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Application of Derivatives

- Dr. Ching I Chen

4.1 Extreme Values of Functions (1) Absolute

(Global) Extreme Values

4.1 Extreme Values of Functions (2) Absolute

(Global) Extreme Values (Example 1)

4.1 Extreme Values of Functions (3) Absolute

(Global) Extreme Values (Example 2-a)

4.1 Extreme Values of Functions (4) Absolute

(Global) Extreme Values (Example 2-b)

4.1 Extreme Values of Functions (5) Absolute

(Global) Extreme Values (Example 2-c)

4.1 Extreme Values of Functions(6, Example 2-d)

Absolute (Global) Extreme Values

4.1 Extreme Values of Functions (7) Absolute

(Global) Extreme Values (Theorem 1)

Maximum and minimum at interior points

4.1 Extreme Values of Functions (8) Absolute

(Global) Extreme Values (Theorem 1)

Maximum and minimum at endpoints

4.1 Extreme Values of Functions (9) Absolute

(Global) Extreme Values (Theorem 1)

Maximum at interior point, minimum at endpoint

4.1 Extreme Values of Functions (10) Absolute

(Global) Extreme Values (Theorem 1)

Minimum at interior point, maximum at endpoint

4.1 Extreme Values of Functions (11) Local

(Relative) Extreme Values

4.1 Extreme Values of Functions (12) Local

(Relative) Extreme Values

4.1 Extreme Values of Functions (13) Finding

Extreme Values (Theorem 2)

4.1 Extreme Values of Functions (14) Finding

Extreme Values

4.1 Extreme Values of Functions (15) Finding

Extreme Values (Example 3-1)

Absolute maximum value of about 2 at x 3 and

absolute minimum value of 0 at x 0

4.1 Extreme Values of Functions (16) Finding

Extreme Values (Example 3-2)

4.1 Extreme Values of Functions (17) Finding

Extreme Values (Example 4)

4.1 Extreme Values of Functions (18) Finding

Extreme Values (Example 5-1)

4.1 Extreme Values of Functions (19) Finding

Extreme Values (Example 5-2)

4.1 Extreme Values of Functions (20) Finding

Extreme Values (Example 6)

4.1 Extreme Values of Functions (21) Finding

Extreme Values (Exploration 1)

4.1 Extreme Values of Functions (22) Finding

Extreme Values (Exploration 1-2)

4.1 Extreme Values of Functions (23)

Exercise 1 , 4, 7, 10, 13, 16, 19, 22, 25, 28,

31, 34, 37, 40, 43

4.2 Mean Value Theorem (1) Mean Value Theorem

4.2 Mean Value Theorem (2) Mean Value Theorem

4.2 Mean Value Theorem (3) Mean Value Theorem

4.2 Mean Value Theorem (4, Example 1) Mean Value

Theorem

4.2 Mean Value Theorem (5, Example 2) Mean Value

Theorem

4.2 Mean Value Theorem (6, Example 3) Physical

Interpretation

4.2 Mean Value Theorem (7) Increasing and

Decreasing Functions

4.2 Mean Value Theorem (8, Example 4) Increasing

and Decreasing Functions

4.2 Mean Value Theorem (9, Example 5) Increasing

and Decreasing Functions

4.2 Mean Value Theorem (10) Other Consequences

4.2 Mean Value Theorem (11, Example 6) Other

Consequences

4.2 Mean Value Theorem (12) Other Consequences

4.2 Mean Value Theorem (13, Example 7) Other

Consequences

4.3 Connecting f ? and f ? with the Graph of f

(1) First Derivative Test for Local Extrema

4.3 Connecting f ? and f ? with the Graph of f

(2) First Derivative Test for Local Extrema

(Theorem 4)

4.3 Connecting f ? and f ? with the Graph of f

(3) First Derivative Test for Local Extrema

(Theorem 4)

4.3 Connecting f ? and f ? with the Graph of f

(4) First Derivative Test for Local Extrema

(Theorem 4)

4.3 Connecting f ? and f ? with the Graph of f

(5) First Derivative Test for Local Extrema

(Theorem 4)

4.3 Connecting f ? and f ? with the Graph of f

(6) First Derivative Test for Local Extrema

(Theorem 4)

4.3 Connecting f ? and f ? with the Graph of f

(7) First Derivative Test for Local Extrema

(Example 1)

4.3 Connecting f ? and f ? with the Graph of f

(8) First Derivative Test for Local Extrema

(Example 2)

4.3 Connecting f ? and f ? with the Graph of f

(9) Concavity

4.3 Connecting f ? and f ? with the Graph of f

(10) Concavity

4.3 Connecting f ? and f ? with the Graph of f

(11) Concavity (Example 3)

4.3 Connecting f ? and f ? with the Graph of f

(12) Concavity (Example 4)

4.3 Connecting f ? and f ? with the Graph of f

(13)Points of Inflection

4.3 Connecting f ? and f ? with the Graph of f

(14) Points of Inflection (Example 5-1)

4.3 Connecting f ? and f ? with the Graph of f

(15) Points of Inflection (Example 5-2)

4.3 Connecting f ? and f ? with the Graph of f

(16) Points of Inflection (Example 5-3)

4.3 Connecting f ? and f ? with the Graph of f

(17) Points of Inflection (Example 6)

4.3 Connecting f ? and f ? with the Graph of f

(18) Points of Inflection (Example 6)

4.3 Connecting f ? and f ? with the Graph of f

(19) Second Derivative Test for Local Extrema

4.3 Connecting f ? and f ? with the Graph of f

(20) Second Derivative Test for Local Extrema

(Ex. 7)

4.3 Connecting f ? and f ? with the Graph of f

(21) Second Derivative Test for Local Extrema

(Ex. 8)

4.3 Connecting f ? and f ? with the Graph of f

(22) Second Derivative Test for Local Extrema

(Ex. 8-a,b)

4.3 Connecting f ? and f ? with the Graph of f

(23) Second Derivative Test for Local Extrema

(Ex. 8-c,d)

4.3 Connecting f ? and f ? with the Graph of f

(24) Learning about Function From Derivatives

4.3 Connecting f ? and f ? with the Graph of f

(25) Learning about Function From Derivatives

(Explo. 2)

4.4 Modeling and Optimization (1, Example 1)

Example from Business and Industry

4.4 Modeling and Optimization (2, Example 2)

Example from Business and Industry

4.4 Modeling and Optimization (3, Example 3)

Example from Mathematics

4.4 Modeling and Optimization (4, Example 4)

Example from Mathematics

4.4 Modeling and Optimization (5) Example from

Mathematics(Exploration 1)

4.4 Modeling and Optimization (6) Example from

Mathematics (Exploration 1-1)

4.4 Modeling and Optimization (7) Example from

Mathematics (Exploration 1-2)

4.4 Modeling and Optimization (8) Example from

Mathematics (Exploration 1-3)

4.4 Modeling and Optimization (9) Example from

Mathematics (Exploration 1-4)

4.4 Modeling and Optimization (10) Example from

Mathematics (Exploration 1-5)

4.4 Modeling and Optimization (11) Example from

Economics

4.4 Modeling and Optimization (12) Example from

Economics

4.4 Modeling and Optimization (13, Example 5)

Example from Economics

4.4 Modeling and Optimization (14, Example 6)

Example from Economics

4.5 Linearization and Newtons Method (1) Linear

Approximation (Exploration 1-1,2)

4.5 Linearization and Newtons Method (2) Linear

Approximation (Exploration 1-3)

4.5 Linearization and Newtons Method (3) Linear

Approximation

4.5 Linearization and Newtons Method (4) Linear

Approximation (Example 1)

4.5 Linearization and Newtons Method (5) Linear

Approximation (Example 2)

4.5 Linearization and Newtons Method (6) Linear

Approximation (Example 3)

4.5 Linearization and Newtons Method (7) Linear

Approximation (Example 4)

4.5 Linearization and Newtons Method (8)

Newtons Method

Newtons method is a numerical technique for

approximating a zero of a function of with zeros

of its linearizations. Under favorable

circumstances, The zeros of the linearizations

converge rapidly to an accurate approximation.

Many calculators use the method because it

applies to a wide range of functions and usually

gets results in only a few steps.

4.5 Linearization and Newtons Method (9)

Newtons Method

4.5 Linearization and Newtons Method (10)

Newtons Method

4.5 Linearization and Newtons Method (11)

Newtons Method (Example 5)

4.5 Linearization and Newtons Method (12)

Newtons Method

Fig 4.43

4.5 Linearization and Newtons Method (13)

Newtons Method

4.5 Linearization and Newtons Method (14)

Differentials

4.5 Linearization and Newtons Method (15)

Differentials (Example 6)

4.5 Linearization and Newtons Method (16)

Differentials (Example 7)

4.5 Linearization and Newtons Method (17)

Estimating Change with Differentials

4.5 Linearization and Newtons Method (18)

Estimating Change with Differentials

4.5 Linearization and Newtons Method (19)

Estimating Change with Differentials (Example 8)

4.5 Linearization and Newtons Method (20)

Absolute, Relative, and Percentage Change

4.5 Linearization and Newtons Method (21)

Absolute, Relative, and Percentage Change (Ex. 9)

4.5 Linearization and Newtons Method (22)

Absolute, Relative, and Percentage Change (Ex.

10)

4.5 Linearization and Newtons Method (23)

Absolute, Relative, and Percentage Change (Ex.

11)

4.5 Linearization and Newtons Method (24)

Absolute, Relative, and Percentage Change (Ex.

12)

4.5 Linearization and Newtons Method (25)

Sensitivity to Change (Ex. 13)

4.6 Related Rates (1) Related Rate Equations

4.6 Related Rates (2, Example 1) Related Rate

Equations

4.6 Related Rates (3, Example 2-1) Solution

Strategy

4.6 Related Rates (3, Example 2-2) Solution

Strategy

4.6 Related Rates (4, Example 2-3) Solution

Strategy

4.6 Related Rates (5, Example 2-4) Solution

Strategy

4.6 Related Rates (6, Example 2-5) Solution

Strategy

4.6 Related Rates (7) Solution Strategy

4.6 Related Rates (8) Solution Strategy

4.6 Related Rates (8, Example 3-1) Related Rate

Equations

4.6 Related Rates (10, Example 3-2) Related Rate

Equations

4.6 Related Rates (11, Example 3-3) Solution

Strategy

4.6 Related Rates (12, Example 3-4) Solution

Strategy

4.6 Related Rates (13, Example 3-5) Solution

Strategy

4.6 Related Rates (14, Example 4-1) Related Rate

Equations

4.6 Related Rates (15, Example 4-2) Related Rate

Equations

4.6 Related Rates (16, Example 4-3) Solution

Strategy

4.6 Related Rates (17, Example 4-4) Solution

Strategy

4.6 Related Rates (18, Example 4-5) Solution

Strategy

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