SuperMath: Calculus Course

supbookCopyright by V. Mazzeo

Calculus is a branch of mathematics that deals with rates of change. Its roots go back as far as Ancient Greece, but calculus as we know it today began with Newton and Leibnitz, in the 17th century. Basic ideas of calculus include the idea of limit , derivative , and integral. In the sciences, many processes involving change( or related variables) are studied. Calculus is a powerful tool to study the ways in which the variables interact.

RATIONAL AND IRRATIONAL NUMBERSNumbers are classified according to type. The first type of number is the first type you ever learned about: the counting, or “natural” numbers (N):

1, 2, 3, 4, 5 and so on

If we include 0 (zero), we have the “whole” numbers (W):

0, 1, 2, 3, 4, 5 …etc…etc…

Then come the “integers” (Z), if we include their algebraic negatives (zero, the natural numbers, and the negatives of the naturals):

… –5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5 …

But the necessary numbers are the rationals and irrationals.
Real numbers are either rational or irrational. The word “rational” (Q)comes from the word “ratio.” A number is rational if it can be expressed as the quotient, or ratio, of two whole numbers. Any whole number is rational. Its denominator is 1.
Any rational number can be expressed as the quotient of two integers in many ways. For example,

8=8/1=16/2=32/4=…

-> As a fraction a
b
, where a and b are integers (b 0).

An irrational number is a real number that cannot be expressed as the ratio of two integers. For example:
\sqrt(2)= 3,14...

A real number has a decimal representation.  It gives the approximate location of the number on the real line.

Most commonly, “real line” is used to mean real axis, a line with a fixed scale so that every real number corresponds to a unique point on the line. The term “real line” is also used to distinguish an ordinary line from a so-called imaginary line.

Any real number except 0 (zero) is either negative or positive. Numbers to the left of zero on the number line are negative numbers. They are written with a minus sign (e.g., -2).
Numbers to the right of 0 on the number line are positive numbers. They may be written with a plus sign (e.g., +2), but usually the plus sign is omitted. Thus, real numbers may be referred to as signed numbers.

+C_0,C_1,C_2...C_n.... or -C_0,C_1,C_2...C_n...

THE DOMAIN AND RANGE OF A FUNCTION

A function is a relation that uniquely associates members of one set with members of another set.  More formally, a function from A to B is an object f such that every a \in A is uniquely associated with an object f(b) \in B. A function is therefore a many-to-one relation. The domain of a function is the complete set A of possible values of the independent variable in the function. The range of a function is the complete set of all possible resulting values of the dependent variable of a function, after we have substituted the values in the domain. Domain and range can be seen clearly from a graph.

For example: consider the function f(x)= \frac{1}{x-2}.

Notice that there is no output number when the input number equals 2, since division by zero is meaningless. Thus x can have any value except 2. We say that 2 is not in the domain of the function f.

Here is another example. Let f(x)= \sqrt(x-4)

Since the quantity under the radical must be non-negative x-4>=0. Thus the implicit domain is [4,+oo[

Another one: let f(x)= \frac{1}{x\sqrt(x-1)}.

Since we cannot divide by 0, we know that x cannot equal to either 0 or 1.  And since we cannot take the square root of a negative number (remember, output numbers must be real), then x-1>=0. Taken together, x-1>=0 and x \neq0, these requirements mean that x-1>0. Thus the implicit domain is (1, +oo)

Example: Find the range of function f defined by: f(x)= x^2-2

The domain of this function is the set of all real numbers. The range is the set of values that f(x) takes as x varies. If x is a real number, x2 is either positive or zero. Hence we can write the following: x^2>=0. Subtract -2 to both sides to obtain x^2-2>=-2. The last inequality indicates that x^2- 2 takes all values greater that or equal to -2. The range of f is given by [-2, +oo( . A graph of f also helps in interpreting the range of a function. Below is shown the graph of function f given above. Note the lowest point in the graph has a y (= f (x) ) value of -2.

Example: f(x)= \sqrt (x+4). We notice that there are only positive y-values. There is no value of x that we can find such that we will get a negative value of y. We say that the range for this function is y >=0.

A REVIEW OF TRIGONOMETRY

Historically, trigonometry was developed for astronomy and geography, but scientists have been using it for centuries for other purposes, too. Besides other fields of mathematics, trigonometry is used in physics and engineering.

One statement of plane geometry states that a triangle is determined by a side and two angles. In other words, given one side of a triangle and two angles in the triangle, then the other two sides and the remaining angle are determined. Trigonometry includes the methods for computing those other two sides. The remaining angle is easy to find since the sum of the three angles equals 180°.

RIGHT TRIANGLE TRIG

Trigonometry is the study of the relations between the sides and angles of triangles. The word “trigonometry” is derived from the Greek words τρίγωνον, meaning “triangle”, and  μέτρον, meaning “measure”.

Angles

—————————-

Question: ” Why did the 30°-60°-90° triangle marry the 45°-45°-90° triangle?”

Answer:  ” They were right for each other”.

—————————-

Recall the following definitions from elementary geometry:

(a) An angle is acute if it is between 0◦ and 90◦.
(b) An angle is a right angle if it equals 90◦.
(c) An angle is obtuse if it is between 90◦ and 180◦.

(d) An angle is a straight angle if it equals 180◦.

In elementary geometry, angles are always considered to be positive and not larger than 360◦.

(1)  Two acute angles are complementary if their sum equals 90◦.

(2)  Two angles between 0◦ and 180◦ are supplementary if their sum equals 180◦.

(3) Two angles between 0◦ and 360◦ are conjugate (or explementary) if their sum equals 360◦.

You learned that the sum of the angles in a triangle equals 180◦, and that an isosceles triangle is a triangle with two sides of equal length. Recall that in a right triangle one of the angles is a right angle. Thus, in a right triangle one of the angles is 90◦ and the other two angles are acute angles whose sum is 90◦.

In a right triangle, the side opposite the right angle is called the hypotenuse, and the other two sides are called its legs.

————————–

Question: “What did the complementary angle say to the isosceles triangle?”

Answer:  “Nice legs”. 

————————–

By knowing the lengths of two sides of a right triangle, the length of the third side can be determined by using the Pythagorean Theorem:

“The square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of its legs”.

Thus, if a right triangle has a hypotenuse of length c and legs of lengths a and b, then the Pythagorean Theorem says:

a^2+b^2=c^2

Recall that two triangles ABC and DEF are similar if their corresponding angles are equal, that is, angle A equals angle D, angle B equals angle E, and angle C equals angle F, and their sides are proportional, that is, the ratios of the three corresponding sides are equal:

AB


DE

 = BC


EF

 = CA


FD


TRIGONOMETRIC FUNCTION OF AN ACUTE ANGLE
Consider a right triangle ABC, with the right angle at C and with lengths a, b, and c, as in the figure:
For the acute angle A, call the leg BC its opposite side, and call the leg AC its adjacent side. Recall that the hypotenuse of the triangle is the side AB.  We define the six trigonometric functions of A as follows:
where \theta=A.
  • sine A (sinA): opposite side/ hypotenuse: a/c
  • cosine A (cosA): adjacent side/ hypotenuse: b/c
  • tangent A (tgA): opposite side/ adjacent side: a/b
  • cosecant A (cscA): hypotenuse/opposite side: c/a
  • secant A (secA): hypotenuse/ adjacent side: c/b
  • cotangent A (cotA): adjacent side/opposite side: b/a
The pairs sin A and csc A, cos A and sec A, and tan A and cot A are reciprocals:
\csc A= \frac{1}{\sin A}
\sin A= \frac{1}{\csc A}
\sec A= \frac{1}{\cos A}
\cos A= \frac{1}{\sec A}
\cot A= \frac{1}{\tan A}
\tan A= \frac{1}{\cot A}
TRIGONOMETRIC FUNCTIONS OF ANY ANGLE
We need a more general definition of an angle.
We say that an angle is formed by rotating a ray OA about the endpoint O ( the vertex), so that the ray is in a new position, denoted by the ray OB. The ray OA is called the initial side of the angle, and OB is the terminal side of the angle. We say that the angle is positive, and the angle is negative if the rotation is clockwise

We can define the trigonometric functions of any angle in terms of Cartesian coordinates. Recall that the xy-coordinate plane consists of points denoted by pairs (x,y) of real numbers. The first number, x, is the point’s x coordinate, and the second number, y, is its y coordinate. The x and y coordinates are measured by their positions along the x-axis and y-axis, respectively, which determine the point’s position in the plane. This divides the x y-coordinate plane into four quadrants.

Now let θ be any angle. We say that θ is in standard position if its initial side is the positive x-axis and its vertex is the origin (0, 0). We then define the trigonometric functions of θ as follows:

sin θ  = y
r
csc θ  = r
y
cos θ  = x
r
sec θ  = r
x
tan θ  = y
x
cot θ  = x
y
According to the Pythagorean theorem:
r= \sqrt {x^2+y^2}

Summarizing the signs (positive or negative) for the trigonometric functions based on the angle’s quadrant:

The Law of Sines

It is an equation relating the lengths of the sides of an arbitrary triangle to the sines of its angles. According to the law,

 \frac{a}{\sin A} \,=\, \frac{b}{\sin B} \,=\, \frac{c}{\sin C}

where ab, and c are the lengths of the sides of a triangle, and AB, and C are the opposite angles. Sometimes the law is stated using the reciprocal of this equation.

Another way of stating the Law of Sines is: ” The sides of a triangle are proportional to the sines of their opposite angles”. 

The Law of Cosines

It relates the lengths of the sides of a plane triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines says

c^2 = a^2 + b^2 - 2ab\cos\gamma\ ,

where γ denotes the angle contained between sides of lengths a and b and opposite the side of length c.

By changing which sides of the triangle play the roles of ab, and c in the original formula, one discovers that the following two formulas also state the law of cosines:

a^2 = b^2 + c^2 - 2bc\cos\alpha\,
b^2 = a^2 + c^2 - 2ac\cos\beta\,

TRIGONOMETRIC IDENTITIES

We will now derive one of the most important trigonometric identities.Let \theta be any angle with a point (x,y) on its terminal side a distance r>0 from the origin.

By the Pythagorean Theorem, r^2=x^2+y^2, dividing both sides of the equation by  r^2 gives:

\frac{r^2}{r^2}= \frac{x^2+y^2}{r^2}= \frac{x^2}{r^2} + \frac{y^2}{r^2}= (\frac{x}{r})^2+ (\frac{y}{r})^2

Since: \frac{r^2}{r^2}=1, \frac{x}{r}= \cos \theta and \frac{y}{r}= \sin \theta,we can rewrite this as:

\cos^2 \theta +\sin^2 \theta= 1

We can think of this as sort of a trigonometric variant of the pythagorean Theorem.

From the above identity we can derive more identities. For example:

Solving for either the sine or the cosine:

\sin\theta = \pm \sqrt{1-\cos^2\theta} \quad \text{and} \quad \cos\theta = \pm \sqrt{1 - \sin^2\theta}. \,

Also,from the inequalities 0<= \sin^2(\theta) = 1 − \cos^2(\theta) <= 1 and 0 <= \cos^2(\theta)=1 − \sin^2(\theta) <= 1 ,taking square roots gives us the following bounds on sine and cosine:

−1 <=\sin(\theta)<= 1

−1 <=\cos(\theta)<= 1

ADDITION FORMULAS, SUBTRACTION FORMULAS

For the sum of any two angles A and B, we have the addition formulas:

cos(A+B) = cos A cos B − sin A sin B

sin(A+B) = sin A cos B + cos A sin B

It is simple to verify that they hold in the special case of A = B = 0◦. For general angles, we will need to use the relations which involve adding or subtracting 90◦:

sin(θ+90◦) = cos θ

sin(θ−90◦) = −cos θ

cos(θ+90◦) = −sin θ

cos(θ−90◦) = sin θ

Replacing B by −B in the addition formulas and using the relations sin (−θ) = − sin θ and cos (−θ) = cos θ , we have the subtraction formulas:

sin(A−B) = sin A cosB − cos A sinB

cos(A−B) = cos A cosB + sin A sinB

Using the identity tan θ = sin θ , and the addition formulas for sine and cosine, we can  derive the addition the addition and subtraction formulas for tangent:

tan(A+B) = (tan A + tan B) / (1 − tan A tan B)

tan(A−B) = (tan A − tan B) / (1 + tan A tan B)

Double-Angle and Half-Angle Formulas    

A special case of the addition formulas is when the two angles being added are equal, result- ing in the double-angle formulas:

To derive the sine double-angle formula, we see that:
sin2θ = sin(θ+θ) = sinθ cosθ + cosθ sinθ = 2sinθ cosθ.

Likewise, for the cosine double-angle formula, we have:
cos2θ = cos(θ+θ) = cosθ cosθ − sinθ sinθ =latex (\cos)^2 \theta − (\sin)^2 \theta,

and for the tangent we get:
tan2θ = tan(θ+θ) = (tanθ + tanθ)/(1 − tan θ tan θ) =latex 2\tan \theta / (1 − \tan^2 \theta)$

RADIAN MEASURE

We have been using degrees as our unit of measurement for angles.

So far we have been using degrees as our unit of measurement for angles. However, there is another way of measuring angles that is often more convenient. The idea is simple: associate a central angle of a circle with the arc that it intercepts. Associating the central angle with its intercepted arc, we could say, for example, that

360◦ “equals” 2π r (or 2π ‘radiuses’)

We use the term radians: 360◦ = 2π radians.

Converting radians to degrees:

To convert radians to degrees, multiply by 180/p, like this:

Converting degrees to radians:

To convert degrees to radians, multiply by p/180, like this:

                         

GRAPHING THE TRIGONOMETRIC FUNCTIONS   

The first function we will graph is the sine function. We will describe a geometrical way to create the graph, using the unit circle.

This is the circle of radius 1 in the x y-plane consisting of all points (x, y) which satisfy the equation x^2 + y^2 = 1.

We see in Figure  that any point on the unit circle has coordinates (x, y) = (cos θ, sin θ), where θ is the angle that the line segment from the origin to (x, y) makes with the positive x-axis (by definition of sine and cosine). So as the point (x, y) goes around the circle, its y-coordinate is sin θ.

We thus get a correspondence between the y-coordinates of points on the unit circle and the values f (θ) = sin θ, as shown by the horizontal lines from the unit circle to the graph of f(θ)=sinθ. To graph the cosine function, we could again use the unit circle idea (using the x-coordinate of a point that moves around the circle):

To graph the tangent function:

Properties of Graphs of Trigonometric Functions    

Recall that the domain of a function f (x) is the set of all numbers x for which the function is defined. The range of a function f (x) is the set of all values that f (x) can take over its domain.

A function f (x) is periodic if there exists a number p > 0 such that x + p is in the domain of f (x) whenever x is, and if the following relation holds:

f(x+p)=f(x) for all x

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