Calculus 1  Hour Exam 1  (continued)   -   Math 220     Fall, 2010

 

 

5.

 

 

Let  f(x)  =  x3  -  5x.  Use the definition of the derivative as a limit to show that

f ‘(x)  =  3x2 – 5.  Show each step of your calculation and be sure to use proper

terminology.    

 

 

 

6.

 

For a given angle  θ, it is known that  cos θ ≈ 0.927,  sin θ ≈0.375 and  tan ≈ 0.404.

What is the value of  cos (π/2 + θ)?

                                                                                               Answer:   - 0.375

 

 

7.

 

 

Evaluate and simplify  tan (cos-1 (2/3) ).                        Answer:  √5 / 2

 

 

 

8.

 

 

Determine the real numbers  a  and  b  so that the expression  8 csc2 θ  -  5 cot2 θ  can

be written as   a csc2 θ  + b.             

                                                                                     Answers:  a = 3 ,   b  =  5

 

 

 

 

 

 

 

 

 

 

 

9.

 

 

Evaluate the following limits and simplify each answer.  An answer ‘does not exist’

Is not sufficient.  It the limit is infinite then you must state if it is  ∞  or  -∞.

 

a.  lim  (2 / (ex + 3)                                 b.  lim ( 1000 + 5 ln(x – 2) )     

   xà0                                                       xà2+                                        

 

 

c.  lim  (4x2  - 9) / (2x – 3)                     d,  lim (5 – 3x) / (x – 2)

    xà3/2                                                xà2-

 

 

e.  lim  (2x + 1)2 / (3x + 1)2                    f.  lim [ 1 / (x – 5)  -  10 / (x2 – 25) ]

    xà∞                                                    xà5

 

 

 

answers:  a.  ½         b.  -∞         c.  6            d.  ∞           e.  4/9             f.  1/10