Chapter 3 | Derivatives
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Figure 3.22 The new value of a changed quantity equals the original value plus the rate of change times the interval of change: f ( a + h ) ≈ f ( a )+ f ′( a )h.
Here is an interesting demonstration (http://www.openstax.org/l/20_chainrule) of rate of change.
Example 3.33 Estimating the Value of a Function
If f (3) =2 and f ′(3) =5, estimate f (3.2).
Solution Begin by finding h . We have h =3.2−3=0.2. Thus,
f (3.2) = f (3+0.2) ≈ f (3)+(0.2) f ′ (3) = 2 + 0.2(5) = 3.
Given f (10) =−5 and f ′(10) =6, estimate f (10.1).
3.21
Motion along a Line Another use for the derivative is to analyze motion along a line. We have described velocity as the rate of change of position. If we take the derivative of the velocity, we can find the acceleration, or the rate of change of velocity. It is also important to introduce the idea of speed , which is the magnitude of velocity. Thus, we can state the following mathematical definitions.
Definition Let s ( t ) be a function giving the position of an object at time t . The velocity of the object at time t is given by v ( t ) = s ′( t ). The speed of the object at time t is given by | v ( t ) | . The acceleration of the object at t is given by a ( t ) = v ′( t ) = s ″( t ).
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