Lecture12 - 3.2 Differentiation Second and Higher-Order Derivatives If y = f(x is a differentiable function then its derivative f(x is also a

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Unformatted text preview: 3.2 Differentiation Second- and Higher-Order Derivatives If y = f (x) is a differentiable function, then its derivative f (x) is also a function. If f is differentiable, then we can differentiate f to get a new function of x denoted f . So f = (f ) . Notations Example y = x6 . Find y Example 10 Find the first 4 derivatives of y = x3 - 3x2 + 2 1 Example Find the horizontal tangent lines of y = 2x2 - 12x + 5. Where does y have the smalles value? 2 3.3 The Derivative as a Rate of Change DEFINITION Velocity ( instantaneous velocity ) is the derivative of position with respect to time. If a body's position at time t is s = f (t), then the body's velocity at time t is v(t) = ds f (t + t) - f (t) = lim . dt t0 t DEFINITION Speed is the absolute value of velocity. Speed = |v(t)| = | ds | dt DEFINITION Acceleration is the derivative of velocity with respect to time. If a body's position at time t is s = f (t), then the body's acceleration at time t is a(t) = dv d2 s = 2. dt dt Jerk is the derivative of acceleration with respect to time: j(t) = da d3 s = 3. dt dt 3 Exercises 3.2 Find y a) by applying the Product Rule and b) by multiplying the factors 14 y = (x - 1)(x2 + x + 1) 16 1 y = (x + x )(x - 1 x + 1) Find the derivatives of the functions: 4 17 y= 2x+5 3x-2 19 g(x) = x2 -4 x+0.5 5 20 f (t) = t2 -1 t2 +t-2 6 ...
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This note was uploaded on 03/31/2008 for the course MATH 1205 taught by Professor Fbhinkelmann during the Spring '08 term at Virginia Tech.

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Lecture12 - 3.2 Differentiation Second and Higher-Order Derivatives If y = f(x is a differentiable function then its derivative f(x is also a

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