Midterm Review - University of Toronto · Midterm Review What to do: Read lectures 1 -6 (pay...
Transcript of Midterm Review - University of Toronto · Midterm Review What to do: Read lectures 1 -6 (pay...
![Page 1: Midterm Review - University of Toronto · Midterm Review What to do: Read lectures 1 -6 (pay attention to the in - class examples) and sections 1.1 -1.6, 2.1 - 2.8, 3.1 -3.6, 4.1](https://reader036.fdocuments.net/reader036/viewer/2022070920/5fb9463ee5382e2d6a65f081/html5/thumbnails/1.jpg)
Midterm Review
What to do:
Read lectures 1-6 (pay attention to the in-
class examples) and sections 1.1-1.6, 2.1-
2.8, 3.1-3.6, 4.1-4.4 from the textbook
Do the assigned exercises from the textbook
Go over the quiz questions
Use sample test to practice
Use extra TAs' office hours
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Topics to review:
Functions and Models
o Functions
Domain/range
Odd/even/neither
Increasing/decreasing
Transformations
Inverse functions
Inverse trigonometric
Logarithmic
o Models
Linear function
Polynomial function
Power function
Rational function
Algebraic function
Trigonometric function
Exponential function
![Page 3: Midterm Review - University of Toronto · Midterm Review What to do: Read lectures 1 -6 (pay attention to the in - class examples) and sections 1.1 -1.6, 2.1 - 2.8, 3.1 -3.6, 4.1](https://reader036.fdocuments.net/reader036/viewer/2022070920/5fb9463ee5382e2d6a65f081/html5/thumbnails/3.jpg)
Example: Find domain of
( ) √
![Page 4: Midterm Review - University of Toronto · Midterm Review What to do: Read lectures 1 -6 (pay attention to the in - class examples) and sections 1.1 -1.6, 2.1 - 2.8, 3.1 -3.6, 4.1](https://reader036.fdocuments.net/reader036/viewer/2022070920/5fb9463ee5382e2d6a65f081/html5/thumbnails/4.jpg)
Example: Given function
( ) ( )
Is it odd, even, or neither?
![Page 5: Midterm Review - University of Toronto · Midterm Review What to do: Read lectures 1 -6 (pay attention to the in - class examples) and sections 1.1 -1.6, 2.1 - 2.8, 3.1 -3.6, 4.1](https://reader036.fdocuments.net/reader036/viewer/2022070920/5fb9463ee5382e2d6a65f081/html5/thumbnails/5.jpg)
Example: Solve
( )
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Limits
o One-sided/two-sided limits
o Limit laws
o Infinite limits
o Limits at infinity
o Vertical/horizontal asymptotes
o Squeeze Theorem
o definition of limit
o Continuity
Types of discontinuities
Dirichlet example
Continuity laws
IVT
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Example: Calculate
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(√ )
![Page 9: Midterm Review - University of Toronto · Midterm Review What to do: Read lectures 1 -6 (pay attention to the in - class examples) and sections 1.1 -1.6, 2.1 - 2.8, 3.1 -3.6, 4.1](https://reader036.fdocuments.net/reader036/viewer/2022070920/5fb9463ee5382e2d6a65f081/html5/thumbnails/9.jpg)
Example: Apply the Squeeze theorem to find
![Page 10: Midterm Review - University of Toronto · Midterm Review What to do: Read lectures 1 -6 (pay attention to the in - class examples) and sections 1.1 -1.6, 2.1 - 2.8, 3.1 -3.6, 4.1](https://reader036.fdocuments.net/reader036/viewer/2022070920/5fb9463ee5382e2d6a65f081/html5/thumbnails/10.jpg)
Example: Use definition of a limit to prove that
( )
![Page 11: Midterm Review - University of Toronto · Midterm Review What to do: Read lectures 1 -6 (pay attention to the in - class examples) and sections 1.1 -1.6, 2.1 - 2.8, 3.1 -3.6, 4.1](https://reader036.fdocuments.net/reader036/viewer/2022070920/5fb9463ee5382e2d6a65f081/html5/thumbnails/11.jpg)
Example: Use IVT to show that there is a root of the equation
√
on interval (0, 1).
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Derivatives
o Definition of derivative
o Functions differentiable/ not differentiable
o Higher derivatives
o Differentiation rules
Power rule
Product rule
Quotient rule
Chain rule
o Derivatives of exponential functions
o Derivatives of trigonometric functions
o Implicit differentiation
o Derivatives of inverse trigonometric functions
o Derivatives of logarithmic functions
o Logarithmic differentiation
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Example: Use the definition of the derivative to find the derivative of
( )
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Example: Find .
√ ( )
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( )
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Applications of Differentiation
o Minimum/maximum values
o Fermat's theorem
o Rolle's theorem
o MVT
o Increasing/decreasing test
o First derivative test
o Concavities/inflection points
o Second derivative test
o L'Hospital's rule
o Sketching the graph of a function
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Example: Calculate
( )
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Example: Sketch ( )
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