test: d = f fxx yy ) xy2 o If d > 0 and fxxa, b) > 0, then f has a relative minimum at the point (a, b) o If d > 0 and f (a, b) < 0, then f has a relative maximum at the point xx (a, b) o If d < 0, then point (a, b, f(a, b)) is a saddle point. o If d = 0, the test is inconclusive.  Theorem: Suppose f(x, y) is continuous on ????,???? × [????,????], then ???? ???? ???? ???? ∫ ∫ ???? ????,???? ???????????????? = ∫ ∫ ???? ????,???? ???????????????? = ∬ ????(????,????)???????? ???? ???? ???? ???? ????,???? × ????,???? o Note: This theorem only works on functions defined over a rectangle. Examples:

×

×

### 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

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."

Jennifer McGill UCSF Med School

#### "Selling my MCAT study guides and notes has been a great source of side revenue while I'm in school. Some months I'm making over \$500! Plus, it makes me happy knowing that I'm helping future med students with their MCAT."

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!"

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