ASSIGNMENT 15 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, clarity and comprehensiveness. 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 Calculus: Early Transcendentals:
From section 4.3, complete questions: 2, 6, 8, 10, 12, 14, 16, 34, 36, 38, 40, 44, 46, 48, 50, 80
Part B
1. Determine if each of the following statements is true. If so, explain why; if not, provide a counterexample.
1. (a) The sum of two concave up functions is concave up.
1. (b) A function which is increasing and concave down everywhere must have a horizontal asymptote.
2. Find the critical points and inflection points for f (x) =
x2
−x2
if x < 0
.
if x ≥ 0
3. Find the number of inflection points for f (x) = x2 + cos(kx), where k is a real number. (There is more
than one possibility.)
4. (a) Find a and b such that f (x) = ax3 + bx2 + 1 has an inflection point at (−1, 2).
4. (b) Using the constants a and b determined in part (a), sketch the graph of f (x).
5. Let f (x) have an inflection point at 0. Prove that the curve y = f (x) crosses the tangent line at 0. (Hint:
let g(x) be the vertical distance between the function and the tangent line. What do the graphs of g 0 (x)
and g(x) look like around 0?)