1.(10 points)  State the Fundamental Theorem of Calculus.  

2.(30)  Compute derivatives.  (No algebraic simplification is necessary.)
  a)  j(x) = ln(e^ax+b)          [(dj)/( dx)] =
 b)  G(x) = [(sin²x+1)/( cos²x+1)]    G ^¢(x) =
 c)  w = 2 ^{- 4z} sin(pz)    dw/dz =
 d)  T(u) = arctan([(u)/(1+u)])    T ^¢(u) =
 e)  g(x) = [(x²+Öx+1)/( x^3/2)]    g¢(x) =

3.(10)  Suppose that the expression 4x²-3y⁵ = - x³y⁴ defines a function in an interval around the point (-1,1).  Find the slope of the tangent line to this function at the point (-1,1).  (Hint:  substitute the point after one step of your calculation.)

4.(15)  Consider the function f(x) = [Ö(5+4x)] .  
a)  Find an equation for the line tangent to f at the point where x = 1.
b)  The formula for f says that f(1.5) = [Ö(5+4(1.5))] = [Ö11].  Use (a) to estimate the value of [Ö11]
c)  Using your answer to (a), estimate [Ö9.2]  (Hint:  you need to select an appropriate number (near 1!) to substitute into your answer to (a).)
d)  Which estimate (b) or (c) do you think is more accurate and why?

5.(20)  In addition to the famous (continuous) functions zot(x) and zing(x) from a recent quiz, there is another function:  zap(x).  The relationship between these functions is:

(zot(x)) ^¢ = zing(x)      (zot(x)) ^¢¢ = zap(x).

Here are some values for these functions:

zot(1) = 7	zot(3) = 2
zing(1) = -1	zing(3) = 0
zap(1) = 1	zap(3) = 4

a)  If f(x) = x² zot(x), find f ¢(1).
b)  If g(x) = zing(x³), find g ¢(1).
c)  ò₁³ zing(x) dx =
d)  Does zot(x) have a maximum, a minimum, or neither at x = 3?  Explain.

6.(15)  A turkey is put into a 400 degree F oven.  Suppose that after 30 minutes in the oven, the turkey is 25023 F and is increasing at the rate of  2 degrees per minute.  Newton's law of cooling (well, heating, in this case) says that the temperature at time t will be given by T(t) = 400-a e^-bt, where a and b are positive constants.

a)  Explain in a sentence or two why this equation ``makes sense.''   (Things to mention:  the 400, why a and b should be positive, the two negative signs.)
b)  Find a and b.  (You do not need to find any decimals here!)
c)  What is the initial temperature of the turkey?