School of Mathematics and Statistics
Junior
The University of Sydney
Maths & Stats Home / Teaching program / Intermediate / MATH2067 / Quizzes / Quiz 1
The University of Sydney
Usyd Home
MyUni
MyUni
Library
Sitemap
spcr
Discussion
   Questions


right first
    attempt
right
wrong

Quiz 1: Assumed knowledge

Last unanswered question  Question  Next unanswered question
 

Discussion

 
 

This self-assessment quiz should be attempted after having read, and where necessary revised, the items in Chapter 1 of the printed notes.

The quiz is intended to be first attempted in a single pass – and should take less than 10 minutes.

A self-assessed score of 3 or less should be interpreted to mean additional work is required. Any of the questions can be redone at any time.


 

Question 1

 
 
Which if any of the following equations are invalid? Equations involving x are identities for all x, and f ′ represents df∕dx etc.

(A) (cos2x)′ = 2sin2x, (B) (sin2x)′ = 2cos2x, (C) sinx = tanx cosx,     
(D) (cos4x)′′ = 16cos4x,(E) (sin4x)′′ = -16cos4x,(F) cosh0 = 0,
(G) sinh0 = 0, (H) cosh(-1) = cosh(1), (I) sinh(-1) = sinh(1)
a) ACEFG   b) BCDH   c) ABFI
d) BD   e) ADEFI   f) All A,…,I are correct.

 

Not correct. Choice (a) is false.
See response (e).
Not correct. Choice (b) is false.
See response (e).
Not correct. Choice (c) is false.
See response (e).
Not correct. Choice (d) is false.
See response (e).
Your answer is correct.

(A) should be (cos2x)′ = -2sin2x
(D) should be (cos4x)′′ = -16cos4x
(E) should be (sin4x)′′ = -16sin4x
(F) should be cosh0 = 1
(I) should be sinh(-1) = -sinh(1)
[sinh(x) is odd]
Not correct. Choice (f) is false.
See response (e).
 

Question 2

 
 
Consider the functions
(A) sinx (B) sin2x (C) sin(x - π ∕ 4)
(D) cos(x - π ∕ 4)(E) 2coshx(F) cosh(x + 2)
(G) sinhx (H) x3 (I) x2 + 1
Which of these functions match the following graphs in the order shown?
PIC
a) AIGD   b) BIHD   c) BEGD   d) BHGC   e) AEHD

 

Not correct. Choice (a) is false.
See response (c).
Not correct. Choice (b) is false.
See response (c).
Your answer is correct.

Graph 1 resembles a sine function. Of possibilities A,B,C, it matches only B, in having zeros at x = ±π∕2 (and slope greater than 1 at x = 0, but you need to allow for the different x,y scales to estimate that).
Graph 2 resembles a cosh function, and a quadratic. Of possibilities E,F,I, it matches only E in taking the value 2 at x = 0. [cosh0 = 1]
Graph 3 resembles a sinh function, and a cubic. Of possibilities G,H, it matches only G, in having non-zero slope at x = 0. [dsinhx∕dx = coshx = 1 at x = 0.]
Graph 4 resembles a shifted cosine or sine function. Of possibilities C,D, it matches only D, in having a maximum at x = π∕4 and zeros at x = -π∕4,3π∕4. [C is zero at x = +π∕4.]
Not correct. Choice (d) is false.
See response (c).
Not correct. Choice (e) is false.
See response (c).
 

Question 3

 
 
Indicate, using the order shown, whether the following functions are even (e), odd (o) or neither (n).
(A) e|x| sinx, (B) e|x| cosx, (C) ex sinx.
a) e,e,o   b) o,n,n   c) o,n,e   d) o,e,n

 

Not correct. Choice (a) is false.
See response (d).
Not correct. Choice (b) is false.
See response (d).
Not correct. Choice (c) is false.
See response (d).
Your answer is correct.
e|x| is even; sinx is odd; cosx is even; ex is neither. So:
A is e × o → o
B is e × e → e
C is n × o → n
 

Question 4

 
 
Match the functions (A) e|x| sinx, (B) e|x| cosx, (C) ex sinx,
to the following graphs, in the order shown.
PIC
a) ABC   b) ACB   c) BAC
d) CBA   e) None of above.

 

Not correct. Choice (a) is false.
See response (b).
Your answer is correct.
The graphs show functions that are o,n,e respectively.
Not correct. Choice (c) is false.
See response (b)
Not correct. Choice (d) is false.
See response (b).
Not correct. Choice (e) is false.
See response (b).
 

Question 5

 
 
Find c1, c2 given that y = e-x(c1 cos2x + c2 sin2x) and
π- ′′ y(2 ) = 0, y (0) = 1.
(BC1,2)
To apply (BC2), preferably use Leibnitz’s rule in Ch.1 of the printed notes.
a) c1 = 0, c2 = 0   b) c1 = 0, c2 = 1   c) c1 = 0, c2 = 2
d) c1 = 0, c2 = -1∕2   e) None of the above.

 

Not correct. Choice (a) is false.
See response (e).
Not correct. Choice (b) is false.
See response (e).
Not correct. Choice (c) is false.
See response (e).
Not correct. Choice (d) is false.
See response (e).
Your answer is correct.
c1 = 0, c2 = -1∕4
(BC1) ⇒ y(π∕2) = e-π∕2(c1(-1) + c2 sinπ) = 0
I.e. c1 = 0, and hence y = c2 e-xsin2x.
(BC2) ⇒ y′′(0) = c2 e0 (-4sin0 - 4cos0 + sin0) = -4c2 = 1.
I.e. c2 = -1∕4.