ASSIGNMENT 4 for SECTION 001
This assignment is to be handed in. There are two parts: Part A and Part B.
Part A will be graded for completeness. You will receive full marks only if every question has been completed.
Part B will be graded for correctness. You will receive full marks on a question only if your answer is correct
and your reasoning is clear. In both parts, you must show your work.
Please submit Part A and Part B separately, with your name on each part.
Part A
From The Mathematics Survival Kit:
Complete questions 1 and 2 on the following pages: 182, 184, 186
Complete question 1 on page 188
From Calculus: Early Transcendentals:
From section 2.4, complete questions: 2, 4
From section 2.5, complete questions: 2, 4, 6, 8, 10, 12, 14, 16, 18, 22, 24, 32, 36, 40, 46, 48, 60, 64
Part B

 |cx − 3| if x < 1
1. Let f (x) =
2(cx)−1 if x = 1 . Find c such that f is continuous everywhere.
 √
c + 3x if x > 1
2. Determine where f (x) =
x2 + 5
is discontinuous.
cos x
3. Find an example of a function f whose domain is all real numbers, and which is:
1. (a) continuous everywhere
1. (b) continuous nowhere
1. (You must justify your answers.)
4. Given a continuous function f , note that
g(a) = the slope of the tangent line to y = f (x) at x = a
is a function. Find an example of a continuous function f such that g, as defined above, is discontinuous.
5. Let f (x) be a polynomial function of odd degree. (The degree of a polynomial is the largest degree of any
one term in the polynomial — for example, 4x3 + x − 1 is a polynomial of degree 3.) Prove that f (x) has
at least one real root.