Section 4.5 Indeterminate Forms and L'Hospital's Rule 1. L'Hospital's Rule - Suppose f and g are differentiable and g (x) Suppose that and lim f ( x) 0 lim g ( x) 0
x a x a
0
lim f ( x)
x a
and
lim g ( x)
x a
Then lim
x z
f ( x) g ( x)
lim
x
MA 241 Test 2 Form B Spring 2007
Begin each new problem on the top of a new page, back or front.
NO calculators, No PDA, No cell phones , No notes, no etc! !!
1. (l0 points each) For each of the following, show every step required to set up the
denite i
MA 241 Test 1 Form E )
Spnng
9. Set 11 ONLY a b and c below. Insert all numerical values so that if you had a
calculator, you could nish the arithmetic.
a. Use Simpsons Rule with n=4 subintervals to approximate
3 x
fdx
1 1 + (sin x)2
b. Find the a
3" A 241 Test 1 Form
Spring 2 0
9. . Integrate.
x
a fdx
c. fxzexsdx
d. fxln(5x)aix
x2
. dx
e f(x24x+1)2
Continue on next page.
Form D continued
10. Set up ONLY a, b, and c below. Insert all numerical values so that if you had a
calculator, you cou
MA 241Test4 Fall2007 FormB *Showall work clearlY. * Begineach problemon a newPa!e. * NO CALCULATORS, CELL PHONES, PDAS. NO NO namethe test,showthe criteria is met, or When proving convergence divergence, and makea conctudingstatement. - *(-1)'(-r+2)'
MA24l Test4 Fall 2007 FormA on Showall work clearly. Begineachproblem a newpage. PDAs. NO NO CALCULATORS, CELL PHONES,NO namethe test,showthe criteria is metn or When proving convergence divergence, and makea concludingstatement. fo, l. (20points)Fin
MA 241 Test 2 Form B
Fall 2007
*No Calculators, No Cell Phones, No PDAs.
*Show all work neatly. Begin each problem on a new page in the blue book, please!
*Circle your nal answer.
* Please do not leave if you have to pass over someone on your row. Wait
Final Exam Study Guide 50 questions + 5 extra credit questions. If you know and can apply the following, you should do well on the final. I. Chapters 1, 2, 3, 4, 5, 6, 7, 8.
The basic language of chemistry: 1. Can you identify a substance as element,
Related Rates Example1 A ladder 10 feet long rests against a vertical wall. If the bottom of the ladder slides away from a wall 1 ft/se, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 ft. from the wall?
St
Sample Test #4 1. Find the
most
general
antiderivative
of
the
function.
f ( x) 5 x
1 4
sec2 ( x)
1 x
8x
3
2
f (1) 0 .
2. Find f(t) if f (t )
32 and f (0) 24 and
3. Evaluate the Riemann Sum of f ( x) x 2 where 2 x 10 with four subinter
Old Test #2 from another section of Math 141 No Calculators. Show all work. 1. From the following graph of the derivative, (I'll put on the board)
dy , dx
a. b. c. d.
On what intervals is y increasing? At what values of x are the maxima of y? Is y
Section 4.6 Optimization
To optimize is to find the BEST way of doing something. Example. We can maximize things like area, volume, profit, height. We can minimize things like time, distance, cost.
Steps
1. Understand the problem. Read it carefully
Class Wednesday, October 17 No class Thursday; Friday section 4.3 Derivatives and shapes of curves I. Do more related rate problems
1. #10 page 267 At noon, ship A is 159 km west of ship B. Ship A is sailing east at 35 km/h and ship B is sailing nort
Math 141 Old Test #2 1. Find an equation of the tangent line to the graph of f ( x) . 2. If f ( x)
6 x 2 12x 10 , find f 4 ( x ) .
1 x 1
at the point (3, )
3. Make a conjecture about the derivative by calculating the first few derivatives and obse
Math 141 Test#2 from another teacher 1. Use the definition of derivative of the function below. In other works, find the derivative of the function below "the long way". f ( x) x x 2. Use the graph below of the derivative to answer the following ques
Section 4.3 Derivatives and Shapes of Curves 1. The Mean Value Theorem
If f is differentiable on the interval [a, b], then there exists a number c between a and b such that
f (c )
Sketch:
f (b) f (a) b a
Example: Given f ( x ) x 2 , find the c gua
Class Newton's Method Review Problems first 1. Evaluate the following limits tan x x a. lim x 0 x3
b. lim
x
sin x 1 cos x
2. Optimization A can is to be made to hold 1 Liter of oil. Find the dimensions that will minimize the cost of metal to manu