VV156 Homework 3 Solved

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 3.1 [Ste10, p. 205]
√

1. (i)  (1 pt) The curve y = |x|/ 2 − x2 is called a bullet-nose curve. Find an equation of the tangent line to this curve at the point (1, 1).
2. (ii)  (1 pt) Illustrate part (i) by sketch the curve and the tangent line on the same coordinate system.

(2 pts)

Exercise 3.2 [Ste10, p. 208] Use the Chain Rule to prove the following. (i) (1 pt) The derivative of an even function is an odd function.

(ii) (1 pt) The derivative of an odd function is an even function. (2 pts)

Exercise 3.3 [Ste10, p. 208] If y = f(u) and u = g(x), where f and g are twice differentiable functions, show that d2y d2y􏰉du􏰊2 dy d2u

dx2 =du2 dx +dudx2
Exercise 3.4 [Ste10, p. 215] Use implicit differentiation to find an equation of the tangent line to the curve at the

(2 pts)

given point.
(i) (1pt)(cardioid)x2+y2 =(2×2+2y2−x)2 at(0,12).

(ii) (1 pt) (astroid) x2/3 + y2/3 = 4 at (−3√3, 1).

(iii) (1 pt) (lemniscate) 2(x2 + y2)2 = 25(x2 − y2) at (3, 1).

(iv) (1 pt) (devil’s curve) y2(y2 − 4) = x2(x2 − 5) at (0, −2). (4 pts)

Exercise 3.5 [Ste10, Sec. 3.11] Given the following hyperbolic functions defined as

ex −e−x sinhx = 2

cschx = 1 sinh x

(i) (3 pts) Show that
d (sinhx) = coshx

coshx = sechx =

ex +e−x 2

1 cosh x

sinhx tanhx = coshx

cothx = coshx sinh x

d (coshx) = sinhx
dx dx dx

d (tanhx) = sech2 x
d (cschx) = −cschxcothx d (sechx) = −sechxtanhx d (cothx) = −csch2 x

dx dx dx (ii) (3 pts) and show that1

d (sinh−1x)= √ 1

d (cosh−1x)= √ 1
dx x2 − 1

d (tanh−1x)= dx

1
1 − x2 1 1−x2

dx
d (csch−1x)=−

1 + x2 √1

d (sech−1x)=− √ 1
dx x 1−x2

d (coth−1x)= dx

dx
1Notice that the formulas for the derivatives of tanh−1 x and coth−1 x appear to be identical. But the domains of these functions have

|x| x2+1
no numbers in common: tanh−1 x is defined for |x| < 1, whereas coth−1 x is defined for |x| > 1.

Page 1 of 2

(6 pts)
Exercise 3.6 [Ste10, p. 223] Find the derivative of the following functions

(i) (1pt)y=(sinx)lnx (ii) (1 pt) y = (tan x)1/x

(2 pts)
Exercise 3.7 [Ste10, p. 272] If

Show that y′ = 1 . a+cosx

y=√ x
a2 −1

−√ 2
a2 −1

arctan

a+

√ sinx
a2 −1+cosx

(2 pts)
Exercise 3.8 [Ste10, p. 282] If f has a local minimum value at c, show that the function g(x) = −f(x) has a local

maximum value at c. (2 pts)

Exercise 3.9 [Ste10, p. 289] Suppose f is an odd function and is differentiable everywhere. Show that for every positive number b, there exists a number c ∈ (−b,b) such that f′(c) = f(b)/b.

(2 pts)

Exercise 3.10 [Ste10, p. 300] Show that the inflection points of the curve y = x sin x lie on the curve y2(x2 +4) = 4×2. (2 pts)

Exercise 3.11 [Ste10, p. 309] Evaluate

(2 pts)
Exercise 3.12 [Ste10, p. 309] Let

f(x) = (i) (1 pt) Show that f is continuous at 0.

(ii) (1 pt) Calculate f ′ (0). (2 pts)

􏰅 2 􏰉1+x􏰊􏰆

lim x−x ln x→∞

x

√

􏰧|x|x,
1, x=0

Exercise 3.13 [Ste10, p. 309] Show that the shortest distance from the point (x1, y1) to the straight line Ax + By + C = 0

is2
(2 pts)

References

|Ax1 +By1 +C|

A2 + B2

x ̸= 0

[Ste10] J. Stewart. Calculus: Early Transcendentals. 7th ed. Cengage Learning, 2010 (Cited on pages 1, 2).