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Midterm

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Mathematics

MATH 180

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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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