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Calculus I                                                                           Test #1                                              October  6, 2003

Name____________________                                   R.  Hammack                                              Score ______
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(1) (12 points)

(a)  sin(-(3π)/4) =

(b) tan(-(3π)/4) =

(c) Find all solutions of  the equation    2cos(x) = 3^(1/2).

 



(2) (8 points)

(a) Sketch the graph of the equation    2x - 4y = -8.

[Graphics:HTMLFiles/T1F03Csol_12.gif]



(b) Find the equation of the line that is perpendicular to the graph of   2x - 4y = -8 (from Part a, above) and which has an x-intercept of 1.  Put your final answer in slope-intercept form, and simplify as much as possible.

 



(3) (40 points)  The problems on this page concern the functions f(x) = x^2/(3x^2 - 9x) and   g(x) = cos(x) .

(a) f∘g(x) =

(b) g∘f(x) =

(c) State the domain of f(x) = x^2/(3x^2 - 9x)

(d)  List the values of x at which f is not continuous

(e)     Underscript[lim , x1] x^2/(3x^2 - 9x) =

(f)     Underscript[lim , x0] x^2/(3x^2 - 9x) =  

(g)   Underscript[lim , x3^+] x^2/(3x^2 - 9x) =  

(h)   Underscript[lim , x∞] x^2/(3x^2 - 9x) =

(i) List the vertical asymptote(s) of f  (if any).  (Feel free to use any relevant information from parts a-d above)



(j) List the horizontal asymptote(s) of f  (if any).   (Feel free to use any relevant information from parts a-d above)

 


(4)  (16 points)  Evaluate the following limits.

(a)    Underscript[lim , x3] (x^2 - 3x + 4) =

(b)  Underscript[lim , x0] sin((x + π)/2) =

(c)    Underscript[lim , x0] sin(x)/x =

(d)   Underscript[lim , w5] (1/5 - 1/w)/(w - 5) =

(5) (20 points)  This problem concerns the function f(x) = {   2 1 - x if  x<1 x - 1 if  1≤x
(a) f(1) =

(b) Underscript[lim , x1^-] f(x) =

(c) Underscript[lim , x1^+] f(x) =

(d) Sketch a graph of the function f.
[Graphics:HTMLFiles/T1F03Csol_51.gif]

(e)  Is the function f continuous at x = 1?  



(6) (4 points)
Underscript[lim , x -∞] x/(4x^2 - 6 )^(1/2) =