Name
Student ID #
Class Section
Instructor
Math 1210
Spring 2007
EXAM
Dept. Use Only
Exam Scores
Problem Points Score
1.
20
2.
20
3.
20
4.
20
5.
20
TOTAL
Show all your work and make sure you justify all your
answers.
Math 1210
Exam
1. I want to warn you again there maybe a few typos (to keep you on your
toes).
2. these problems involve derivatives
(a) State the product rule.
(b) Dierentiate f (x) = cos(x) (x + x + 1) Also nd the equation
of the tangent line at x = 1.
2
(c) Dierentiate f (x) = cos(x) (x + x + 1) sin(x)
2
(d) State the chain rule.
(e) Dierentiate f (x) = cos(x + x + 1)
2
(f) Dierentiate f (x) = sin(cos(xsin (x)))
2
(g) State the quotient rule.
(h) Dierentiate f (x) = cosx2x
( )
1
3. these problems involve higher derivatives. You should dierentiate
these problems until you feel comfortable dierentiating functions. For
deniteness of the problem nd the third derivative. If you want more
practice feel free to nd the fth derivative ( or if your really up for a
work out nd the tenth derivative it's good practice).
(a) f (x) = x cos(sin(x ))
2
2
(b) f (x) = x2x x
+ +1
1
2
4. If the position of an object at time t is given by p(t) = 16t +100t +70.
Find it's velocity and acceleration.
2
3
dy for two of the
5. These problems involve implicit derivatives. Find dx
following equations. (If you do not clearly indicate which two you want
graded I reserve the right to give you a zero score.)
(a) y + y x + x + 1 = 0
3
2
2
(b) cos(x)(xy) + y = 0
5
(c) x + y + x y + y x = (x + y )
3
3
2
2
2
2 3
(d) cos(ysin(x ))x + x = 0.
2
2
4
6. This question is on related rates and will come from your homework
(please rewrite your solutions to the homework problems from this section if you want full credit on this assignment). That is rewrite solutions
to Section 2.8: 5, 6, 14, 17,21, 22.
5
7. This question is on dierentials.
(a) Dene a dierential dy where y = f (x).
(b) Use dierentials to approximate the increase in area of a soap
bubble when it radius increases from 3 inches to 3.025 inches.
(Hint area of the buble is A = 4r
2
(c) The period1 of a simple pendulum of length L feet is given by
T = 2 Lg 2 seconds assume g is the acceleration due to gravity
(g = 32ft=sec). If the pendulum is that of a clock that keeps
good time when L = 4ft how much time will the clock gain in 24
hours if the length of the pendulum is decreased to L = 3:97ft?
(d) Find dy if y = x + x, also nd the change in y when x = 1, x = 2.
2
6
8. Find the maxium and minium value of each of the following functions
if they exist. If they exist you must state why it exists and if it does
not exist you must state why it does not exist.
(a) f (x) = x2x2 x on [ 1; 1]
+
2
+1
(b) f (x) = x on [0; 1]
2
4
(c) f (x) = cos(x) on [ 2; 1]
(d) f (x) = x + 4x + 4 on [ 10; 10]
2
(e) f (x) = x on [ 1; 1]
1
7