12.3 The Tangent Line Problem

12.3, The Calc in Pre-Calc

The art of calculus serves two primary purposes.  These are finding the slope of a point on a curve, and finding the area under a curve.  The ability to take these measurements is essential to modern engineering, physics, and economics.  The first of these two abilities will be covered here.  This blog post contains the basics of tangent lines and derivatives.

Warning:
Calculus may seem like a fun tool, but it is nothing to take lightly.  Its capacity for calculation is great, but so is its ability to destroy.  Only read this post if you are willing to risk being corrupted by the immense power you are about to wield.




The idea behind derivatives is to find the slope of a function at a given input.  This means it is finding the slope at a single point.  The slope at any point can be easily figured with algebra if the function is a straight line, but if it is anything else then calculus becomes necessary.

The way to find the slope of a point on a curved line is to find the slope of its tangent line.  A tangent line is a line that intersects that part of the function at exactly one spot.  It can cross the path of the function elsewhere, but it can only touch the curve you are evaluating once.  The slope of this line will be the slope of the function at that particular point.

Derivatives can be used to find the slope of these tangent lines.  This is achieved by first using the difference quotient.

The difference quotient is the formula for the function that will tell you the slope of the line between f(x) and f(x+h).  To find the slope at f(x) for any point x, one must first plug the f(x) into the difference quotient formula.  Let's do this for f(x)=x^2

This simplified function now tells us the slope between the points f(x) and f(x+h) in the function f(x)=x^2 for any value x and h.  The way to use this information to find the slope at f(x) is to apply the concept of limits.  Since h is the difference in x values between the two points, h must be made as small as possible without being 0 to find the slope of the tangent line of the single point f(x).  This can be done by taking the limit of the simplified difference quotient as it approaches 0.

The resulting limit is known as the derivative of f(x).  For any x value that is put into this derivative, the resulting value will be the slope at f(x).  For example, this means where x = 0, the slope of the function is equal to 2(0),  or 0.  Where x=2, the slope of the function is 2(2), or 4.

This same process can be used to find the slope of any point on any graph.  Just find the derivative by taking the difference quotient as h approaches 0, and plug in any x value to find the slope at that exact point.


Ian Hoeck

12.2 Techniques for Evaluating Limits

A function is continuous at c if exists and
It can also be written as

Example of Continuous:

Types of Noncontinuous

           


Holes                 






Vertical Asymptotes 







 Piecewise Functions

Techniques for Evaluating Limits

Direct Substitution:

Dividing Out:

Rationalizing:

12.1 Introduction to Limits

 

Definition of Limits- If f(x) becomes arbitrarily close to a unique number L as x approaches c from either side, the limit of f(x) as x approaches c is L.

 

 

 

 

 

*Note- the limit doesn't care what is actually happening at c

 

Example-

 

 

Example-

 

 

Even though this graph has a hole at x=1, the graph still approaches 2 from both sides so the limit is still 2. The limit doesn't care what is actually happening at f(1).



 

Example-

 

This limit does not exist because as x approaches 0, f(x) oscilalates between 1 and -1.



Properties of Limits- 

10.5 Parametric Equations

Parametric Equations are a useful way to model certain realistic scenarios mathematically, especially situations involving motion-over-time.

• In fact, time is so important to the equation, it often takes over the entire role of the dependent variable, creating two separate equations where 'x' and 'y' (coordinates) are functions of the new variable 't', which is called the parameter. In some cases, 't' may not represent time.
   
• The parameter 't' is frequently restricted by the real-world conditions it is modeling. For example, if you are modeling a projectile path and ignoring air resistance, 't' will have a domain of [0, t where y=0, x ≠ 0] (from 0 to the time of landing.)

• Therefore, parametric functions are rarely continuous. In fact, they are most commonly fragments of curves, or sometimes circles when sin/cos are involved.

•  In some interesting cases, two parametric equations, once restricted, will appear to graph the exact same curve. Unless they are the same equation, this indicates their equations only change the speed at which the motion occurs, not its direction (see page 732 for example.)

Definition: A curve in the plane is said to be parameterized if the set of coordinates on the curve, (x,y), are represented as functions of a variable t. Namely, 

x = f(t),  y = g(t)   The variable t is called a parameter and the relations between x, y and t are called parametric equations. The set of ordered pairs (f(t),g(t)) on the interval 'I' is a plane curve (denoted 'C'.)

Key Idea: Because parametric equations model real physical motion, the way their graphs are sketched must be realistic. Specifically, because motion occurs along the path of the parametric curve, the curve itself is sketched in a specific direction. 

• The direction in which a parametric curve is sketched is called its orientation. This can be left to right, right to left, clockwise, or counter-clockwise.
• Orientation is determined by simply comparing coordinates that result from subsequent values of the variable 't'
  
  Eliminating the Parameter
  
  
  
  Often enough we find the need to model parametric equations in the more classical f(x) type of equation. While usually possible, restrictions on the parameter won't always transfer over to the new equation, so you must determine the domain and range of the parametric equations before eliminating the parameter.
  
  
  
  
  

 Note: While the graph above shows the new function, it fails to model the actual path including the restrictions on t, as mentioned in the first step. These should be considered to determine the domain (think about limits of x and y...)

• Trigonometric Parametric Equations are special, as solving for t and substituting is generally a poor method and will not necessarily yield a useful equation. Instead, one should recognize and attempt to apply basic trigonometric relationships, such as the always-true Pythagorean Identity, as shown below.

• By solving for cos(t) and sin(t), we can quickly use an identity to eliminate the trigonometry and the parameter in, shall we say, one fell swoop. You should probably review your other identities for more advanced equations people...

10.6 Polar Coordinates

Polar Coordinates are only half as scary as the name sounds. No, it's not like the "Polar Vortex" or the "Polar Express", it is instead a method of describing distance and direction using a radius and an angle.

In a standard (X,Y) grid, points are based upon distance X and distance Y, in a polar grid, none of that applies, and points are based upon the hypotenuse of (X,Y) and the angle the line forms with the horizontal.

The hypotenuse is easy to find, just use the Pythagorean Theorem! The square root of (X^2)+(Y^2)

is the first part of a polar coordinate, the radius (r).

The second part of the polar coordinate is the degree measure that the coordinate is based upon the measurement the hypotenuse of (X,Y) forms with the horizontal, using a right triangle, we can determine that this is arcsin(Y/X). This gives us the degree measure for a Polar Coordinate, Theta.

Congratulations! You may now know how to convert between Rectangular Coordinates and Polar Coordinates, you should celebrate with cake or other delicious pastries.

9.5 The Binomial Theorem

The Binomial Theorem is used when finding the coefficients of expanded binomials.

The Binomial Theorem in the expansion of  :



The coefficient of can be found using:

Another way to write: is.


Example : Find the binomial coefficients of :

Step 1: Plug in to formula=

Step 2: Solve====


Example: Find the binomial coefficients of :

Step 1: Plug into formula =

Step 2: Solve ====


Pascal's Triangle:

Another way to find binomial coefficients is through Pascal's Triangle.

In the triangle, each number is found by adding together the two numbers above it. Example: 3+3=6 These numbers are the same as the coefficients of expanded binomials:

Example: Expand the binomial 

    

    Step 1: Find Coefficients

            The coefficients from the 4^th row of Pascal’s Triangle are 1,4,6,4,1

   

    Step 2: Expand

   

    Step 3: Simplify 

When expanding binomials that have subtraction signs instead of addition you alternate the signs in front of each coefficient.

   Example: Expand the binomial 

   The coefficients are the same but the signs alternate: