ASSIGNMENT 7 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 Calculus: Early Transcendentals:
From section 3.3, complete questions: 2, 4, 6, 8, 10, 12, 14, 16, 18, 22, 24, 30, 32, 34, 40, 42, 44, 46, 48
Part B
sin
1. Find lim
x→0
π
6x + π
− sin
6
6
, if it exists.
x
2. Given a thrice-differentiable function f (x), its Maclaurin polynomial of degree 3 is the polynomial of the
form
g(x) = a0 + a1 x + a2 x2 + a3 x3
satisfying the conditions g(0) = f (0), g 0 (0) = f 0 (0), g 00 (0) = f 00 (0) and g (3) (0) = f (3) (0). Find the
Maclaurin polynomials of degree 3 for the following functions:
2. (a) f (x) = sin x
2. (b) f (x) = cos x
2. (c) f (x) = ex
3. Determine how often the curve y =
sin x
has a horizontal tangent line on the interval (0, 100).
x
4. Determine how often the curve y =
cos x − 1
has a horizontal tangent line on the interval (0, ∞).
x
5. Prove that
d
cos x = − sin x.
dx
(You may use, without proof, the fact that cos(α + β) = cos α cos β − sin α sin β.)