Antiderivatives
Sometimes we are given a function f (x), and wish to find a function for which f (x) is the derivative,
that is, we want to find a function F (x) with F 0 (x) = f (x). For example if I knew the velocity of a
particle over a time interval,
Lectures 17/18 Derivatives and Graphs
When we have a picture of the graph of a function f (x), we can make a picture of the derivative f 0 (x)
using the slopes of the tangents to the graph of f . In this section we will think about using the derivative
f
Names
Math 10550 Worksheet 4
1. (a) Calculate
x2 3x 4
x2 9
and
lim
.
x4 x2 + 5x + 4
x3 x2 2x 3
lim
9
6
3
Solution: We see that x2x2x3
= x+3
x+1 , so when x 3 we get that the limit is 4 = 2 , and
1
when x 4 then x + 1 3 but x+4 goes to plus or minus infint
Name:
Instructor:
Math 10550, EXAM II
October 13, 2016
The Honor Code is in effect for this examination. All work is to be your own.
No calculators.
The exam lasts for 1 hr. and 15 min.
Be sure that your name is on every page in case pages become detached
Newtons Method
In this section we will explore a method for estimating the solutions of an equation f (x) = 0 by a
sequence of approximations that approach the solution.
Note that for a quadratic equation ax2 +bx+c = 0, we can solve for the solutions usin
Name:
Math 10550, Final Exam: Solutions
December 18, 2016
Instructor:
The Honor Code is in effect for this examination, including keeping your answer
sheet under cover.
No calculators are to be used.
The exam lasts for two hours.
Be sure that your nam
Name:
Math 10550, Final Exam Fall 2014:
December 18, 2016
Instructor:
The Honor Code is in effect for this examination, including keeping your answer
sheet under cover.
No calculators are to be used.
The exam lasts for two hours.
Be sure that your nam
Name: *;> (~3me 0 N S
Math 10550, Final Exam: Instructor:
December 17, 2008
o The Honor Code is in effect for this examination, including keeping your answer
sheet under cover.
0 N o calculators are to be used.
a The exam lasts for two hours.
0 Be sure th
Names
Date
Mth 10550 Worksheet 9
1. The following shows the graph of a function f (t):
2
1
2
4
6
8
-1
-2
(a) Give a rough sketch of the graph of the function g(x) =
FTC.
Rx
0
f (t) dt. Fix g(0) = 0 and use the
Solution: We expect to have a graph that look
Differentiation Formulas Sums and Differences The Product Rule Special Case of The quotient Rule The Quotient Rule Example General Power Fun
Differentiation Formulas
As we did with limits and continuity, we will introduce several properties of the
derivat
4
16
8
6
Lecture 14 :Linear Approximations and Differentials
2
Consider a point on a smooth curve y = f (x), say P = (a, f (a), If we draw a tangent line to the curve
at the point P14
6, we can see from the pictures below that as we zoom in towards the po
Lecture 3 : Limit of a Function
Click on this symbol
to view an interactive demonstration in Wolfram Alpha.
Limit of a Function
Consider the behavior of the values of f (x) = x2 as x gets closer and closer . and closer . to 3.
Example
Let f (x) = x2 . The
Names
Date
Math 10550 Worksheet 2 Solution
1. Calculate the following limits. If the limit is finite, give its value, otherwise say whether the limit
is +, , or does not exist and is not equal to + or .
(a)
x5
x5 x2 10x + 10
lim
Solution: Since lim (x2 10
Lecture 5 : Continuous Functions
Definition 1 We say the function f is continuous at a number a if
lim f (x) = f (a).
xa
(i.e. we can make the value of f (x) as close as we like to f (a) by taking x sufficiently close to a).
Example Last day we saw that i
Lecture 15 :Maxima and Minima
In this section we will study problems where we wish to find the maximum or minimum of a function.
For example, we may wish to minimize the cost of production or the volume of our shipping containers
if we own a company. Ther
The Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus shows that differentiation and Integration are inverse processes.
Consider the function f (t) = t. For any value of x > 0, I can calculate the definite integral
Z x
Z x
f (t)dt =
tdt.
Lecture 7 : Derivative AS a Function
In the previous section we defined the derivative of a function f at a number a (when the function f is
defined in an open interval containing a) to be
f (a + h) f (a)
h0
h
f 0 (a) = lim
when this limit exists. This gi
Indefinite Integrals/Applications of The Fundamental Theorem
We saw last time that if we can find an antiderivative for a continuous function f , then we can evaluate
the integral
Z
b
f (x)dx.
a
Indefinite Integrals
In light of the relationship between th
Volumes by Disks and Washers
Volume of a cylinder A cylinder is a solid where all cross sections are the same. The volume of a
cylinder is A h where A is the area of a cross section and h is the height of the cylinder.
For a solid S for which the cross se
Lecture 10 : Chain Rule
Here we apply the derivative to composite functions. We get the following rule of dierentiation:
The Chain Rule : If g is a dierentiable function at x and f is dierentiable at g(x), then the
composite function F = f g dened by F (x
Lecture 4 : Calculating Limits using Limit Laws
Using the denition of the limit, limxa f (x), we can derive many general laws of limits, that help us to
calculate limits quickly and easily. The following rules apply to any functions f (x) and g(x) and als
Lecture 3 : Limit of a Function
Limit of a Function
Consider the behavior of the values of f (x) = x2 as x gets closer and closer . and closer . to 3.
Example
Let f (x) = x2 . The table below shows the behavior of the values of f (x) as x approaches
3 fro
Lecture 5 : Continuous Functions
Denition 1 We say the function f is continuous at a number a if
lim f (x) = f (a).
xa
(i.e. we can make the value of f (x) as close as we like to f (a) by taking x suciently close to a).
Example Last day we saw that if f (
Lecture 2 : Tangents
Functions
The word Tangent means touching in Latin. The idea of a tangent to a curve at a point P , is a
natural one, it is a line that touches the curve at the point P , with the same direction as the curve.
However this description
Lecture 12 :Rates of Change in The Natural and Social Sciences
dy
When y = f (x), dx
denotes the rate of change of the function f (x) with respect to x.
Recall the average rate of change of y with respect to x in the interval [x1 , x2 ] is
f (x2 ) f (x1 )
The Definite Integral
If f is continuous for a x b, we divide the interval [a, b] into n subintervals of equal length,
x = ba
. Let x0 = a, x1 , x2 , . . . , xn = b be the end points of these subintervals and let x1 , x2 , . . . , xn be
n
any sample point
Lecture 4 : Calculating Limits using Limit Laws
Click on this symbol
to view an interactive demonstration in Wolfram Alpha.
Using the definition of the limit, limxa f (x), we can derive many general laws of limits, that help us to
calculate limits quickly
Limits at Infinity
When graphing a function, we are interested in what happens the values of the function as x becomes
very large in absolute value. For example, if f (x) = 1/x then as x becomes very large and positive, the
values of f (x) approach zero.
Integration by Substitution
In this section we reverse the Chain rule of differentiation and derive a method for solving integrals
called the method of substitution. Recall the chain rule of differentiation says that
d
f (g(x) = f 0 (g(x)g 0 (x).
dx
Rever