Vv156 Honors Calculus II  Assignment 2

Vv156 Honors Calculus II 
Assignment 2

This assignment has a total of (40 points).
Note: Unless specified otherwise, you must show the details of your work via logical reasoning for each exercise.
Simply writing a final result (whether correct or not) will receive 0 point.
Exercise 2.1 [Ste10, p. 189]
(i) (2 points) The curve y = 1/(1 + x
2
) is called a witch of Maria Agnesi. Find an equation of the tangent
line to this curve at the point (−1,
1
2
).
(ii) (2 points) The curve y = x/(1 + x
2
) is called a serpentine. Find an equation of the tangent line to this
curve at the point (3, 0.3).
(4 points)
Exercise 2.2 [Ste10, p. 190] If f is a differentiable function, find an expression for the derivative of each of the
following functions.
y = x
2
(i) f(x) y =
f(x)
x
2
(ii) y =
x
2
f(x)
(iii) y =
1 + xf(x)
√
x
(iv)
(4 points)
Exercise 2.3 [Ste10, p. 191]
(a) If g is differentiable, the Reciprocal Rule says that
d
dx
1
g(x)

= −
g
0
(x)
[g(x)]2
Use the Quotient Rule to prove the Reciprocal Rule.
(b) Use the Reciprocal Rule to verify that the Power Rule is valid for negative integers, that is,
d
dx(x
−n
) = −nx−n−1
(4 points)
Exercise 2.4 [Ste10, p. 197] Calculate the first and second derivatives of the following functions.
f(x) = √
(i) x sin x f(x) = sin x +
1
2
(ii) cot x (iii) f(x) = 2 sec x − csc x f(x) = x
2 − tan x
(iv)
f(x) = sec x
1 + sec x
(v) f(x) = x sin x
1 + x
(vi) f(x) = 1 − sec x
tan x
(vii) f(x) = x
2
(viii) sin x tan x
(8 points)
Exercise 2.5 [Ste10, p. 197]
(a) Use the Quotient Rule to differentiate the function
f(x) = tan x − 1
sec x
(b) Simplify the expression for f(x) by writing it in terms of sin x and cos x, and then find f
0
(x).
(c) Show that your answers to parts (a) and (b) are equivalent.
(3 points)
Exercise 2.6 [Ste10, p. 198] Find the limit (use whatever method you like, but show the details of your work)
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limx→0
sin 3x
x
(i) limx→0
sin 4x
sin 6x
(ii) limt→0
tan 6t
sin 2t
(iii) lim
θ→0
cos θ
sin θ
(iv)
limx→0
sin 3x
5x
3 − 4x
(v) limx→0
sin 3x sin 5x
x
2
(vi) lim
θ→0
sin θ
θ + tan θ
(vii) limx→0
2x
x + sin x
(viii)
lim
x→π/4
1 − tan x
sin x − cos x
(ix) limx→1
sin(x − 1)
x
2 + x − 2
(x)
(10 points)
Exercise 2.7 [Ste10, p. 198]
(a) Evaluate limx→∞
x sin
1
x
.
(b) Evaluate limx→0
x sin
1
x
.
(c) Illustrate parts (a) and (b) by graphing y = sin(1/x).
(3 points)
Exercise 2.8 [Ste10, p. 198] Find constants A and B such that the function y = A sin x + B cos x satisfies the
differential equation y
00 + y
0 − 2y = sin x.
(2 points)
Exercise 2.9 Given function f satisfying |f(x)| ≤ x
2
, calculate f
0
(0).
(2 points)
References
[Ste10] J. Stewart. Calculus: Early Transcendentals. 7th ed. Cengage Learning, 2010 (Cited on pages 1, 2).
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