2. 2 Differentiation 2.1 TANGENT LINES AND VELOCITY 2.2 THE DERIVATIVE
2.3
COMPUTATION OF DERIVATIVES: THE POWER RULE
2.4
THE PRODUCT AND QUOTIENT RULES
2.5
THE CHAIN RULE
2.6
DERIVATIVES OF TRIGONOMETRIC FUNCTIONS
2.7
IMPLICIT DIFFERENTIATION
2.8
THE MEAN VALUE THEOREM
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3. 2.2
THE DERIVATIVE
2.1
The derivative of the function f at the point x = a is defined as
provided the limit exists. If the limit exists, we say that f is differentiable at x = a.
An alternative form is
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4. 2.2 THE DERIVATIVE 2.1 Finding the Derivative at a Point Slide 4
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5. 2.2 THE DERIVATIVE 2.1 Finding the Derivative at a Point Slide 5
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6. 2.2 THE DERIVATIVE 2.2 Finding the Derivative at an Unspecified Point
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7. 2.2 THE DERIVATIVE 2.2 Finding the Derivative at an Unspecified Point
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8. 2.2
THE DERIVATIVE
2.2
The derivative of the function f is the function f‘ given by
The domain of f is the set of all x’s for which this limit exists.
The process of computing a derivative is called differentiation. Further, f is differentiable on an open interval I if it is differentiable at every point in I .
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9. 2.2
THE DERIVATIVE
2.3
Finding the Derivative of a Simple Rational Function
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10. 2.2
THE DERIVATIVE
2.3
Finding the Derivative of a Simple Rational Function
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11. 2.2
THE DERIVATIVE
2.3
Finding the Derivative of a Simple Rational Function
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12. 2.2 THE DERIVATIVE 2.5 Sketching the Graph of f’ Given the Graph of f
Given the graph of f in the figure, sketch a plausible graph of f’.
Keep in mind that the value of the derivative function at a point is the slope of the
tangent line at that point.
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13. 2.2 THE DERIVATIVE 2.5 Sketching the Graph of f’ Given the Graph of f
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14. 2.2 THE DERIVATIVE 2.6 Sketching the Graph of f Given the Graph of f’
Given the graph of f’ in the figure, sketch a plausible graph of f.
Keep in mind that the slope of the tangent line of f at a point is the value of f’ at that point.
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15. 2.2 THE DERIVATIVE 2.6 Sketching the Graph of f Given the Graph of f’
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16. Alternative Derivative Notations
2.2
THE DERIVATIVE
Alternative Derivative Notations
If we write y = f (x), the following are all alternatives for denoting the derivative:
The expression
is called a differential operator and tells you to take the derivative of whatever expression follows.
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17. Differentiability Implies Continuity
2.2
THE DERIVATIVE
Differentiability Implies Continuity
We observed that f (x) = |x| does not have a tangent line at x = 0 (i.e., it is not differentiable at x = 0), although it is continuous everywhere.
Thus, there are continuous functions that are not differentiable.
But, are there differentiable functions that are not continuous? No.
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18. 2.2
THE DERIVATIVE
2.1
If f is differentiable at x = a, then f is continuous at x = a.
Note that Theorem 2.1 says that if a function is not continuous at a point, then it cannot have a derivative at that point.
It also turns out that functions are not differentiable at any point where their graph has a “sharp” corner, as is the case for f (x) = |x| at x = 0.
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19. 2.2
THE DERIVATIVE
2.7
Showing That a Function Is Not Differentiable at a Point
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20. 2.2
THE DERIVATIVE
2.7
Showing That a Function Is Not Differentiable at a Point
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21. 2.2
THE DERIVATIVE
2.7
Showing That a Function Is Not Differentiable at a Point
Since the one-sided limits do not agree (0 = 2), f’ (2) does not exist (i.e., f is not differentiable at
x = 2).
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22. Examples of Points of Non-Differentiability
2.2
THE DERIVATIVE
Examples of Points of Non-Differentiability
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23. Examples of Points of Non-Differentiability
2.2
THE DERIVATIVE
Examples of Points of Non-Differentiability
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24. Numerical Differentiation
2.2
THE DERIVATIVE
Numerical Differentiation
There are many times in applications when it is not possible or practical to compute derivatives symbolically.
This is frequently the case where we have only some data (i.e., a table of values) representing an otherwise unknown function.
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25. 2.2 THE DERIVATIVE 2.8 Approximating a Derivative Numerically Slide 25
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26. 2.2 THE DERIVATIVE 2.8 Approximating a Derivative Numerically Slide 26
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27. 2.2 THE DERIVATIVE 2.9 Estimating Velocity Numerically
Suppose that a sprinter reaches the following distances in the given times. Estimate the velocity of the sprinter at the 6-second mark.
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28. 2.2 THE DERIVATIVE 2.9 Estimating Velocity Numerically
Based on the table, conjecture
that the sprinter was reaching a peak speed at about the 6-second mark. Thus, accept an estimate of 35.2 ft/s. (There is not a single
correct answer to this question.)
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