Calculus I                                                       Test #3                                       November 25, 2002

Name____________________                R.  Hammack                                         Score ______

(1) The following function is one-to-one. Find its inverse.  f(x) = e^( x^3 + 1)

(2) Decide if the function  f(x) = e^x + 50x is invertible.  Explain your reasoning.

(3)   Solve the equation   7 = 5e^(x + 2)   for x.

(4) Simplify the following expressions as much as possible.

(a)    8^(-4/3) =

(b)     e^(ln(3) + ln(2)) =   

(c)     log_2(4^(-3/2)) =

(d)     ln(1/e^9) =

(e)     ln(25) + 2 ln(e/5) =

(f)   sin^(-1)(-1/2) =

(g)    sin ( cos^(-1)(x)) =

(5) The graph of the derivative f ' (x) of a function f(x) is sketched.


Answer the following questions about f(x).

(a)  Find the interval(s) on which  f  is increasing.

(b)  Find the interval(s) on which  f  is concave down.

(c)  List the x-coordinates of all inflection points of  f.

(d)  List all the critical numbers of f.

(e)  List the x-coordinates of the relative minima of  f  (if any).

(6) Find the following derivatives.
(a)   d/dx[  e^(1 - x e^(-x)) ] =

(b)   d/dx[  tan^(-1)(x^3)   ] =

(c)   d/dx[    ( cos^(-1)(x) )^2   ] =

(d)  d/dx[  x^( cos(x)) ]  =


(7) Consider the function f(x) = ln(x^4 - 2x^2 + 20)

(a) Find all the critical numbers of f.



(b) Find the locations of all the relative extrema of f, and classify them as relative maximums or minimums.


(8) A formula from physics states that an object which is propelled up or down with an initial velocity of v_0feet per second from a height of s_0feet has a height of  s(t) = s_0 + v_0t - 16t^2 feet at time t seconds.

Suppose you are on top of a 48 foot tall building and toss a ball straight up with an initial velocity of 32 feet per second. (Assume the ball is 48 feet above the ground when it leaves your hand.)

(a) When does the object reach its highest point?


(b) When does the object strike ground?


(c) What is the object's velocity at the instant it strikes the ground?


(d) When does the object have a velocity of 8 feet per second?