Download Keep in mind that high school TMSCA is

Calculus Handout PARTY!!! Week
Preparation for High School
Keep in mind that high school TMSCA is completely different from that of middle school.
Although this may seem to be a disadvantage, there is actually a positive outcome to the test
restructuralization. Those who did not excel in certain aspects (Number Sense, General Math,
Calculator) are now given a chance to start fresh with everybody else.
As many may know from looking at some practice tests, the high school TMSCA contains many
calculus problems! 30% of the general math, 60% of the calculator, and 10% of the number sense
exams contain calculus problems.
Lastly, if you want to have a starting advantage against your competitors, PRACTICE, PRACTICE,
AND PRACTICE! With sufficient exercise (and looking up solutions online or asking Me/Tony),
your scores can dramatically increase.
Below are some guidelines for scoring and a projected TMSCA state winning score for a 9th grader:
Number of
Scoring
Highest
State-Winning
Questions
Possible Score Score
Number
80
(questions completed)*5 – (questions
400
260
Sense
incorrect or skipped)*9
Calculator
35 Number
(questions completed)*5 – (questions
350
240
Crunchers
incorrect or skipped)*9
14 Geometry
21 Stated
General Math 60
(questions completed)*6 – (questions
360
270
incorrect)*7
Science
20 Biology
20 Chemistry
20 Physics
No penalty for skipped
(questions completed)*6 – (questions
incorrect)*7
360
270
No penalty for skipped
Note that when competing in any UIL competitions (local, regional, or state) grade divisions are not
taken into account. All grades compete with each other for each contest type.
The top 3 at the local competition and the top team for each contest type advance to regionals.
The top 3 at the state competition and the top team for each contest type advance to state.
For more info and resources (many tests and links to important online resources) go to the high
school math club website: www.chhsmathclub.wikispaces.com
Overview:
Calculus is the mathematical study of change, in the same way that geometry is the study of shape
and algebra is the study of operations and their application to solving equations. It has two major
branches, differential calculus (concerning rates of change and slopes of curves), and integral
calculus (concerning accumulation of quantities and the areas under curves); these two branches are
related to each other by the fundamental theorem of calculus. Both branches make use of the
fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit.
Calculus has widespread uses in science, economics, and engineering and can solve many problems
that algebra alone cannot.
Calculus is a major part of modern mathematics education. A course in calculus is a gateway to other,
more advanced courses in mathematics devoted to the study of functions and limits, broadly
called mathematical analysis.
Calculus has historically been called "the calculus of infinitesimals", or "infinitesimal calculus". The
word "calculus" comes from Latin (calculus) and means a small stone used for counting. More
generally, calculus (plural calculi) refers to any method or system of calculation guided by the symbolic
manipulation of expressions. Some examples of other well-known calculi are propositional
calculus, calculus of variations, lambda calculus, and process calculus.
Derivatives:
The derivative is a measure of how a function changes as its input changes. Loosely speaking, a
derivative is basically a way to measure a point’s slope (the slope of the point’s tangent line).
The process of finding a derivative is called differentiation. The reverse process is
called antidifferentiation. The fundamental theorem of calculus states that antidifferentiation is the
same as integration. Differentiation and integration constitute the two fundamental operations in
single-variable calculus.
There are several methods to denote the derivative:

In Leibniz's notation (infinitesimal change in x is denoted by dx, and the derivative of y with
respect to x)

Shorthand notation (derivative of a function)
*the presence of multiple tick marks indicates the number of derivatives you perform
the derivative of the function f(x) taken once is shown below
There are multiple ways to calculate a derivative (with respect to x):

Common Way (using exponents)
o Functions of Addition and Subtraction
1. Separate the function by terms
2. Take the exponent and place it in front of the term
3. Subtract the exponent by 1
o Examples
1. f(x) = 5x2 + 30x - 25
f ’(x) = (2)5x2-(1) + (1)30x1-(1) – (0)250-(1)
f ’(x) = 10x + 30
2. f(x) = 3x3 + 5
f ’(x) = (3)3x3-(1) + (0)50-(1)
f ’(x) = 9x2
f ’’(x) = (2)9x2-(1)
f ”(x) = 18x
o Functions of Multiplication (Product Rule)

Let h(x) = f(x) * g(x)
h ‘(x) = f ‘(x) * g(x) + f(x) * g ‘(x)
o Examples
1. f(x) = (2x3 + 1)(5x2)
f ‘(x) = (2x3 + 1)’ * (5x2) + (2x3 + 1) * (5x2)’
f ‘(x) = (6x2)(5x2) + (2x3 + 1)(10x)
f ‘(x) = 30x4 + 20x4 + 10x
f ‘(x) = 50x4 + 10x
2. f(x) = sin(x)cos(x)
*Refer to the list of derivatives on the next page
f ‘(x) = (sin(x))’ * (cos(x)) + (sin(x)) * (cos(x))’
f ‘(x) = cos(x) * cos(x) + sin(x) * -sin(x)
f ‘(x) = cos2(x) – sin2(x)
o Functions of division (quotient rule)

Let h(x) =
h ‘(x) =
𝑓(𝑥)
𝑔(𝑥)
𝑓 ′ (𝑥)∗𝑔(𝑥)−𝑓(𝑥)∗𝑔′ (𝑥)
𝑔(𝑥)2
o Examples
𝑥 2 +1
1. f(x) = 𝑥 2 −234
f ‘(x) =
f ‘(x) =
(𝑥 2 +1)′ ∗(𝑥 2 −234)−(𝑥 2 +1)∗(𝑥 2 −234)′
(𝑥 2 −234)2
(𝟐𝒙) ∗(𝒙𝟐 −𝟐𝟑𝟒)−(𝒙𝟐 +𝟏)∗(𝟐𝒙)
(𝒙𝟐 −𝟐𝟑𝟒)𝟐

Formula Bashing
o h indicates the derivative you wish to calculate (explained ahead) with limits
o f() is the function you are differentiating
The application to finding the derivative function is that you can determine the slope of a curved
line using this function.
For example, you are given the function f(x) = x2 and wish to calculate the instantaneous slope of
this quadratic curve at the x-coordinate 10.
1. Find f’(x)
f(x) = x2
f ‘(x) = (2)x2-(1)
f ‘(x) = 2x
2. Plug the value at x from which you want to solve the slope for
f ‘(10) = 2(10)
f ‘(10) = 20
Note that derivatives are closely related to limits (mentioned later) which you might have saw under
the bullet point above labeled “formula bashing.”
Also note that the formulas listed above do not work for specific functions such as trigonometric or
logarithmic. Several of these special function’s derivatives are located on the next page:
Simple Functions
Exponential and Logarithmic Functions
Trigonometric Functions
Integrals:
An integral (also called the antiderivative, primitive integral or indefinite integral) of a function f is a
differentiable function F whose derivative is equal to f (F ′ = f).
The process of solving for antiderivatives is called antidifferentiation (or indefinite integration) and
its opposite operation is called differentiation, which is the process of finding a derivative.
Antiderivatives are related to definite integrals through the fundamental theorem of calculus: the
definite integral of a function over an interval is equal to the difference between the values of an
antiderivative evaluated at the endpoints of the interval.
Indefinite Integral:
The indefinite integral is the exact opposite of the derivative. Its notation is as follows:
∫ 𝑓(𝑥)𝑑𝑥
The integral is calculated as exactly opposite of that of the integral (with respect to x):
1. Separate the function by terms
2. Add one to the exponent
3. Take the new exponent value, inverse it, then place it in front of the term
Examples:
1. ∫ 2𝑥 3 + 𝑥 2 − 5𝑥 + 3
1
1
1
1
= ∫ ( ) 2𝑥 3+(1) + ( ) 𝑥 2+(1) − ( ) 5𝑥1+(1) + ( )30+(1)
4
3
2
1
𝟏 𝟒)
𝟏 𝟑
𝟓 𝟐
= ∫( )𝒙 + ( )𝒙 − ( )𝒙 + 𝟑
𝟐
𝟑
𝟐
Definite Integral:
Here’s where things get exciting!!! Definite integrals are used to calculate 2-d areas under a curve.
The notation is as follows:
a represents the lower bound of the curved area, b represents the upper bound of the curved area
the functions F() represent indefinite integrals
Indefinite Integral
(no specific values)
Definite Integral
(from a to b)
Just like derivatives, there are several integrals that can’t be integrated using traditional methods.
These integrals are located below:
Limits:
In mathematics, a limit is the value that a function or sequence "approaches" as the input or index
approaches some value. Limits are essential to calculus (and mathematical analysis in general) and
are used to define continuity, derivatives, and integrals.
The concept of a limit of a sequence is further generalized to the concept of a limit of a topological
net, and is closely related to limit and direct limit in category theory.
In formulas, limit is usually abbreviated as lim as in lim(an) = a, and the fact of approaching a limit is
represented by the right arrow (→) as in an → a.
Limit of a Function
Suppose f is a real-valued function and c is a real number. The expression
means that f(x) can be made to be as close to L as desired by making x sufficiently close to c. In that
case, the above equation can be read as "the limit of f of x, as x approaches c, is L".
Note that the above definition of a limit is true even if f(c) ≠ L. Indeed, the function f need not even
be defined at c.
For example, if
then f(1) is not defined (see division by zero), yet as x moves arbitrarily close to
1, f(x) correspondingly approaches 2:
f(0.9) f(0.99) f(0.999) f(1.0)
1.900 1.990 1.999
f(1.001) f(1.01) f(1.1)
⇒ undefined ⇐ 2.001
2.010 2.100
Thus, f(x) can be made arbitrarily close to the limit of 2 just by making x sufficiently close to 1.
In other words,
This can also be calculated algebraically, as
for all real numbers
.
Now since
is continuous in
at 1, we can now plug in 1 for , thus
In addition to limits at finite values, functions can also have limits at infinity. For example, consider



f(100) = 1.9900
f(1000) = 1.9990
f(10000) = 1.99990
As x becomes extremely large, the value of f(x) approaches 2, and the value of f(x) can be made as
close to 2 as one could wish just by picking x sufficiently large. In this case, the limit
of f(x)as x approaches infinity is 2. In mathematical notation,
Limit of a Sequence
Consider the following sequence: 1.79, 1.799, 1.7999,... It can be observed that the numbers are
"approaching" 1.8, the limit of the sequence.
Formally, suppose a1, a2, ... is a sequence of real numbers. It can be stated that the real number L is
the limit of this sequence, namely:
to mean
For every real number ε > 0, there exists a natural number n0 such that for all n > n0, |an − L| < ε.
Intuitively, this means that eventually all elements of the sequence get arbitrarily close to the limit,
since the absolute value |an − L| is the distance between an and L. Not every sequence has a limit; if
it does, it is called convergent, and if it does not, it is divergent. One can show that a convergent
sequence has only one limit.
The limit of a sequence and the limit of a function are closely related. On one hand, the limit
as n goes to infinity of a sequence a(n) is simply the limit at infinity of a function defined on
the natural numbers n. On the other hand, a limit L of a function f(x) as x goes to infinity, if it exists,
is the same as the limit of any arbitrary sequence an that approaches L, and where an is never equal to
L. Note that one such sequence would be L + 1/n.
Other Notes:
Calculus Theorems
______________________________________________________________________________
The first fundamental theorem of calculus states that, if is continuous on the closed
interval
and is the indefinite integral of on
, then
This result, while taught early in elementary calculus courses, is actually a very deep result connecting
the purely algebraic indefinite integral and the purely analytic (or geometric) definite integral.
______________________________________________________________________________
The second fundamental theorem of calculus holds for a continuous function on an open
interval and any point in , and states that if is defined by
then
at each point in .
______________________________________________________________________________
The extreme value theorem states that if a real-valued function f is continuous in
the closed and bounded interval [a,b], thenf must attain its maximum and minimum value, each at
least once. That is, there exist numbers c and d in [a,b] such that:
A related theorem is the boundedness theorem which states that a continuous function f in the
closed interval [a,b]
is bounded on that
interval. That is, there exist
real
numbers m and M such that:
______________________________________________________________________________
The intermediate value theorem states that for each value between the least upper bound and
greatest lower bound of the image of a continuous function there is at least one point in
its domain that the function maps to that value.
Version I. The intermediate value theorem states the following: If f is a real-valued continuous
function on the interval [a, b], and u is a number between f(a) and f(b), then there is a c ∈ [a, b] such
that f(c) = u.
______________________________________________________________________________
The mean value theorem states, roughly: given a planar arc between two endpoints, there is at least
one point at which the tangent to the arc is parallel to the secant through its endpoints.
The theorem is used to prove global statements about a function on an interval starting from local
hypotheses about derivatives at points of the interval.
More precisely, if a function f is continuous on the closed interval [a, b], where a < b, and
differentiable on the open interval (a, b), then there exists a point c in (a, b) such that
[1]
______________________________________________________________________________
Rolle’s theorem states that: If a real-valued function f is continuous on a closed
interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists a c in the
open interval (a, b) such that
This version of Rolle's theorem is used to prove the mean value theorem, of which Rolle's theorem
is indeed a special case. It is also the basis for the proof of Taylor's theorem.
______________________________________________________________________________
The squeeze/sandwich/pinching theorem is formally stated as follows.
Let I be an interval having the point a as a limit point. Let f, g, and h be functions defined on I,
except possibly at a itself. Suppose that for every x in I not equal to a, we have:
and also suppose that:
Then



The functions g and h are said to be lower and upper bounds (respectively) of f.
Here a is not required to lie in the interior of I. Indeed, if a is an endpoint of I, then the
above limits are left- or right-hand limits.
A similar statement holds for infinite intervals: for example, if I = (0, ∞), then the
conclusion holds, taking the limits as x → ∞.