=========================preview======================
(MATH013)0786cf - 2008final-selected.pdf
Back to MATH013 Login to download
======================================================
HKUST MATH 013 Calculus 1
Final Examination Name: 13th Dec 2008 Student I.D.: 16:30C19:30 Tutorial Section:

1.
Do not open the exam until instructed to do so.

2.
When instructed to open the exam, please check that you have 15 pages of questions in addition to the cover page.

3.
Write your name, and other information in the space provided above.

4.
Show an appropriate amount of work for each problem. If you do not show enough work, you will get only partial credit.

5.
This is a closed book and notes examination.

6.
You may write on the backside of the pages, but if you use the backside, clearly indicate that you have done so.

7.
Please turn o. all phones and pagers and remove headphones.

8.
Cheating is a serious o.ense. Students caught cheating are subject to a zero score and other penalties.


Question No. Points Out of
Q. 1 12
Q. 2 10
Q. 3 10
Q. 4 9
Q. 5 14
Q. 6 13
Q. 7 6
Q. 8 18
Q. 9 10
Q. 10 16
Q. 11 12
Total Points 130

1. (12 = 3+3+3+3points) Hereisa tableof some valuesof three di.erentiable functions a, b, and

c. The derivatives a , b and c are assumed to be continuous.
x a(x) a (x) b(x) b (x) c(x) c (x)
0 .2 1 2 5 1 .1 1
2 0 0 9 2 0 .1
5 .1 .1 2 19 3 2 3

Answer the following: An answer of true means true in all situations An answer of false means there is a case where the assertion is false.
(i) (3 points) The function c has an inverse.
True False Brief reason.


(ii) (3 points) The derivative a takes on the value 1 in the interval 0 x 2. True False Brief reason.
Compute the following:
(iii) (3 points) g(2) and g (2), where g(x)= b(a(x))
(iv) (3 points) Suppose b is an increasing function, and h is the inverse function to b. Compute h(9) and h (9).
2. (10 = 5+5points)A 6-ft-tall maniswalkingaway froma 24-ft-tall lamppostata rateof3 ft/s.
(i) (5 points) At what rate is the length of his shadow changing when he is 16 ft away from the lamppost?
24 ft
6 ft xs

shadow
(ii) (5 points) At what rate is the distance between the lamp and the tip of his shadow changing when he is 16 ft away from the lamppost?
2
3. (10 = 5+5points) Consider the parabola y = x.
(i) (5 points) Write down the equations of the tangent and normal lines to the parabola at the point (b,b2).
Tangent line equation is y =
Normal line equation is y =
(ii) (5 points) The .gure shows a circle of radius 2 inscribed in the parabola. Determine the center of the circle.

Center is located at(x,y)= 4. (9 =3+6points) Consider the equation lnx =4sinx, for 0 <x 3.
(i)
(3 points) How many solutions to the equation are there in the given interval? Give brief reason for your answer.

(ii)
(6 points) Starting with the point x =2.5, use Newtons method to approximate the smallest solution to an accuracy of 0.001. Retain .ve decimal places in your calculation.