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Midterm

# MATH180 Term Test 1 Analysis

18 Pages
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Department
Mathematics
Course
MATH 180
Professor
All Professors
Semester
Fall

Description
Calculus Midterm I – Analysis 1) Structure breakdown & what to look out: - Atypical midterm exam has 4 to 5 problems, each problem consists of up to 3-4 sub- problems. - What is usually on each problem: 1. Limit/continuity/differentiation such as evaluating the limits of functions, finding the derivative, the slope of tangent at some point, find the point where the slope of tangent equals to 0, determine whether a piece-wise function is continuous, etc. Usually the EASIEST part of the test 2-4(or 5). Longer Problems/TheoremApplied/Real-Life Problem The order may vary, but these types of problems are commonly used:  Finding the equation of the line tangent to the function 0t point x  Sketching the graph of a function, or sketching the derivative of the function  Using IVT or Definition of Derivative to solve problems (eg. To compute a limit, or to compute a derivative, this could be a HARDEST problem for most students)  Real-life problems such as evaluating the velocity/acceleration of an object given its position etc. - Every minute counts. Since a midterm only has 50 minutes, that means you have only about 10 minutes to finish each problem, so once you have spent half of the time you are supposed to spend on for each question and you are still stuck, skip it and come back if you have time. Don’t be afraid to skip problems. - Do easier questions first. Usually we feel less stressed when realizing we have done some problems, even if they are easy ones. 2) Test Statistics Topics 2008 2009 2010 2011 2012 Total Evaluation of Limits 3 2 3 3 4 15 Derivatives/Differentiability 5 3 5 5 3 21 Continuity 2 2 1 1 2 8 Definition of Derivative 1 1 1 1 1 5 Intermediate Value Theorem 1 1 2 Information from the Graph 1 1 2 Graphing the Derivative Func- 2 2 tion Velocity and Acceleration 1 2 3 Piece-wise Functions 1 1 1 1 4 3) Knowledge Summary  Evaluation of Limits  Finding the limit of a function, especially rational functions, where we use the variety of factorization to solve 𝑖  The facts i , i , and the square difference are very commonly used toward these types of problems Vincent Huang University of British Columbia  Continuity  In general, first realize the difference between 𝑖 𝑓 and 𝑓  Then, in the situation where the function is continuous at x=a, we have the two expression above being equal, ie. 𝑖 𝑓 𝑓  A function that is continuous at x=a also has limit existing as x approaches to a.  The summation or composite of continuous functions is also continuous  Derivatives/Differentiability  General Power Rule, Product Rule, Quotient Rule and Chain Rule are the core techniques that the professors require you to be able to demostrate  The derivatives of basic trigonomotric functions (at least sinx, cosx, and tanx) are also important to know  So farALL the midterms in the past 5 years include this question that asks you to find the equation of the tangent line to a function at some po0nt x  Students are to know how to compute the second derivative of a function, and know that relates to acceleration  Realize that differentiability of a function is the strongest condition among existence of limit, continuity, and differentiability. That is, if a function is differentiable at point x=a, that means the function is continuous at x=a, which implies the limit exists as x approaches to a.  Definition of Derivative  Definition: f x i 𝑕 0 𝑓 :𝑕 ;𝑓 𝑕  The main-stream problem ask students to find the derivative of a function using ONLY the definition of derivative.  Since the derivative is defined using limit, consider the problem as a special type of limit problem  Intermediate Value Theorem  Can be applied only when the function is continuous in some interval  Special case is using IVT to find the root (zero) of a function  The Graph  Sketching the derivative funtion given the graph of the original  Given the graph of a function, read off from the graph for some information  Velocity andAcceleration  Some physics applied problem, eg. position function of an moving object etc  Given position function f, think of velocity as f , acceleration as f  Piece-wise Functions  Continuity of a piece-wise function Q: is a piecewise function continuous at the boundary points? Q: For what constant value is the piece-wise function continuous everywhere?  Derivatice of a piece-wise function Q: differentiable at the boundary points? Vincent Huang University of British Columbia 4) Problems from Past Midterms  Evaluation of Limits  Easy 2008 #1 2009 #1 2011 #1 2010 #1 2012 #1  Medium 2008 #1 2009 #1 2010 #1 2010 #1 2011 #1 2012 #1  Hard 2011 #2 2012 #1 Vincent Huang University of British Columbia  Continuity  Easy 2011 #1  Hard 2008 #3 *Many concepts regarding continuity are somehow integrated into other types of questions.  Derivatives/Differentiability  Easy 2009 #1 2009 #1 2011 #1 2011 #1 2011 #1 2011 #3 2012 #1 2012 #1  Medium 2008 #1 2008 #1 2010 #1 2010 #1 2012 #1 Vincent Huang University of British Columbia 2010 #2  Hard 2009 #3 2008 #1  Definition of Derivative  Easy 2008 #4  Medium 2011 #4  Hard 2009 #4 2010 #4 2012 #3  Intermediate Value Theorem  Easy 2009 #2  Hard 2010 #3 Vincent Huang University of British Columbia  The Graph  Easy 2008 #2  Hard 2008 #3  Velocity andAcceleration  Easy 2010 #1  Medium 2012 #2 Vincent Huang University of British Columbia  Hard 2012 #2 2008 #5  Piece-wise Functions  Easy 2008 #1  Medium 2010 #4 Vincent Huang University of British Columbia 2011 #5  Hard 2012 #4 Vincent Huang University of British Columbia 5) Solutions (answers are highlighted in blue)  Evaluation of Limits  2008#1a: Fastest way to solve this question is compare the dimensions(degrees) of the denominator and numerator.  Here the degree of denominator is 2 (deg(D) = 2) while the degree of numerator is 3 (deg(N) = 3).As x ∞, in general, the limit tends to zero as deg(D)  Here 4 2 2  The common factor on top and bottom cancels  Left with i = : 4  2011#1a: Same technique as 2009#1a, factorize the denominator and numerator and cancel out the term (x+3). 5  Ans: 6  2010#1 a: Realize the cos0 = 1, so the as x , cosx , we can even just pretend cosx is not there 2  Left with , again, factorizing and cancel ; 3; 2  Ans:  2012#1b: Using and Trig. Identity t x 𝑖 𝑠𝑖𝑛𝑥 𝑡 𝑐𝑜𝑠𝑥 𝑖  i 0 i 0 i 0 ∙ ∙  2008#1b: i 4 , it does not affect the whole evaluation, let’s just pretend it is not there.  . The answer is 1/4  2009#1b:  2010#1b: Vincent Huang University of British Columbia The answer is 6/7.  2010#1c:  2011#1f: Factorize the denominator first |; |  i − ; : , now
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