2. Create both cost and revenue functions for
running the eye surgery operating room as it
currently operates (i.e. without considering the new
machine or the advertising) by identifying revenue
per case and annual fixed and variable costs.
Expenses
utili
1 Executive Summary
This project involved Crazy Als Car Rentals hiring our group as consultants to evaluate how
many of his cars should be at each of his three locations at the start of each week so that the same
number of cars will be there at the end of
MAT 220: FIRST AND SECOND DERIVATIVE TEST PRACTICE
For each function below use the first and/or second derivative tests to determine:
a. All critical values.
b. All relative and absolute maximum or minimums over the indicated interval. ( Write as coordina
GUIDELINES FOR USING THE FIRST AND SECOND DERIVATIVE TEST
FIRST DERIVATIVE TEST: Tests for possible relative maximums, minimums and intervals of increase and decrease.
Determine f ' x , set it equal to zero and solve to determine the critical values. The
MAT 220
FIRST AND SECOND DERIVATIVE TEST ASSIGNMENT
THIS WORKSHEET IS DUE AT THE START OF CLASS ON MONDAY, MARCH 25TH.
POINTS WILL BE DEDUCTED IF INSTRUCTIONS ARE NOT FOLLOWED.
YOU MUST SHOW ALL WORK IN A NEAT AND ORGANIZED MANNER TO RECEIVE CREDIT.
SIGN
MAT 220
FIRST AND SECOND DERIVATIVE TEST HOMEWORK
THIS WORKSHEET IS DUE AT THE START OF CLASS ON MONDAY, OCTOBER 13TH.
FOLLOW ALL INSTRUCTION S CAREFULLY, POINTS WILL BE DEDUCTED IF INSTRUCTIONS
ARE NOT FOLLOWED.
YOU MUST SHOW ALL WORK IN A NEAT AND ORGAN
MAT 220
EXAM 1 STUDY GUIDE
LIMITS
Determine left limits, right limits and limits of functions if they exist through use of the table, algebraic, graphical
and direct substitution method.
The definition/conceptual understanding of Limit
x c
Limits of piec
MAT 220: CONTINUITY AND DIFFERENTIABILTY HOMEWORK
ALL WORK AND SOLUTIONS ON SEPARATE PAPER
PENCIL ONLY. FSAs
1. For problems a c:
Determine the domain and continuity of each function.
Use the Alternative Form of the Derivative Determine the differentiabil
Chain Rule Practice 2 Solutions
1.
f x e1ln x
2.
3e3x
e3x - 8
-e1-lnx
x
3.
x 10 x2e2 x
4.
20x e2x + 20x2 e2x
5.
7.
f x ln e3 x 8
f x
5 e
x 3
30ex
4
5 - ex
x 2 2 x3
f x ln
2
2e x
6.
2x - 6x 2
f x = 2
- 2x
3
x - 2x
3
6x - 1
f x ln x
8.
6
MAT 220 CHAIN RULE PRACTICE 2
Determine the derivative of each function.
Positive exponents only in all final answers. Show all work separate paper. Only solutions on
this worksheet.
1.
f x e1ln x
2.
f x ln e3 x 8
3.
x 10 x2e2 x
4.
f x
5.
x 2 2 x3
f x
MAT 220 CHAIN RULE HOMEWORK 2: TRIGONOMETRIC DERIVATIVES
Determine the derivative of each function. Positive exponents only in all final answers. Show all work separate
paper. Only solutions on this worksheet. This worksheet is due at the start of class o
MAT 220: CHAIN RULE PRACTICE 3 SOLUTIONS
1.
f x 3x 4 32 x
2.
f x 4
3.
f x 5x 52 x
x
f' x = 12x 3 - 2 ln3 32x
3ln 2 x 2
2
2
2
6
2
1 3 x2
5 e3 x
4
5.
f x
6.
f x 2
7.
e 2 x 22 x
f x
2
8.
f x 5 2 x
x
2
2 x
ln4 4
x
+
x2
+ 2
2
2
f x =10xe x + 10x ln3 35x
MAT 220 CHAIN RULE PRACTICE 3
Determine the derivative of each function.
Positive exponents only in all final answers. Show all work separate paper. Only solutions on this
worksheet.
1.
f x 3x 4 32 x
2.
f x 4
3.
f x 5x 52 x
4.
f x 5e x 35 x
5.
f x
6.
f
MAT 238: SURFACE AREA CLASS EXAMPLES
Let y = f have a continuous derivative on the interval[a , b]. The area S of the surface of revolution termed
by revolving the graph off about a horizontal or verticai axis is given by:
S 223!:1'(x)"l+[f'(x) 2 dx ( yts
MAT 230: PARAMETRIC EQUATIONS AND CALCULUS
y
2
1
x
-2
-1
1
2
-1
-2
Objectives:
a. Evaluate a parametric equation at a point and write as a coordinate.
b. Determine
dx
dy
and
, evaluate at a point and interpret their meaning.
dt
dt
c.
dy
evaluate at a poin
Mat 230: EXAM 2 MIXED REVIEW
1.
sin
2.
sin x cos x
3.
3cos 5x dx
4.
sec
5.
cos 15x sin 4 x dx
6.
x3 2 x 2 x 1
x 4 5x 2 4 dx
7.
3
x dx
2
dx
2
9
x tan 5 x dx
tan 3 x
dx
sec x
Integrate using Integration by Parts Non-Tabular Method
8.
x cos 3x
9.
x
TRIGSUB
1/x2(a2-x2)
cos2 = 1- sin2
1. x=a sin dx=a cos
2. solve for x. x/a = sin
3. Draw your triangle
4. Cos side should match up with original integral.
5. Make sure you add the a cos d to the integral.
6. Sub in a sin for all axs. cos d / sin2(a2-(a si
This is for tangent to an even power and sec is to an odd
power.
tan(even)(x) secodd(x) dx
Convert the tangents to secants and use the reduction
formula for powers of the secant (sec2x-1)k Then follow the secant
rules.
1. Rewrite integral so that there i
This is for tangent to an even power and sec is to an odd power.
tan(even)(x) secodd(x) dx
Convert the tangents to secants and use the reduction formula
for powers of the secant (sec2x-1)k Then follow the secant rules.
1. Rewrite integral so that there i
This is for tangent to an odd power and positive (and the
power of secant is odd).
1. Rewrite integral so that there is the following:
tan(odd)(x) secodd(x)(tan2(x)left over powerstanK(x) secpower-1(x) sec(x)tan(x)dx
power example x10 = (x2)5=( x2)5-1(x2)
This is for tangent to an even power and positive (and the
power of secant is odd).
1. Convert the tangents to secants and use the reduction
formula for powers of the secant (
Rewrite integral so that there is the following:
tan(odd)(x) secodd(x)(tan2(x)l
David Livingston
With Cindy Brace
MAT230
1A. A ball is dropped from M meters above a flat surface. Each time the ball hits the
surface after falling a distance, h, it rebounds a distance rh, where 0< r < 1. Find the
formula that represents the total dista
This is for cos to an odd power and positive.
1. Rewrite integral so that there is the following:
cos(odd power)(x) sinany power(x)(cos2(x)Power over twosinK(x) cos(x) dx
2. Sub 1-sin2(x) for cos2(x) (1-sin2(x)Power over twosinK(x) cos(x) dx
3. Do u sub w
This is for sin to an odd power and positive.
1. Rewrite integral so that there is the following:
sin(odd power)(x) cosany power(x)(sin2(x)Power over twocosK(x)
sin(x) dx
Use(1-cos2(x) for the (sin2(x).
2. Sub 1-cos2(x) for sin2(x) (1-cos2(x)Power over tw
This is for secant to an even power and positive.
1. Rewrite integral so that there is the following:
sec(n=2k even)(x) tanany power(x)(sec2(x)power-1tanK(x) sec2(x) dx
power example x10 = (x2)5=( x2)5-1(x2)1
2. Sub 1+tan2(x) for sec2(x) (1+tan2(x)Power o
This is for cos and sin to an even power.
Use the identities as many times needed sin2u=1/2
cos(2u) and cos2u= - cos (2u) then use the reduction formula
below.
1. Rewrite integral so that there is the following:
sin(odd power)(x) cosany power(x)(sin2(x)P
MAT 217 Chapter 4 Project:
Optimization
You have been hired as a consultant to advise Sandbaggers, Inc., a
company that purchases silica sand, cleans it, and then sells the silicon
dioxide to manufacturers of computer chips. Sandbaggers, Inc., operates
tw