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Alternate Definition of a Derivative
Section 2.1A Calculus AP/Dual, Revised ยฉ2018 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative
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ยง2.1A: Alternate Definition of a Derivative
Alternate form of the Derivative Equation: ๐ โฒ ๐ = ๐ฅ๐ข๐ฆ ๐โ๐ ๐ ๐ โ๐ ๐ ๐โ๐ 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: ๐ฅ๐ข๐ฆ ๐โ ๐ โ ๐ ๐ โ๐ ๐ ๐โ๐ = ๐ฅ๐ข๐ฆ ๐โ ๐ + ๐ ๐ โ๐ ๐ ๐โ๐ 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative
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ยง2.1A: Alternate Definition of a Derivative
Review Find the slope of the tangent line to the graph of the function ๐ ๐ =๐โ ๐ ๐ , at the given point ๐, ๐ 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative
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ยง2.1A: Alternate Definition of a Derivative
Review Find the slope of the tangent line to the graph of the function ๐ ๐ =๐โ ๐ ๐ , at the given point ๐, ๐ 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative
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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 =๐โ ๐ ๐ , at the given point ๐, ๐ 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative
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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 =๐โ ๐ ๐ , at the given point ๐, ๐ 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative
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ยง2.1A: Alternate Definition of a Derivative
Example 2 Use the alternative form to find the derivative of ๐ ๐ = ๐ ๐ when ๐=๐ 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative
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ยง2.1A: Alternate Definition of a Derivative
Example 2 Use the alternative form to find the derivative of ๐ ๐ = ๐ ๐ when ๐=๐ 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative
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ยง2.1A: Alternate Definition of a Derivative
Example 3 Use the alternative form to find the derivative of ๐ ๐ = ๐ ๐ +๐๐ when ๐=๐ 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative
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ยง2.1A: Alternate Definition of a Derivative
Your Turn Use the alternative form to find the derivative of ๐ ๐ = ๐ ๐ +๐, when ๐=โ๐ 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative
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When I Say Derivative, You Sayโฆ
SLOPE 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative
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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, ๐โฒ is increasing If ๐ has a negative slope, ๐โฒ 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
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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
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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
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ยง2.1A: Alternate Definition of a Derivative
Example 5 Sketch the derivative of the function given. Pos. Slope Pos. Pos. โ3 โ1 1/2 2 Neg. Neg. Inc. Inc. Inc. โ3 โ1 1/2 2 Dec. Dec. 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative
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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
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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
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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
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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
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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 โcuspsโ (i.e. ๐(๐)=|๐โ๐|) Vertical Tangents (i.e. ๐ ๐ = ๐ ๐/๐ ) 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
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ยง2.1A: Alternate Definition of a Derivative
Example 7 Use the graph to determine the derivative of ๐ ๐ = ๐โ๐ . If it does not exist, explain why. 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative
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ยง2.1A: Alternate Definition of a Derivative
Example 7 Use the graph to determine the derivative of ๐ ๐ = ๐โ๐ . If it does not exist, explain why. 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative
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ยง2.1A: Alternate Definition of a Derivative
Example 7 Use the graph to determine the derivative of ๐ ๐ = ๐โ๐ . If it does not exist, explain why. ๐ is not differentiable at ๐=๐ because ๐ has a sharp turn at ๐=๐. 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative
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ยง2.1A: Alternate Definition of a Derivative
Example 8 Use the graph to determine the derivative of ๐ ๐ = ๐ ๐/๐ at ๐ =๐. If it does not exist, explain why. ๐ is not differentiable at ๐=๐ because ๐ has a sharp turn at ๐=๐. 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative
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ยง2.1A: Alternate Definition of a Derivative
Example 9 Use the graph to determine the derivative of ๐ ๐ = ๐โ๐ ๐/๐ . If it does not exist, explain why. ๐ is not differentiable at ๐=๐ because ๐ has a cusp at ๐=๐. 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative
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AP Multiple Choice Practice Question 1 (non-calculator)
Let ๐ be a function such that ๐ฅ๐ข๐ฆ ๐โ๐ ๐ ๐+๐ โ๐ ๐ ๐ =๐. Which of the following must be true? ๐ is continuous at ๐=๐ ๐ is differentiable at ๐=๐ The derivative of ๐ is continuous at ๐=๐ (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
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AP Multiple Choice Practice Question 1 (non-calculator)
Let ๐ be a function such that ๐ฅ๐ข๐ฆ ๐โ๐ ๐ ๐+๐ โ๐ ๐ ๐ =๐. 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
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ยง2.1A: Alternate Definition of a Derivative
Assignment Worksheet 11/16/ :51 AM ยง2.1A: Alternate Definition of a Derivative
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