121616949-math.64 - 50 Chapter 3 Rules For Finding...

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50 Chapter 3 Rules For Finding Derivatives Exercises Find the derivatives of the functions in 1–6. 1. 5 x 3 + 12 x 2 - 15 2. - 4 x 5 + 3 x 2 - 5 /x 2 3. 5( - 3 x 2 + 5 x + 1) 4. f ( x ) + g ( x ), where f ( x ) = x 2 - 3 x + 2 and g ( x ) = 2 x 3 - 5 x 5. ( x + 1)( x 2 + 2 x - 3) 6. 625 - x 2 + 3 x 3 + 12 (See section 2.1 .) 7. Find an equation for the tangent line to f ( x ) = x 3 / 4 - 1 /x at x = - 2. 8. Find an equation for the tangent line to f ( x ) = 3 x 2 - π 3 at x = 4. 9. Suppose the position of an object at time t is given by f ( t ) = - 49 t 2 / 10 + 5 t + 10. Find a function giving the speed of the object at time t . The acceleration of an object is the rate at which its speed is changing, which means it is given by the deriviative of the speed function. Find the acceleration of the object at time t . 10. Let f ( x ) = x 3 and c = 3. Sketch the graphs of f , cf , f , and ( cf ) on the same diagram. 11. Suppose that f and g are differentiable at x . Show that f - g is differentiable at x using the two linearity properties from this section. 12. The general polynomial P of degree n in the variable x has the form P ( x ) = n X k =0 a k x k = a 0 + a 1 x + . . . + a n x n . What is the derivative (with respect to
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Unformatted text preview: x ) of P ? ⇒ 13. Find a cubic polynomial whose graph has horizontal tangents at (-2 , 5) and (2 , 3). ⇒ 3.3 The Product Rule Consider the product of two simple functions, say f ( x ) = ( x 2 + 1)( x 3-3 x ). An obvious guess for the derivative of f is the product of the derivatives of the constituent functions: (2 x )(3 x 2-3) = 6 x 3-6 x . Is this correct? We can easily check, by rewriting f and doing the calculation in a way that is known to work. First, f ( x ) = x 5-3 x 3 + x 3-3 x = x 5-2 x 3-3 x , and then f ± ( x ) = 5 x 4-6 x 2-3. Not even close! What went “wrong”? Well, nothing really, except the guess was wrong. So the derivative of f ( x ) g ( x ) is NOT as simple as f ± ( x ) g ± ( x ). Surely there is some rule for such a situation? There is, and it is instructive to “discover” it by trying to do...
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