The Mean Value Theorem
Consider a function f that is continuous on [61, b] and
differentiable on (a, b) . Furthermore, let f (a): f (b)
i ' C <3 = 0
What occurs for at least one value of x between a and b? (Hint:
think of critical numbers) Rolle's Theor
Continuity
A function is continuous at x = a if lim f (x) = f (a)
1)0
In order for the above statement to hold true,
the following three conditions must be met:
1. f(a) is dened
2. limf(x) exists
1)(1
3. limf(x)= f(a)
1)(1 Find points of discontinuity, if
Linear Approximations and Differentials
Consider a function y = f (x)
Now consider its tangent line at x : a
The slope of the tangent line is mu =f'(a) 3 Fotm (Q-QLQ)
so the equation of the tangent line is y f (a) = f ' (a)(x a)
or y : f'(a)(xa)+f(a)
If y
Volumes
The volume of a simple solid is found (generally) by multiplying
the area of a cross-section (the base for simple gures) by the
height. For example, the volume of a cylinder is the area of the
circle multiplied by the height of the cylinder.
For s
The Denite Integral
If f is a continuous lCthIl dened for [a,b], divide the interval
[a,b] into n subintervals of equal Width. If x; is a sample x-value
on the ith subinterval, we dene the definite integral of f from a
tob as
i=1
am
A : I:f(x)dx=1i_zn:f(x
Calculating Limits Using the Limit Laws
Limits for more complicated functions can be found using the limit laws.
Assuming that lim f (x) and $307) exist, and that c is a constant,
1)0
1- ljm[1"(16)+g(x)] = 93f (x)+1_r38(x)
1)
2. 'imD" (10-806)] =1i$f (x)-
Related Rates
When an equation can be written expressing how two (or
more) quantities are related to each other, we can
differentiate to discover how the rates of change in the
two variables with respect to time are related.
If at2 +3:3 :17 and 24 ,nd 3
Graphing with Calculus and Calculators
Use technology (appropriately) to produce a hand-drawn sketch
that displays all important aspects of the curve. In particular, use
graphs of the rst and second derviatives to estimate the intervals
of increase/decrea
Integration of Rational Functions by Partial Fractions
2 5 23(4(4 SKQO
_ _ : + -
Add x4+x+2 (WMMZ) 1X45(X+2)
7xIQ.
[Krl)(><+1>
'7 K . | (p
K1 *Qx -' 8
Can this process be reversed?
Reversing it is known as "resolving into partial fractions and you
learn
Rates of Change in the Natural and Social Sciences
Open your textbook and read pages 221-230 with the intent of
noticing how pervasive the function-derivative relationship is in
various elds and to take note of the major examples. You
should be able to re
Average Value of a Function
We know how to nd the average value given a set of numbers.
Apply this knowledge to nd the average value of a function over
a given interval. First, we want to average the y-values, so for a
nite number of y-values,
y1+y2 +y3 +
Areas Between Curves
We know how to nd the area below a curve (and above the x-
axis). Today we will explore nding the area between two curves.
Consider the following graph. How can we nd the area between
the curves?
9x) = 36)
MM +3- (H) 41
(x 4w) +3: sfV
The Fundamental Theorem of Calculus
We dene the following function as the accumulation of area
under a curve from t = a to t = x: I
g(x) = L f(t) dt
1. Find g (2), g (4), g (8), and g (-3) ifwe def1ne g (x) as
follows: g(x)=I:f(t)dt 1 \
- ff (2) : L g m a
New Functions from Old Functions
Transformational graphing allows us to build new functions from
old ones by translating (shifting), stretching/ shrinking, and reecting.
We also get new functions by combining two or more functions using
addition, subtract
Derivatives and Rates of Change
In order to nd the tangent line to a curve, we use points
P (a, f(a) and Q (x, f(x) to obtain a formula for secants
from xed point P to movable point Q.
= f (x)f (a)
xa
As point Q approaches point P, the secant slopes appro
h.
The Substitution Rule
We know how to nd the derivative of a composition of
functions, such as nding the derivative for
y=oos(x3+2)
2L9 _ . 3 1
AK - Stn(x +Q>3K.
We use the chain rule. Now we need to know how to nd
the antiderivative when there is a co
AP Calc
WarmUp
Answer the following questions in your
notes
1. What is an inverse function?
2. How do you find an inverse? How do
you know it even exists?
3. How can you verify that two functions
are inverses?
x Functions - Inverses (5.3)
Objectives:
- De
Newton's Method
Newton wanted to nd a way to approximate irrational numbers,
such as nding the positive value of x that is a solution for at:2 = 2
Newton knew how to nd the equation of a tangent line to a curve,
and he noticed that when he graphed the tan
The Chain Rule
Recall the homework problem #55 on page 189 where we
found the derivative of y : [f (10]3 to be % :3[ f (1.0]2 f '(x)
This leads us to an intuitive understanding of the chain rule, which
states that if we have two dierentiable functionsf an
Volumes Using Cylindrical Shells
Picture one of those devices one uses to remove lint and pet hair
from clothing. It is constructed to have concentric lengths of tape
(sticky side outward), and when one considers all of those
concentric pieces of tape, on
Maximum and Minimum Values
A function can take on an absolute maximum value or an
absolute minimum value. For example, a parabola will have an
ab solute minimum value of k if the vertex is at (h, k) and the
parabola points up. Any continuous function that
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Math 20: Midterm 1
Monday, 01/30/2012
Complete the following problems. You may use any result from class you like, but if
you cite a theorem be sure to verify the hypotheses are satised.
This is a closed-book exam. No notes allowed. No calculators or ot
Math 20: Spring 2013
Midterm 2
NAME:
LECTURE:
Time: 75 minutes
This is a closed book and closed notes exam. Calculators and any electronic aid are not
allowed.
For each problem, you should write down all of your work carefully and legibly to receive full
Math 20: Midterm 2
Tuesday, 02/22/2011
Complete the following problems. You may use any result from class you like, but if
you cite a theorem be sure to verify the hypotheses are satised.
This is a closed-book exam. No calculators or other electronic ai
AP Calc
WarmUp
Identify three types of behavior associated
with the nonexistence of a limit. Illustrate
each with a graph or a function.
5. 1.3 Evaluating Limits
Analytically
Objectives
1. Evaluate a limit using properties
of limits
2. Develop and use t
How Derivatives Affect the Shape of the Graph
The rs]; derivative
Notice that whenever the function has a positive rst derivative,
it is increasing, and when the rst derivative is negative, the
function decreases.
Suppose that f is continuous and c is a c
Approximate Integration
Now that we know how to integrate (nd the antiderivative,
F (x) and evaluate it, F (b) - F(a), it seems like a step
backwards to consider approximating an integral. Keep in
mind, however, that a particular function f(x) may not be
The Product and Quotient Rules
Let y = i:2 -x3 Find % using two methods: 1) by guessing a rule
for the product of two functions and 2) simplifying the expression
and than differentiating to yield the correct result.
i) 2&2 2X: 3K2. -_ xg FEinCOVC-i
a) nd-
Implicit Dierentiation
An explicit function (the majority of the functions you have ever
encountered are explicit) is one for which one variable (typically y)
can be expressed in terms of the other (typically x).
An implicit function is one that shows a r