Name:

1. Determine the following limits (8 pts. each):

(a): lim_{x® 3^-}[(x²+5)/(x²-9)] =

(b): lim_{x® 4}[(2x²-9x+4)/(x²-x-12)]=

(c): lim_{x® 5}[(3-[Ö(2x-1)])/(x-5)]=

2. Let f(x) = 2x⁴+x²-x-7.

Show that f(x) has at least one root somewhere on the real line. (15 pts.) (Hint: Try to find one somewhere between, oh, I don't know, -3 and 3! Trust me, you can't tell me what the root is, just that there is one!)

3. (a): Find, using (one of) the (limit) definitions of the derivative, the derivative of the function

at the point x=1. (15 pts.)

(b): Find the equation for the tangent line to the graph of y=f(x)=3x²-5x+6 at the point (1,f(1)). (8 pts.)

4. Find, using any method, the derivatives of the following functions (7 pts. each):

(a): f(x) = 3x⁵-6x^[3/4]+[5/(x²)]

(b): f(x) = (x²+x+2)(3sinx -5)

(c): f(x) = [(x⁴+3)/(2x³-4x)]

(d): h(x) =sin(x^[1/3]-x)

5. Find the slope of the tangent line to the graph of the equation

at the point (2,1). (10 pts.)