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The Derivative

#1. Define f: R\{0} ® R  by .  Use the definition of derivative (page 1 of Differentiation notes, or Def. 4.1.1, Lebl)  to find ,  the derivative of f at c.

#2.   Let    Use the definition of derivative (page 1 of Differentiation notes, or Def. 4.1.1, Lebl) and the result of Week 5 Homework #1(b)  to find , the derivative of f at 0.

#3.   Is     differentiable at 1?   Explain carefully and justify your answer.

#4. Fill in the blanks to use the Mean Value Theorem and the known facts that   and  to carefully show that             

                                            ,  i.e.,  

Proof:   [Indicate what belongs in each of the blanks)

Let    on the interval [36, 39].  

 

Since     ___________________________________________________________________

 

the Mean Value Theorem  can be applied, and so there exists c Î _________  (state the interval) such that

 

            

 

                   

 

                                       Now we want a lower bound and an upper bound for .  

 

    =   _________________.   (State the derivative, in terms of c.)

 

Since c > 36,          ____________.   (State an appropriate fraction, a "nice" rational number.)

 

Since c < 39 < 49,   ___________.   (State an appropriate fraction, a "nice" rational number. Make use of the fact that we know ; we don't know an approximation for ; indeed that is what we are trying to determine).

 

Youmay assume relevant derivative formulas from calculus for powers of x, polynomials, sine, cosine, tangent, exponentials, and logs.

 

#5.  Let f(x) = - 4x2 + 4x + 9 for x in [0, 2].

#5 (a) Use the derivative to find where f is increasing and where f is decreasing. (Show some work.)

#5 (b) Find the maximum and the minimum of f on [0, 2]. Show work/explanation.

#6. For each of the following scenarios, find a function f and a domain [a, b] which satisfies the scenario and for which there is NO point c in (a, b) with f ´(c) = 0. (You can have a different function and interval for each part). By carrying out this exercise, you are verifying that all of the hypotheses for Rolle's Theorem must be satisfied in order to guarantee the conclusion that f ´(c) = 0 for some c in (a, b). (Explanations not required)

 

#6(a)  f is continuous on [a, b] and f is not differentiable on (a, b) and f(a) = f(b), andthere is no point c in (a, b) with f ´(c) = 0.

#6 (b)  f is continuous on [a, b] and differentiable on (a, b) and f(a) ¹ f(b), andthere is no point c in  (a, b) with f ´(c) = 0.

# 6(b)  f is not continuous on [a, b] and f is differentiable on (a, b) and f(a) = f(b), andthere is no point c in (a, b) with f ´(c) = 0.

L'Hopital's Rule

You may assume relevant derivative formulas from calculus for powers of x, polynomials, sine, cosine, tangent, exponentials, and logs.

                                                                             

#7.  Determine the following limits (if they exist). Show work.

 

#7(a). 

#7(b). 

#7(c).        Hints: . First find

#8.  Critique the following work. Is it correct? If not, explain what has been done wrong, and correctly determine the limit.

 

Infinite Series

 

#9. For each of the following six series, decide if it is convergent or divergent. Justify your answer with explanation/work. Reference appropriate examples,  theorems, or tests  from the Week 6 Infinite Series notes or Section 2.5 of Lebl.

 

#9(a) 

 

 

 

 

 

 

 

 

 

 

 

#9(b) 

#9 (c)      

#9(d) 

#9 (e)      

#9 (f)    

 

 

  • 6 years ago
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