1.2 Essential Functions

Highlights from the lecture. We will plot some of these by hand in class (polynomials and rational functions for sure). Remember to use Shift-Enter or the Enter key on the number pad to evaluate cells. We will be doing more work with Mathematica next week, so this is just a bit of foreshadowing for how Mathematica can help us learn calculus.

Polynomials: Linear 
y-(Subscript[y, 0])=m(x-(Subscript[x, 0])) or y=m(x-(Subscript[x, 0]))+(Subscript[y, 0])

Linear functions will be important in calculus, as tangent lines to curves. A linear function is characterized by slope m=((Δy)/(Δx)) and the most useful form given above does not have the y-intercept but includes a point the line passes through ((Subscript[x, 0]),(Subscript[y, 0])).

Plot[-2/3(x-3)-2,{x,-5,5}]

The above example has slope -2/3 and passes through the point (3,-2)

Polynomials: Quadratic 
y=a (x^2)+b x + c
y=((a(x-h))^2)+k

Quadratics typically require the use of the quadratic formula, and can be sketched by
1.) Finding Zeros
2.) Opens up if a>0 and opens down if a<0
If there are no real zeros, the x coordinate of the vertex can be found by taking the average of the zeros (even when the zeros are complex!).

You can also complete the square to get the vertex form y=((a(x-h))^2)+k, which will locate the vertex of the quadratic (h,k).

Plot[1/20(x-1)^2-13,{x,-5,5}]

Notice that Mathematica does not always put the origin of the axes at (0,0).

Polynomials

Plotting a polynomial. If you can factor the polynomial, you can plot these by hand without much trouble. To plot a polynomial by hand you have to:
1.) Factor the polynomial, which can be difficult if it is high degree.
2.) Determine the zeros with multiplicity.
3.) Determine the end behaviour of the polynomial.

Plot[(2x^2-3x+1)(x+1)^2,{x,-5,5}]

We have to zoom in to see the zeros.

Plot[(2x^2-3x+1)(x+1)^2,{x,-2,2}]

Here is another polynomial. We would not be able to determine the shape of this graph without using calculus!

Plot[12x^3+3x^2-4x-2,{x,-1,1}]

Since the zeros are not rational numbers, we could not sketch this easily by hand.

NSolve[12x^3+3x^2-4x-2==0,x]

Monomials

Monomials are the building blocks for polynomials, and it is helpful to have an understanding of their end behaviour.

Plot[{x^2,x^3,x^4,x^5,x^6,x^7},{x,-2,2}]

All monomials approach positive infinity as x approaches positive infinity. 
Monomials with even power approach positive infinity as x approaches negative infinity.
Monomials with odd power approach negative infinity as x approaches negative infinity.

Understanding monomials will be helpful in sketching, and evaluating certain kinds of limits.

Square Root Function

Plot[x^(1/2),{x,-2,2}]

Cube Root Function

Mathematica does not plot the cube root function correctly.

Plot[x^(1/3),{x,-2,2}]

The cube root function has domain x∈ℝ, and should look like

Plot[p/.NSolve[p^3==x,p,Reals],{x,-10,10}]

At the moment, I do not expect you to understand the Mathematica code that produced this plot

NSolve solves the equation, but returns all the solutions including the complex valued ones.

Let’s just verify that if x<0 we should get a real number out. Let’s do this for x=-1

NSolve[y^3+1==0,y]

The problem is that when Mathematica solves that equation, it chooses a complex solution

(-1.0)^(1/3)

Rational Functions

Plotting a rational function is similar to plotting a polynomial, with one extra step. To plot a rational function by hand you have to:
1.) Factor the numerator and denominator, which can be difficult if it is high degree.
2.) Determine the zeros with multiplicity.
3.) Determine the vertical asymptotes, with multiplicity.
4.) Determine the end behaviour.

Plot[(2x^4-x^2+1)/(x^2-4),{x,-5,5}]

Plot[(9x^2-12x+4)/(16-x^2),{x,-8,8}]

Rational functions can have horizontal and slant asymptotes, which we will study when we look at limits at infinity.

Algebraic Functions

Generally difficult to plot by hand.

Plot[Sqrt[x^2-2]/(x+(4-x)^(1/3)),{x,-5,5}]

Trig Functions

Trig will not play an important role until Chapter 3, so I will review Trig functions at that point.

Exponential and Logarithmic Functions

We will cover these functions in more detail next week.