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Math 5A Calculus

Test #2

100 Points

Instructions:

1.    For any problem not done simple write problem incomplete-all work must be accounted for

2.    All work must be in order, problem 1 then problem 2 etc,

3.    All work MUST be readable or it will not be graded

4.    Each question worth 5 points

5.    Partial credit is at the sole discretion of the instructor.

6.    If you cannot do a problem simply write problem incomplete. Answers that are mere guesses or clearly unreasonable may be subject to penalty greater than the worth of the problem. Don’t guess.

7.    All work MUST be shown

14.   Starting with x = 4 apply just two steps of   Newtons Method to estimate √2, show work

1.     In four to seven sentences, at a                                                 15.   Prove that √2 is irrational.

university not high school level, describe the                              16.   Prove that  sin x  = cos x

contributions of the following                                                      17.   Prove that  d  sin −1 x =        1      

and humanitarians, or written works whose                           18.   From the formal definition of a limit only

contributions changed and even saved the                                     compute the limit below. Can NOT use

world.                                                                                             L’Hopitals rule, power series, natural logs

a.     Vasily Arkhipov                                                                 or anything other than the formal definition

b.      Leonard Euler                                                                                                                       e ℎ  − 1

c.      Gottfried Leibniz                                                                                                  ℎ(li)        ℎ

d.     Robert Bruce Merrifield                                         19.   From the formal definition of a limit only

e.     Jamie Escalante                                                              compute the derivative below. Can NOT use

2.     Prove using the formal definition only of a                                   L’Hopitals rule, power series, natural logs   derivative prove that                                                                       or anything other than the formal definition

xn  = nxn−1                                                                                                                                 ex  = ex

3.     Compute                                                                                 20.  A lifeguard can run on sand at speed v 1 and

d                                                                              swim at speed v2 and will take path A0B as

dx                                                                                  shown below to minimize the time to rescue

4.     Compute using the chain rule                                                            a swimmer at point B. Let C0D = L, find

 sin(cos ex )                                                                   

5.     As explained in the required reading what is

a function space. Presentation MUST be

from the required reading AND in your own

words. Recall your honors pledge for this

problem and all problems.

6.     Prove differentiability implies continuity.

7.     Prove Rolle's Theorem

8.     Prove the Mean Value Theorem

9.    Use the Mean Value Theorem to prove that if f(x) is differentiable on [a, b] and

f′ (x) = 0 on [a, b] then f(x) is a constant

10.  Use the Mean Value Theorem to prove that if f(x) and g(x)is differentiable on [a, b]   and f′ (x) = g′(x) also on [a, b] then         f(x) − g(x) is a constant

11.  A spherical balloon is being filled with air at a constant rate of 2 cm3/sec. How fast is the radius increasing when the radius is 3 cm.

12.   Let f(x) = x +  find all the critical points  and use the second derivative test to prove if they are max or min.

13.   Let P be the perimeter of a rectangular         fence. Prove that the shape that encloses the maximum area is a square.