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Math 3A/2 Final Study: Extrema, Inflection, Asymptotes, Optimization, Exams of Mathematics

University of California - Los Angeles (UCLA)Mathematics

A study guide for the final exam in math 3a/2, focusing on finding extrema, inflection points, asymptotes, and optimization. It includes explanations, examples, and exercises. The material covers sections 2.2, 3.1-5.5, and 5.8.

Typology: Exams

Pre 2010

Uploaded on 08/30/2009

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Math 3A/2 Final Study Guide
Updated 3/14/07
Final: Friday, March 23, 2007
8:00-11:00 am
Disclaimer: This is just a guide to help you study for the final. The material in this study guide will not
necessarily be on the exam and vice versa. This is just what I would know/study if I were a student in Math
3A. There may be some typos/mistakes so do not just use this to study (i.e. be sure to look at old homeworks,
notes, book, etc.). Some of the exercises have been taken from the text.
Material on Final: Section 2.2, 3.1-5.5, 5.8
Material reviewed here: Section 5.3-5.5, 5.8 (Other sections can be found on previous study guides)
Section 5.3: Finding Extrema, Inflection Points, and Asymptotes
To find the local/global extrema of a function, you need to
1. Find the critical points (i.e. find xsuch that f0(x) = 0 or f0(x) does not exist) and the endpoints. Let c
denote one of these points.
2. Second Derivative Test: If f00(c) exists and f00(c)<0 then fhas a locall maximum at c. If f00(c)>0 then
fhas a local minimum as c.
3. First Derivative Test: If f00(c) does not exist or if f00(c) = 0 then you use the first derivative to determine
what intervals fis increasing/decreasing. If fis increasing for x < c and decreasing for x > c then fhas
a local max at c. Conversely, if fis decreasing for x < c and increasing for x > c then fhas a local min
at c.
4. The global max/min (if they exist) will be one of the local max/min you just found. To determine which
is the global max/min, plug your local extrema into the original function fand find the ones with the
largest/smallest value. If fis defined on an open interval or has points of discontinuity, you also need to
calculate the limit of fas it approaches these points so as make sure those are not “candidates” for you
global max/min.
Exercise 1. Find al l local extrema of f(x) = ((x−1)3,−1≤x < 2
−x+ 2,2≤x≤3. Which of these are the global extrema?
We call x=can inflection point if the concavity of fchanges at x. Just like critical points are candidates
for local extrema, the xsuch that f00(x) = 0 are candidates for points of inflection. To check if the cthat satisfy
f00(c) = 0 are inflection points, check the sign of f00 in the intervals to the left and right of c.
1
pf3
pf4

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Download Math 3A/2 Final Study: Extrema, Inflection, Asymptotes, Optimization and more Exams Mathematics in PDF only on Docsity!

Math 3A/2 Final Study Guide

Updated 3/14/

Final: Friday, March 23, 2007

8:00-11:00 am

Disclaimer: This is just a guide to help you study for the final. The material in this study guide will not necessarily be on the exam and vice versa. This is just what I would know/study if I were a student in Math 3A. There may be some typos/mistakes so do not just use this to study (i.e. be sure to look at old homeworks, notes, book, etc.). Some of the exercises have been taken from the text. Material on Final: Section 2.2, 3.1-5.5, 5. Material reviewed here: Section 5.3-5.5, 5.8 (Other sections can be found on previous study guides)

Section 5.3: Finding Extrema, Inflection Points, and Asymptotes

To find the local/global extrema of a function, you need to

  1. Find the critical points (i.e. find x such that f ′(x) = 0 or f ′(x) does not exist) and the endpoints. Let c denote one of these points.
  2. Second Derivative Test: If f ′′(c) exists and f ′′(c) < 0 then f has a locall maximum at c. If f ′′(c) > 0 then f has a local minimum as c.
  3. First Derivative Test: If f ′′(c) does not exist or if f ′′(c) = 0 then you use the first derivative to determine what intervals f is increasing/decreasing. If f is increasing for x < c and decreasing for x > c then f has a local max at c. Conversely, if f is decreasing for x < c and increasing for x > c then f has a local min at c.
  4. The global max/min (if they exist) will be one of the local max/min you just found. To determine which is the global max/min, plug your local extrema into the original function f and find the ones with the largest/smallest value. If f is defined on an open interval or has points of discontinuity, you also need to calculate the limit of f as it approaches these points so as make sure those are not “candidates” for you global max/min.

Exercise 1. Find all local extrema of f (x) =

(x − 1)^3 , − 1 ≤ x < 2 −x + 2, 2 ≤ x ≤ 3

. Which of these are the global extrema?

We call x = c an inflection point if the concavity of f changes at x. Just like critical points are candidates for local extrema, the x such that f ′′(x) = 0 are candidates for points of inflection. To check if the c that satisfy f ′′(c) = 0 are inflection points, check the sign of f ′′^ in the intervals to the left and right of c.

Exercise 2. Determine the points of inflection of f (x) = (x + 1)^4.

The last thing we need to find before we can sketch the graph of a function f are lines called asymptotes that f approach at points where f does not exist or as x → ±∞. There are three types of asymptotes:

  1. Horizontal Asymptote: f has a horizontal asymptote at y = b if limx→±∞ f (x) = b.
  2. Vertical Asymptote: f has a verical asymptote at x = c if limx→c± f (x) = ±∞.
  3. Oblique Asymptote: f has an oblique asymptote at y = mx + b if limx→±∞[f (x) − (mx + b)] = 0.

One case we can find oblique asymptotes is when f (x) = p q((xx)) where p(x) and q(x) are polynomials and

the degree of p is one greater than the degree of q(x). To find them, you need to use long division to rewrite f (x) = mx + b + R(x) where R(x) is the remainder. That quotient mx + b (without the remainder) could be the line of the oblique asymptote since limx→∞ R(x) = 0.

Exercise 3. Determine the asymptotes of the following functions:

  1. f (x) = (^) x−^11
  2. g(x) = e^2 x^ + 3
  3. h(x) = x

(^2) +2x+ x+

Now we have all the ingredients to sketch the graph of a function (without a graphing calculator!). To sketch a graph:

  1. Find the intercepts of f (x).
  2. Find local extrema and intervals where f is increasing/decreasing.
  3. Determine the intervals f is concave up/down.
  4. Find asymptotes.

Exercise 4. Sketch the graph of f (x) = (^) 1+xx.

Section 5.8: Antiderivatives

Definition: A function F is an antiderivative of f in an interval I if F ′(x) = f (x) for all x ∈ I. There are two types of antiderivatives: general and particular. For general antiderivatives you add a constant C to F. Notice that this will not change F ′(x) since the derivative of a constant is zero. To find a particular antiderivative, you must be given an initial condition, i.e. you must be told that the point (x 0 , y 0 ) lies in your solution. Using that information you can find C in you general solution to get the particular antiderivative. There is a table of antiderivatives for common function on page 328. Antiderivatives are useful in solving differential equations which we can just think of as equations that look like dydx = f (x). So to find y(x) we need to find the antiderivative of f (x). Thus the solution to dydx = f (x) is y(x) = F (x) + C.

Exercise 8. If Tim is standing on a roof that is 100 feet tall and tosses a water balloon with initial velocity 3 f t/s and constant acceleration 32 f t/s^2 , how long will it take the water balloon to hit the ground? How fast is the balloon traveling when it hits the ground?