3.6 Implicit Differentiation
Suppose we have equations like x2 + y 2 - 25 = 0, x3 + y 2 - 9xy = 0, y2 - x = 0
dy ? dx
We say that x, y are implicitly defined. But, how would we find
Example 1
Find d
Math 1226
Test 3
Examples with Detailed Solutions: Test 1
The purpose of this document is to provide a detailed solutions to some common
problems coming from Test 1 material.
Example 1.
Prove that lim
S13
Math 1205 Test 3
May 1, 2013
NAME:
CRN:
Use only methods from class. You must show work to receive credit.
1. (11 pts) Let f (x) = cos(x) + sin(x).
(a) (8 pts) Find the linearization L(x) of f (x)
F12
Math 1205 Test 3
5 Dec 2012
Name:
Key
CRN:
Use only methods from class. Unless indicated otherwise, you must show work to receive credit.
1. [13 pts] Starting with x0 = 5, calculate the rst two it
F12
Math 1205 Test 3
5 Dec 2012
Name:
CRN:
Use only methods from class. Unless indicated otherwise, you must show work to receive credit.
1. [13 pts] Starting with x0 = 5, calculate the rst two iterat
S08
Math 1205 - Final Exam
NAME:_ INSTRUCTOR: _
You will be graded on how well and fully you support your work! Use only methods developed in class.
1.[15] Evaluate the following limits or explain why
S13
Math 1205 Test 3
May 1, 2013
NAME:
CRN:
Use only methods from class. You must show work to receive credit.
1. (11 pts) Let f (x) = cos(x) + sin(x).
(a) (8 pts) Find the linearization L(x) of f (x)
MATH 1205 Differential Calculus
Final Exam Review Topics
Test 1 Material
Definitions
Limit
if and only if for any number epsilon greater than zero, no matter how small, there is a number delta
greate
1
4.7 Newtons Method
Example 1
Solve the equation x ln(x) = 1 (or equivalently x ln(x) 1 = 0)
y
2
1
x
1
2
3
4
1
2
Approximate Solutions
Suppose we need to solve the equation
And we know that the point
1
4.6 Applied Optimization
To optimize something means to maximize or minimize some aspect of it.
Optimization problems have applications in Business, Industry, Mathematics, Physics, Economics, etc.
S
1
4.4 Concavity and Curve Sketching
If the slope of a graph of a function is constant (in other words the graph is
a straight line) then the second derivative must be zero. Thus, if a function
has non
1
4.3 Monotonic Functions and the
First Derivative Test
Example 1
Find the intervals where the function on the graph is increasing
x
0
Denition
1
2
3
4
5
Increasing Functions and Decreasing Functions
1
4.2 The Mean Value Theorem
Example 1
Draw a continuous and dierentiable function that goes trough the suggested points and never has its derivative equal to zero.
y
3.0
2.5
2.0
1.5
1.0
0.5
x
0.5
The
1
4.1 Extreme Values of Functions
Denition
Extreme Values
Let f be a function with domain D Then f has an
(or
)
value on D at point c D if
,
and an
(or
for all x in D
,
Example 1
value on D at c D if
1
3.11 Linearization and Dierentials
A close up look at any smooth function reveals that its behavior is close to
linear on any interval that is small enough. In other words, the graph of
the function
Math 1225
Due MON 09/12/2016
Test 1
Homework 3 (Section 2.5)
All problems are weighted equally.
To receive full credit show all your work on paper.
Honor Pledge. I have neither given nor received unau
Math 1225
Due TUE 09/06/2016
Test 1
Supplementary Problems 2.4a
(1) Let f be the function defined on the interval [0, 2] by
(
2x, if 0 x 1
f (x) =
.
4, if 1 < x 2
The graph of y = f (x) is shown below
Math 1225
Due TUE 09/06/2016
Test 1
Homework 2 (Sections 2.3 and 2.4)
All problems are weighted equally.
To receive full credit shown all your work on paper.
Honor Pledge. I have neither given nor rec
Math 1225
Section 2.1
Test 1
Section 2.1: The Tangent And Velocity Problems
Intuition Coming from Physics
Galileos Law of Free Fall: The distance traveled by a falling body is directly
proportional to
Math 1225
Section 2.5
Test 1
In-Class Assignment 4
Problem 1.
Graph the function and decide if it is continuous at the given points.
(a) x = 1
(b) x = 0
(c) x = 1
(d) x = 2
(e) x = 3
(f) Where is f(x)
Math 1225
Section 2.5
Test 1
Section 2.5: Continuity Part 2.
Continuity of a function.
Definition (continuous function).
We say that a function is continuous if it is continuous at every point of its
S16MATH 1225 Test 1 Feb 10, 2016.NAME:
NUMERIC ANS
Use only methods from class.
You must show work to receive credit.
Whenever dealing with limits, make sure to use proper notation to get full cred
S16
MATH 1225 Test 2
Mar 16, 2016.
NAME:
NUMERIC ANS
CRN:
Use only methods from class.
You must show work to receive credit.
Whenever dealing with limits, make sure to use proper notation to get fu
F15
MATH 1225 Test 2
2015
NAME:
NUMERIC ANS
CRN:
Use only methods from class. You must show work to receive credit.
1. Find y 0 . DO NOT SIMPLIFY.
(a) (11 pts) ln(xy) = ex+y
y0 =
ex+y x1
xyex+y y
=
1
F15 MATH 1225 Test 1 Sept 16, 2015.
CRN:
NAME:
NUMERIC ANS
Use only methods from class.
You must show work to receive credit.
Whenever dealing with limits, make sure to use proper notation to get f
Test1: Technical vocabulary
Word
Meaning
Pronunciation
Secant line
Tangent line
Rate of change
Theorem
Limit
Continuity
Discontinuity
Difference quotient
Math 1225
Due TUE 08/30/2016
Test 1
Homework 1 (Sections 2.1 and 2.2)
All problems are weighted equally.
For full credit all the work must be shown on paper.
Problem 1
The accompanying figure shows th
W ORKING W ITH W EB A SSIGN
IN
M ATH 1225
Instructions
You will not access WebAssign using a class code. Rather, you should have recieved an email
from WebAssign telling you how to log in. If you hav
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EngE 1215 Spring 2015
This program is used to test the Line Following Robot motors. It uses a
while loop that continues as long as the user wishes allowing the user to
change the "power-le
% EngE 1215 Spring 2015
% This program established the connection between the computer and the
% robot.
% Run once at start of session and leave 'a'
% defined in the workspace - DO NOT DELETE 'a'!
% C
classdef arduino_new < handle
% December 14 Log
% This a wrapper class for the newer versions of arduino package
% Unlike previous version, this is not a standalone solution and requires
% installatio
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EngE 1215 Spring 2015
This program is used to test the Line Following Robot sensors. It uses a
while loop to allow the user to take sensor readings a set number of
times. This program assume