Thursday November 29

EXAM 3 Review Exercise solutions are now available. Answers to the Fall 2004 Exam 3 are also available.
Question 11 of the Review Exercises now has the solution up.

Exam 3 will be held  in Fowler 301 and Fowler 302 from 7-10pm. The Study Guide for Exam III (Topics and Review Questions) has been distributed online.

Exam 3 from Math 110 Fall 2004 is offered as a Practice Exam although I would pay more careful attention to the Review Exercises given below.

Exam 3 is now available. As well as Solutions to Exam 3.

Here's the Curve for Exam 3, excerpted from the official report.

Other places to look for good questions are in the Chapter 4 and Chapter 5 Review Exercises.

MATH 110

Ron Buckmire

Fall 2007

Study Guide for Exam 3

            The exam will cover Classes 25 through 32, Homework 25 to 32 and Quizzes 8 through 10.  I try to make the study guides complete, but there are no guarantees.  You should look at homeworks, quizzes, worksheets, labs, and your class notes. The best way to study for the exam is to make up your own practice exam. The Sections of the Book you should be looking at are Sections 4.4, 5.1, 5.2, 5.4, 5.5 and 9.2.

I have not yet decided whether to allow calculators.  It’s possible that there will be two parts – one allowing calculators and the other not.

Notice that many of the topics involve computing derivatives. 
Keep all the basic differentiation rules memorized! 

Topics

·        Differentiate any function, even f(x)^g(x) where f(x)>0.

·        Use L’Hopital’s rule, including changing other indeterminate forms into  or  or .

·        Evaluating other Exotic Indeterminate forms such as “∞^0”, “1^∞”or “0^0” by using
 lim e^{g(x) ln(f(x}) =e ^{lim g(x) ln (f(x))} and then using L’Hopital’s Rule to obtain an answer

·        Definition of critical point, stationary point, inflection point, local extremum, global extremum, concavity, root

·        First Derivative Test and Second Derivative Test to classify local extrema

·        Extreme Value Theorem, its Conditions (continuous function on a closed finite interval) and Conclusion (function must achieve one global maximum value and one global minimum value )

·        Determine graphical behavior of function analytically (↑,↓), (∩,U), (,+,−) and curve sketching

·        Optimization

o       Find critical points.  (In this class, always do that by solving , by hand when possible and otherwise with a calculator; I will not accept any argument that doesn’t use calculus, for example just graphing .)

o       Classify them using the first derivative test

o       Classify them using the second derivative test

o       Applied max-min problems (if there is only one critical point that makes sense, you aren’t required to show that it’s the requested global max or global min)

·        Check whether a given function satisfies an IVP (initial value problem) which consists of a rate equation and an initial condition, i.e. y’=y, y(1)=1.

·        Euler’s Method y_new=y_old + Δy where  Δy=f’(x) Δx. Be able to apply Euler’s Method to approximate the solution of an IVP and explain why your solution is an over- or under-estimate by looking at concavity.

·        Qualitative understanding of slope fields and ability to match a slope field with a given rate equation

·        Use a rate equation to interpret when a solution will be concave up/down and increasing/decreasing

Problems

 (1)  Let .  This means that x=tan y. Use implicit differentiation and the trigonometric identity that 1 + tan²x=sec²x to show that .

 (2)  Determine whether  is the solution to the IVP .

(3)  Compute the following limits.

            (a)      (b)     (c)          (d) 

(4)  If , find .

(5)  For the function :  (a) Find the critical points; (b) Use the first derivative test to classify them; (c) Use the second derivative test to classify them (yes, this is a pain in the butt)

(6)  A rectangular picture frame is to be constructed of material that costs 7 cents per inch for the left and right sides of the frame, and 11 cents per inch for the top and bottom.  What are the dimensions of the cheapest frame that can be constructed to hold a picture with area 100 square inches?

(7)  Given the initial value problem y(0)=1,  use Euler’s method, with , to approximate the solution y(x) at .  Plot your piecewise-defined function. Is your approximated value greater or less than the unknown exact value of the solution to the rate equation at x=1?

(8)  With the help of your calculator, find the critical point of , and use the second derivative test to classify it.

(9)  Given the rate equation , show that the unknown solutions to this differential equation are always concave up and decreasing whenever y>0.

(10)  Chapter 5 Review (page 346) 53. TRUE or FALSE.

(a) IF f is differentiable on the open interval (a,b) and if f has an absolute extremum on that interval, THEN it must occur at a stationary point of f.

(b) IF f is continuous on the open interval (a,b), and if f has an absolute extremum on that interval, THEN it must occur at a stationary point of f.

(11)      Find the minimum value of S=x²+1/x² on (0, ∞). Find the absolute maximum and minimum of S on [1/2,3].