Vv156 Honors Calculus II
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
) is called a witch of Maria Agnesi. Find an equation of the tangent
line to this curve at the point (−1,
(ii) (2 points) The curve y = x/(1 + x
) is called a serpentine. Find an equation of the tangent line to this
curve at the point (3, 0.3).
Exercise 2.2 [Ste10, p. 190] If f is a differentiable function, find an expression for the derivative of each of the
y = x
(i) f(x) y =
(ii) y =
(iii) y =
1 + xf(x)
Exercise 2.3 [Ste10, p. 191]
(a) If g is differentiable, the Reciprocal Rule says that
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,
) = −nx−n−1
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 +
(ii) cot x (iii) f(x) = 2 sec x − csc x f(x) = x
2 − tan x
f(x) = sec x
1 + sec x
(v) f(x) = x sin x
1 + x
(vi) f(x) = 1 − sec x
(vii) f(x) = x
(viii) sin x tan x
Exercise 2.5 [Ste10, p. 197]
(a) Use the Quotient Rule to differentiate the function
f(x) = tan x − 1
(b) Simplify the expression for f(x) by writing it in terms of sin x and cos x, and then find f
(c) Show that your answers to parts (a) and (b) are equivalent.
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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3 − 4x
sin 3x sin 5x
θ + tan θ
x + sin x
1 − tan x
sin x − cos x
sin(x − 1)
2 + x − 2
Exercise 2.7 [Ste10, p. 198]
(a) Evaluate limx→∞
(b) Evaluate limx→0
(c) Illustrate parts (a) and (b) by graphing y = sin(1/x).
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.
Exercise 2.9 Given function f satisfying |f(x)| ≤ x
, calculate f
[Ste10] J. Stewart. Calculus: Early Transcendentals. 7th ed. Cengage Learning, 2010 (Cited on pages 1, 2).
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