MATH 221 Test 1 Name______________________________

Instructions: Answer all the questions on the paper provided and show all your work so that partial credit may be given.

I. Let . Find using the limit definition of the derivative. (16)

II. Let . (16)

a) Find ____________

b) Find ____________

c) Find ____________

d) Is f(x) continuous at x = -1? If not, what kind of discontinuity does it have?

III. Give an example of a function with an essential discontinuity at x = 2. (5)

IV. Find the domain of the function . (5)

V. Let (20)

a) What is

b) Find the equation of the tangent line to y = f(x) at x = -1.

c) Find the equation of the tangent line to y = f(x) at x = 3.

d) Find the point of intersection of the lines in b) and c).

VI. Sketch the graph of a function, f(x), which satisfies the following: (16)

f(x) has a removable discontinuity at x = -2 and a jump discontinuity at x = 2.

f(-2) = 2 , f(0) = 0, and f(2) = 3.

VII. Solve the following inequalities: (22)

a)

b)