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Questions and Answers
What is the primary focus of the chapter indicated in the content?
What is the primary focus of the chapter indicated in the content?
In the context of this chapter, what fundamental concept is likely addressed along with derivatives?
In the context of this chapter, what fundamental concept is likely addressed along with derivatives?
Which of the following is NOT typically involved in the discussion of derivatives?
Which of the following is NOT typically involved in the discussion of derivatives?
What might be a common application of derivatives discussed in this chapter?
What might be a common application of derivatives discussed in this chapter?
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Which of the following statements about derivatives is correct?
Which of the following statements about derivatives is correct?
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Study Notes
Derivatives of a Function
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A derivative of a function, denoted as f'(x), represents the instantaneous rate of change or the slope of the function at a specific point.
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Geometrically, the derivative represents the slope of the tangent line to the function's graph at a given point.
Calculating Derivatives Using the Limit Definition
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The derivative is calculated as the limit of the difference quotient as h approaches 0:
f'(x) = lim (f(x + h) - f(x))/h as h → 0
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This method involves substituting (x + h) into the function's expression, subtracting the original function's value, and dividing by h.
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Then, find the limit as h approaches 0.
Example 1: Differentiating f(x) = x³
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Given f(x) = x³, find f'(x).
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f(x + h) = (x + h)³
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f'(x) = lim (f(x + h) - f(x))/h as h → 0
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f'(x) = 3x²
Example 2: Differentiating f(x) = x² / 2
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Given f(x) = x² / 2, find f'(x).
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f(x+h) = ((x + h)²)/2
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f'(x) = lim ((x + h)²/2-x²/2)/h as h → 0
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f'(x) = x
Alternate Definition of the Derivative at a Point
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f'(a) = lim (f(x) - f(a))/(x-a) as x → a
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This method calculates the derivative at a specific point "a".
One-Sided Derivatives
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Right-hand derivative: lim (f(x + h) - f(x))/h as h → 0+
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Left-hand derivative: lim (f(x + h) - f(x))/h as h → 0-
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These are important for determining if a function has a derivative at a given point if the left and right-hand derivatives are not equal, there is no derivative at that point.
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Description
This quiz covers the concepts of derivatives, including the limit definition and calculations using examples like f(x) = x³ and f(x) = x² / 2. Test your understanding of how to find the instantaneous rate of change of functions by calculating their derivatives.
