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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