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Alternate Definition of a Derivative

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1 Alternate Definition of a Derivative
Section 2.1A Calculus AP/Dual, Revised ยฉ2018 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative

2 ยง2.1A: Alternate Definition of a Derivative
Alternate form of the Derivative Equation: ๐’‡ โ€ฒ ๐’„ = ๐ฅ๐ข๐ฆ ๐’™โ†’๐’„ ๐’‡ ๐’™ โˆ’๐’‡ ๐’„ ๐’™โˆ’๐’„ This equation will allow the showing of the slope of the tangent line at a particular point When investigating the relationship between differentiability and continuity The existence of this limit requires that the one-sided limits exist and are equal: ๐ฅ๐ข๐ฆ ๐’™โ†’ ๐’„ โˆ’ ๐’‡ ๐’™ โˆ’๐’‡ ๐’„ ๐’™โˆ’๐’„ = ๐ฅ๐ข๐ฆ ๐’™โ†’ ๐’„ + ๐’‡ ๐’™ โˆ’๐’‡ ๐’„ ๐’™โˆ’๐’„ 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative

3 ยง2.1A: Alternate Definition of a Derivative
Review Find the slope of the tangent line to the graph of the function ๐’‡ ๐’™ =๐Ÿ“โˆ’ ๐’™ ๐Ÿ , at the given point ๐Ÿ, ๐Ÿ 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative

4 ยง2.1A: Alternate Definition of a Derivative
Review Find the slope of the tangent line to the graph of the function ๐’‡ ๐’™ =๐Ÿ“โˆ’ ๐’™ ๐Ÿ , at the given point ๐Ÿ, ๐Ÿ 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative

5 ยง2.1A: Alternate Definition of a Derivative
Example 1 Use the alternative form to find slope of the tangent line to the graph of the function ๐’‡ ๐’™ =๐Ÿ“โˆ’ ๐’™ ๐Ÿ , at the given point ๐Ÿ, ๐Ÿ 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative

6 ยง2.1A: Alternate Definition of a Derivative
Example 1 Use the alternative form to find slope of the tangent line to the graph of the function ๐’‡ ๐’™ =๐Ÿ“โˆ’ ๐’™ ๐Ÿ , at the given point ๐Ÿ, ๐Ÿ 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative

7 ยง2.1A: Alternate Definition of a Derivative
Example 2 Use the alternative form to find the derivative of ๐’‡ ๐’™ = ๐Ÿ ๐’™ when ๐’„=๐Ÿ’ 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative

8 ยง2.1A: Alternate Definition of a Derivative
Example 2 Use the alternative form to find the derivative of ๐’‡ ๐’™ = ๐Ÿ ๐’™ when ๐’„=๐Ÿ’ 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative

9 ยง2.1A: Alternate Definition of a Derivative
Example 3 Use the alternative form to find the derivative of ๐’‡ ๐’™ = ๐’† ๐’™ +๐Ÿ๐’™ when ๐’„=๐ŸŽ 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative

10 ยง2.1A: Alternate Definition of a Derivative
Your Turn Use the alternative form to find the derivative of ๐’‡ ๐’™ = ๐’™ ๐Ÿ +๐Ÿ‘, when ๐’„=โˆ’๐Ÿ 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative

11 When I Say Derivative, You Sayโ€ฆ
SLOPE 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative

12 ยง2.1A: Alternate Definition of a Derivative
Graphing a Derivative When looking at a graph, identify what type of slope that each point has If ๐’‡ has a positive slope, ๐’‡โ€ฒ is increasing If ๐’‡ has a negative slope, ๐’‡โ€ฒ is decreasing After establishing all of the slopes, determine the derivative graph by its slope. In other words, plot points of the derivative onto the graph 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative

13 ยง2.1A: Alternate Definition of a Derivative
Steps Establish any slopes of zero Establish any steep and flat the slopes are Plot the points 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative

14 ยง2.1A: Alternate Definition of a Derivative
Example 4 Sketch the derivative of the function given. m = 1.5 Neg. Pos. 5 1.5 3 0.5 Negative Slope Positive Slope (1.5, 5) Dec. Inc. Decreasing Increasing 5 1.5 3 0.5 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative

15 ยง2.1A: Alternate Definition of a Derivative
Example 5 Sketch the derivative of the function given. Pos. Slope Pos. Pos. โ€“3 โ€“1 1/2 2 Neg. Neg. Inc. Inc. Inc. โ€“3 โ€“1 1/2 2 Dec. Dec. 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative

16 ยง2.1A: Alternate Definition of a Derivative
Example 6 Sketch the derivative of the function given. 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative

17 ยง2.1A: Alternate Definition of a Derivative
Example 6 Sketch the derivative of the function given. Increases Increases + + 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative

18 ยง2.1A: Alternate Definition of a Derivative
Your Turn Sketch the derivative of the function given. 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative

19 Direction of a Derivative from a Graph
A graph which is differentiable is continuous A graph which is continuous, is not always differentiable A graph is neither continuous or differentiable is discontinuous Hole Vertical Asymptotes Jump discontinuities Kinks/Cusps 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative

20 ยง2.1A: Alternate Definition of a Derivative
Special Cases Differentiable functions are Continuous but continuous functions are not always Differentiable Functions can not be differentiable when: Sharp turns or โ€œcuspsโ€ (i.e. ๐’‡(๐’™)=|๐’™โˆ’๐Ÿ|) Vertical Tangents (i.e. ๐’‡ ๐’™ = ๐’™ ๐Ÿ/๐Ÿ‘ ) Discontinuity A function ๐’‡ is said to be differentiable at ๐’™= ๐’„ if the derivative of ๐’‡ exists at ๐’™=๐’„. For the derivative to exist at ๐’™=๐’„, the graph of ๐’‡ must have local linearity. 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative

21 ยง2.1A: Alternate Definition of a Derivative
Example 7 Use the graph to determine the derivative of ๐’‡ ๐’™ = ๐’™โˆ’๐Ÿ . If it does not exist, explain why. 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative

22 ยง2.1A: Alternate Definition of a Derivative
Example 7 Use the graph to determine the derivative of ๐’‡ ๐’™ = ๐’™โˆ’๐Ÿ . If it does not exist, explain why. 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative

23 ยง2.1A: Alternate Definition of a Derivative
Example 7 Use the graph to determine the derivative of ๐’‡ ๐’™ = ๐’™โˆ’๐Ÿ . If it does not exist, explain why. ๐’‡ is not differentiable at ๐’™=๐Ÿ because ๐’‡ has a sharp turn at ๐’™=๐Ÿ. 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative

24 ยง2.1A: Alternate Definition of a Derivative
Example 8 Use the graph to determine the derivative of ๐’‡ ๐’™ = ๐’™ ๐Ÿ/๐Ÿ‘ at ๐’™ =๐ŸŽ. If it does not exist, explain why. ๐’‡ is not differentiable at ๐’™=๐Ÿ because ๐’‡ has a sharp turn at ๐’™=๐Ÿ. 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative

25 ยง2.1A: Alternate Definition of a Derivative
Example 9 Use the graph to determine the derivative of ๐’‡ ๐’™ = ๐’™โˆ’๐Ÿ ๐Ÿ/๐Ÿ‘ . If it does not exist, explain why. ๐’‡ is not differentiable at ๐’™=๐Ÿ because ๐’‡ has a cusp at ๐’™=๐Ÿ. 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative

26 AP Multiple Choice Practice Question 1 (non-calculator)
Let ๐’‡ be a function such that ๐ฅ๐ข๐ฆ ๐’‰โ†’๐ŸŽ ๐’‡ ๐Ÿ+๐’‰ โˆ’๐’‡ ๐Ÿ ๐’‰ =๐Ÿ“. Which of the following must be true? ๐’‡ is continuous at ๐’™=๐Ÿ ๐’‡ is differentiable at ๐’™=๐Ÿ The derivative of ๐’‡ is continuous at ๐’™=๐Ÿ (A) I only (B) II only (C) I and II only (D) II and III only 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative

27 AP Multiple Choice Practice Question 1 (non-calculator)
Let ๐’‡ be a function such that ๐ฅ๐ข๐ฆ ๐’‰โ†’๐ŸŽ ๐’‡ ๐Ÿ+๐’‰ โˆ’๐’‡ ๐Ÿ ๐’‰ =๐Ÿ“. Which of the following must be true? Vocabulary Connections and Process Answer and Justifications 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative

28 ยง2.1A: Alternate Definition of a Derivative
Assignment Worksheet 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative


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