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(MATH023)[2009](f)final~yzou^_10403.pdf
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Math 023 -Honors Calculus I
Final Examination, Fall Semester, 2009

Time Allowed: 2.5 Hours Total Marks: 100
1. (24 Marks) Suppose f and g are continuously di.erentiable everywhere. Evaluate the following limits:
[f(x + a)]2 . [f(x . a)]2
(a) lim ;
a0 a
sin[f(x) . f(a)] . f(x)+ f(a)

(b) lim ;
xa (x . a)3
f(x)f(a)

eg(a) . eg(x)
(c) lim ;
xa x . a
1 1+(x . a)f(x)

(d) lim ln .
xa x . a 1+(x . a)g(x)
2. (10 Marks) Please give two non-constant continuous functions f and g in R such that
1+ f(0)g(x)
lim .
=1.
x0 1+ f(x)g(0)
3. (10 Marks) Determine the values of a and b such that the function
x2 , if x> 0,
f(x)=
ax + b, if x 0,
is di.erentiable everywhere in R.

4. (10 Marks) Show that the error to the approximation 11
sin 1 1 . +
3! 5!
is smaller than 10.3 .

5. (12 Marks) Sketch the curve y = xe.x2 .
(to be continued)
6.
(10 Marks) Show that if |x| , then

| sin x . cos x + x| 1+ .

7.
(12 Marks) For b>a> 0, prove


b2 . a222ab< <a+ b2 .
ln b . ln a
8. (12 Marks) As in Fig. 1, assume that a L-shaped corridor is a and b in width and c in height (to ceiling). What are the largest possible dimensions of a rectangular whiteboard that can be moved through the corridor? Justify your argument.
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Figure 1: A whiteboard is moved through a corridor.

Solutions
1. (24 Marks) Suppose f and g are continuously di.erentiable everywhere.
Evaluate the following limits:
(a) lim a0 [f(x + a)]2 . [f(x . a)]2 a ;
Solution
lim [f(x + a)]2 . [f(x . a)]2 = lim 2f(x + a)f.(x + a) . 2f(x . a)f.(x . a)(.1)
a0 a a0 1
= 4f(x)f.(x).

sin[f(x) . f(a)] . f(x)+ f(a)
(b) lim ;
xa (x . a)3
Solution
sin[f(x) . f(a)] . f(x)+ f(a) cos[f(x) . f(a)] f.(x) . f.(x)
lim = lim
xa (x . a)3 xa 3(x . a)2 cos[f(x) . f(a)] . 1
= f.(a) lim
xa 3(x . a)2 . sin[f(x) . f(a)] f.(x) = f.(a) lim
xa 6(x . a)
1 sin[f(x) . f(a)]
= . [f.(a)]2 lim
6 xa x . a
1 cos[f(x) . f(a)]f.(x)
= . [f.(a)]2 lim
6 xa 1
= . 1[f.(a)]3 .
6
f(x)f(a)
eg(a) . eg(x)
(c) lim ;
xa x . a
Solution
f(x)f(a)f(a)
eg(a) . eg(x) ef(x)f.(x)g(a) . eg.(x)
lim = lim
xa x . a xa 1
= e f(a)f.(a)g(a) . e f(a)g .(a)
= e f(a)[f.(a)g(a) . g .(a)].
3
1 1+(x . a)f(x)
(d) lim ln .
xa x . a 1+(x . a)g(x)
Solution
1 1+(x . a)f(x) ln[1 + (x . a)f(x)] . ln[1 + (x . a)g(x)]
lim ln =lim
xa x . a 1+(x . a)g(x) xa x . a
f(x)+(x.a)f.(x) . g(x)+(x.a)g.(x) 1+(x.a)f(x) 1+(x.a)g(x)
= lim
xa 1 = f(a) . g(a).
2. (10 Marks) Please give two non-constant continuous