Math 1710 Exam 1

Show all work (i.e., work things out on paper, not in your head).

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

(a): limx 3-[(x2+5)/(x2-9)] =

(b): limx 4[(2x2-9x+4)/(x2-x-12)]=

(c): limx 5[(3-[(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

x4-3xy2+2y5 = 12

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

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