I have a short 4 question math test in about 2 hours. The test will be given to us, and I will send it to you immediately. The topics are Continuity and Intermediate Value Theorem from Intro to Calcul

(b) lim h ! 1 p 4 h 2 4h 3 + 2 h:

(c) lim s ! 1 s 2 5s p 4 s4 + 5 s2 :

2.Use the Intermediate Value Theorem to show the following equations have solutions (a) x5 x2 + 2 x+ 3 = 0 (b) 3 p x = 1 x:

3.Sketch a graph of a function f(x ) which sati es all the following properties.

( a ) lim x! 1 f (x ) = 0 ( b) lim x!1 f (x ) = 2 ( c) lim x! 3 f (x ) = 1 (d ) lim x! 3+ f (x ) = 1 ( e )f (0) = 0 ( f) lim x! 2f (x ) = 3 ( g)f (3) does not exist 4.Use the de nition of the derivative to nd f0 ( a ): Then nd the equation of the tangent line for the given value of a:

(a) f(x ) = x2 + 3 x 4 a= 2 (b) f(x ) = 1 p x 2 a = 6 (c) t t 1 a= 1 5.Find lim x!1 f (x ) if for all x >1 10 ex 21 2 ex < f (x ) < 5p x p x 1:

1 Solutions 1.(a) 3 5 (b)1 (c) 1 2 2.(a)Use f(x ) = x5 x2 + 2 x+ 3 with 1< c < 0.

(b)Use f(x ) = 3 p x + x 1 with 0 < c <1.

3.There are lots of graphs with those properties.

4.(a) y ( 6) = 1( x ( 2)) or y+ 6 = (x + 2) (b) y 1 2 = 1 16 ( x 6) (c) y 0 = 2( x (1)) or y= 2( x+ 1) 5.lim x!1 f (x ) = 5 2