**Unformatted text preview: **SMU Classification: Restricted Part 2: Calculus
A: Differential Calculus
1.The Limits
2.The Derivatives
3.Graph Sketching Using Differentiation
4.Applications of Differentiation Version: 1.0
Date: Sep 20, 2018
Textbook, Chapter 2, Sections 2.4, 2.5; Chapter 3, Sections 3.1, 3.2, 3.3, 3.4, 3.5
Grace in Mathematics Associate Professor Christopher Ting Page 1 of 29 SMU Classification: Restricted The derivative of a function of , is still a function of . Textbook, p. 156 () ′ () domain of () domain of ′ () Grace in Mathematics Associate Professor Christopher Ting Page 2 of 29 SMU Classification: Restricted Textbook, p. 156 Grace in Mathematics Associate Professor Christopher Ting Page 3 of 29 SMU Classification: Restricted Secant line has two points,
two points determine a
straight line, including its
slope.
Tangent line has one point
only. How to determine the
slope? ℎ→0 Textbook, p. 154 Grace in Mathematics Associate Professor Christopher Ting Page 4 of 29 SMU Classification: Restricted ℎ→0 ′ ()
Textbook, pp. 151, 153 Grace in Mathematics Associate Professor Christopher Ting Page 5 of 29 SMU Classification: Restricted 4 − ▪ EXAMPLE 6:
▪ EXAMPLE 7: Textbook, pp. 156-157 Grace in Mathematics 2 ′
1 ′ = 4 − 2 =− +2 1
2
′ = 1 2 Associate Professor Christopher Ting Page 6 of 29 SMU Classification: Restricted ′ → ()
We use formulas,
properties and rules.
Grace in Mathematics Associate Professor Christopher Ting Page 7 of 29 SMU Classification: Restricted Notation (The Derivative)
If = , then
′ , ′
, , all represent the derivative of at .
Textbook, p. 165 Grace in Mathematics Associate Professor Christopher Ting Page 8 of 29 ′ SMU Classification: Restricted → ()
✓
✓
✓
✓ Formulas for Elementary Functions:
Constant Multiple Property:
Sum and Difference Property:
Rule: ✓ Rule: ✓
✓ Rule:
Implicit differentiation: e.g. e.g. e.g. e.g. = 2
= 3 2
= 2 + = 2 3 e.g. = 2
2−1
1+ ln 2 e.g. = e.g. = Grace in Mathematics Associate Professor Christopher Ting Page 9 of 29 SMU Classification: Restricted 1 2
3
4
5 1 (Constant Function Rule)
Textbook, p. 165 2 (Power Rule)
Textbook, p. 167 3 (Exponential Functions)
Textbook, p. 209 4 Textbook, p. 209 5 (Logarithmic Functions)
Textbook, p. 209 6
7
8 This Table will be given in
the “Formula Sheet” in
the final exam. 6
Textbook, p. 209
(Sine and Cosine Functions)
Textbook, p. 520 Q: How to use these formula?
Q: How to determine which formula to use? Grace in Mathematics Associate Professor Christopher Ting Page 10 of 29 SMU Classification: Restricted 1. Constant Function Rule = = constant
→ ′ = 0 Q: If = 2 , then what is ′ ?
Textbook, p. 166 Grace in Mathematics Associate Professor Christopher Ting Page 11 of 29 = ∙ − SMU Classification: Restricted 2. Power Rule Q: If = ,
then what is ′ ? Textbook, p. 167 Grace in Mathematics Associate Professor Christopher Ting Page 12 of 29 SMU Classification: Restricted 3. Exponential Functions Textbook, p. 209 = or ?
Grace in Mathematics Associate Professor Christopher Ting Page 13 of 29 SMU Classification: Restricted 4. Exponential Functions = ln ∙ , > , ≠ Q: If = 2 , then what is ′ ?
Q: If = 3 , then what is ′ ?
Q: If = 3.5 − , then what is ′ ? If you choose a different value for , > 0, ≠ 1,
then you will have a new question to practice.
Textbook, p. 209 Associate Professor Christopher Ting Page 14 of 29 SMU Classification: Restricted 5. Natural Logarithmic Functions ln = log → ln Q: If = ln(2), then what is ′ ? Textbook, p. 209 Grace in Mathematics Associate Professor Christopher Ting Page 15 of 29 SMU Classification: Restricted 6. Logarithmic Functions of Base b
log =
∙
, > , ≠ ln ′ Q: If = log 4 (), then what is ? Q: If = log ln(2) (), then what is ′ ?
If you choose a different value for , > 0, ≠ 1,
then will you have a new question to practice.
Textbook, p. 209 Grace in Mathematics Associate Professor Christopher Ting Page 16 of 29 SMU Classification: Restricted 7&8. Sine and Cosine Functions sin() = cos() cos() = − sin() Textbook, p. 520 Grace in Mathematics Associate Professor Christopher Ting Page 17 of 29 SMU Classification: Restricted More Properties and Rules (before studying the Chain Rule)
▪ Constant Multiple Property [Theorem 3. p. 168] ▪ Sum and Difference Property [Theorem 4. p. 169]
▪ Product Rule [Theorem 1. p. 214]
▪ Quotient Rule [Theorem 2. p. 216] Grace in Mathematics Associate Professor Christopher Ting Page 18 of 29 SMU Classification: Restricted Example: = 3 2 = 3 ∙ , = 2 , ′ = 2 (Power rule) ′ = 3 ∙ ′ = 3 ∙ 2 = 6
3 2 ′ = 3 2 ′ =⋯ Example: = 3 2 + 2 = + (), = 3 2 , = 2 ′ = ′ + ′ 3 2 + 2 Textbook, pp. 168-169 Grace in Mathematics ′ = 3 2 ′ + 2 ′ =⋯ Associate Professor Christopher Ting Page 19 of 29 SMU Classification: Restricted radical form → exponent form
denominator
→ negative exponent form Textbook, p. 169 Grace in Mathematics Associate Professor Christopher Ting Page 20 of 29 SMU Classification: Restricted : first
: second Example: = = () ∙ , = , = ′ = ′ + ′() ′ = Textbook, p. 214 ′ + ′ Q: How about Grace in Mathematics ′ = ′ + ′ Associate Professor Christopher Ting Page 21 of 29 SMU Classification: Restricted Q: Which is simpler?
Textbook, pp. 214-215 Grace in Mathematics Associate Professor Christopher Ting Page 22 of 29 SMU Classification: Restricted SOLUTION:
First, find ′(). for slopes of tangent lines
′ = − + slope equals zero (computation omitted) Then,
(A) find = −45, ′ = 12. Construct the equation of the tangent line at = 3 in the
form of = + , with the point (, ()), and slope = ′().
(a linear function) = − (B) Solve ′ = , for .
6 2 − 18 + 12 = 0 → = , (a quadratic equation)
Textbook, p. 215 Grace in Mathematics Associate Professor Christopher Ting Page 23 of 29 SMU Classification: Restricted Textbook, pp. 215-216 Grace in Mathematics Associate Professor Christopher Ting Page 24 of 29 SMU Classification: Restricted : top
: bottom Example: () =
=
,
2 + 1 () = , = 2 + 1
′ − ′() ′ = 2
′ 2 + 1 ′ − ()(2 + 1)′
=
2 + 1
2 + 1 2
Textbook, p. 216 … Grace in Mathematics Associate Professor Christopher Ting Page 25 of 29 SMU Classification: Restricted Q: Will all of them use quotient rule? Q: Remember we saw an example which did not use product rule for a product? Textbook, p. 217 Grace in Mathematics Associate Professor Christopher Ting Page 26 of 29 SMU Classification: Restricted Textbook, p. 217 Grace in Mathematics Associate Professor Christopher Ting Page 27 of 29 SMU Classification: Restricted Textbook, p. 217 Grace in Mathematics Associate Professor Christopher Ting Page 28 of 29 SMU Classification: Restricted Product Rule: ( ) d
F (x) S (x) = F(x) S (x) + F (x) S(x)
dx
Quotient Rule: d N (x) D(x) N (x) − N (x) D(x)
= dx D(x) [D(x)] 2
Grace in Mathematics Associate Professor Christopher Ting Page 29 of 29 ...

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