Elementary Functions

MAT 229, Spring 2021

Week 3

The function typically used in Calculus courses are called elementary functions.

Calculus I functions

Polynomial functions

Polynomial functions are those that involve only addition, subtraction, and multiplication.

Example

f(x)=5(x^3)-8(x^2)+3x-15

Facts

Easiest functions to work with. Later in the semester we will approximate more complicated functions with polynomials.

Domain: (-∞,∞)

End behavior: Except for constant functions, the limit as x->∞ or x->-∞ will be ±∞.

To differentiate use the sum and power rules: (((x^n)))'=n (x^(n-1)).

To integrate use the sum and power rules: ∫(x^n)d x=((x^(n+1))/(n+1))+C.

Rational functions

Rational functions are those that involve addition, subtraction, multiplication, and division. They can be written as the ratio of two polynomials.

Example

g(x)=((9(x^2)+1)/(4x-7))

Facts

Domain: All real numbers except those that cause division by zero.

End behavior: There may be horizontal asymptotes as x->±∞. There may be vertical asymptotes where the function is undefined.

To differentiate a rational function use the power rule and the quotient rule.

Integration becomes harder. Logarithms and inverse tangents may appear in the integrals of rational functions. For example,

∫(1/x)d x=ln|x|+C

∫(1/(1+(x^2)))d x=(tan^(-1))(x)+C

Trigonometric functions

The six trigonometric functions are

sin(x)

cos(x)

tan(x)=((sin(x))/(cos(x)))

csc(x)=(1/(sin(x)))

sec(x)=(1/(cos(x)))

cot(x)=((cos(x))/(sin(x)))

Facts

Used to model repeating (periodic) behavior.

Domains:

Both sine and cosine have domains (-∞,∞)

The remaining four are defined for all real values except those that cause division by zero.

The range for both sine and cosine is [-1,1].

End behavior: There are vertical asymptotes for tangent, cosecant, secant, and cotangent where they are not defined.

Derivatives:

((sin(x)))'=cos(x)

((cos(x)))'=-sin(x)

((tan(x)))'=(sec^2)(x)

((csc(x)))'=-csc(x)cot(x)

((sec(x)))'=sec(x)tan(x)

((cot(x)))'=-(csc^2)(x)

Integrals:

∫sin(x)d x=-cos(x)+C

∫cos(x)d x=sin(x)+C

∫tan(x)d x will get to in coming weeks

∫csc(x)d x will get to in coming weeks

∫sec(x)d x will get to in coming weeks

∫cot(x)d x will get to in coming weeks

Calculus II functions

Exponential functions

The simple exponential functions can be written as h(x)=(b^x) where b is a positive number other than 1. General exponential functions have the form k(x)=((p(x))^(q(x))).

Facts

Used to model growth and decay behavior.

For calculus always write simple or general functions in base e.

Simple exponential functions: (b^x)=(e^(r x)) where r=ln(b).

General exponential functions: ((p(x))^(q(x)))=(e^(k(x))) where k(x)=q(x)ln(p(x)).

Domains:

For simple exponential functions the domain is (-∞,∞).

For general exponential functions  the domain is that of the exponent function k(x)=q(x)ln(p(x)).

The range for simple exponential functions is (0,∞).

End behavior: One end, either x->-∞ or x->∞, goes to zero. The other end, x->∞ or x->-∞, goes to ∞.

To differentiate exponential functions, use the exponential rule (((e^x)))'=(e^x) along with the chain rule.

Integration: ∫ln(x)d x will get to in coming weeks.

Logarithm functions

Logarithms are the inverse functions to simple exponential functions.
	y=(b^x)⟷x=(Subscript[log, b])(y)

Just as for exponential functions, we prefer to work with base e, so we convert other logs to base e logs, via
	(Subscript[log, b])(y)=(1/(ln(b)))ln(y)

Facts

The domain of (Subscript[log, b])(x) is the range of its inverse (b^x),
	Domain of (Subscript[log, b])(x): (0,∞)

The range of (Subscript[log, b])(x) is the domain of its inverse (b^x),
	Range of (Subscript[log, b])(x): (-∞,∞)

End behavior for natural logarithm ln(x) with domain (0,∞),

(lim_(x->(SuperPlus[0])))ln(x)=-∞

(lim_(x->∞))ln(x)=∞

Differentiation: ((ln(x)))'=(1/x)

Integration: ∫ln(x)d x we will get to in coming weeks.

Inverse trigonometric functions

These are the inverse functions to the trigonometric functions. However, the domains must be restricted.

y=sin(x),-(π/2)≤x≤(π/2)⟷ x=(sin^(-1))(y)

y=tan(x),-(π/2)<x<(π/2)⟷ x=(tan^(-1))(y)

y=cos(x),0≤x≤π⟷ x=(cos^(-1))(y)

Facts

The domains of each is the range of its inverse.

Domain (sin^(-1))(x) is [-1,1].

Domain (cos^(-1))(x) is [-1,1].

Domain (tan^(-1))(x) is (-∞,∞).

The range of each is the restricted domain of its inverse

Range (sin^(-1))(x) is [-(π/2),(π/2)].

Range (cos^(-1))(x) is [0,π].

Range (tan^(-1))(x) is (-(π/2),(π/2)).

(tan^(-1))(x) is the only one of these three with an unbounded domain.

(lim_(x->-∞))(tan^(-1))(x)=-(π/2)

(lim_(x->∞))(tan^(-1))(x)=(π/2)

Differentiation:

(((sin^(-1))(x)))'=(1/(Sqrt[1-(x^2)]))

(((cos^(-1))(x)))'=-(1/(Sqrt[1-(x^2)]))

(((tan^(-1))(x)))'=(1/(1+(x^2)))

Integration: we will get to it in the coming weeks.

Combinations

All of the above functions can be combined in different ways using addition, subtraction, multiplication, division, and composition.

Facts

Differentiation: use the product rule, the quotient rule, and the chain rule along with the above differentiation rules.

Limits: one thing we’ve done recently is slip limits inside of compositions -- which we can do if the functions are continuous.

Integration:

substitution can be used in some cases

we will develop other integration techniques

there are functions that don’t have “nice” integrals, no techniques exist to write their antiderivatives easily. For these we require numerical techniques to approximate the integral values.

Questions

Let f represent the general exponential function f(x)=(x^x).

How would it be written base e?

What is its domain? (Answer only after you switch to base e.)

What are its critical points? For each determine if it is a maximum, a minimum, or neither.

(Video)

(* Define the function: *)
f[x_]:=E^(x Log[x])
(* Let's graph it as variable "p1" on a carefully chosen interval, that shows off all the interesting action (properly zoomed, you might say) -- but suppress plotting with a semi-colon (;) at the end of the command: *)
p1=Plot[f[x],{x,0,2},PlotLabel->HoldForm[y=x^x],PlotRange->{0,4}];
(* Let's find (and report) those critical numbers, saving them to variable "soln": *)
soln=Solve[f'[x]==0,x]
(* The following says "grab that (first) value of x from soln" *)
xmin=x/.soln[[1]] 
f''[xmin] (* Positive, so a minimum *)
Show[p1,ListPlot[{{xmin,f[xmin]}}]]

Let g represent the combined function g(x)=(e^(-x))(tan^(-1))(x).

What is its domain?

Describe its end behavior.

What is its linear approximation at x=0?

(Video)

(* Define the function: *)
f[x_]:=E^(-x) ArcTan[x]
(* Let's graph it as variable "p1" on a carefully chosen interval, that shows off all the interesting action (properly zoomed, you might say) -- but suppress plotting with a semi-colon (;) at the end of the command: *)
p1=Plot[f[x],{x,-1,4},PlotLabel->HoldForm[y=E^(-x) ArcTan[x]]];
(* Set the value of x for which we want the tangent line: by using a parameter it's easier to change to another value later on.... *)
x0=0;
(* Now we'll "plot" that tangent line, defined as p1 -- but, again, suppress plotting with a semi-colon (;) at the end: *)
p2=Plot[f'[x0](x-x0)+f[x0],{x,-1,4},PlotStyle->{Black,Thin,Dashed}];
(* Let's find (and report) those critical numbers, saving them to variable "soln": *)
soln=NSolve[f'[x]==0,x,Reals]
(* Just one real critical number; the following says "grab that (first) value of x from soln" *)
xmax=x/.soln[[1]] 
(* Let's check the 2nd derivative: negative, so we've got a maximum at xmin -- hence the name!:) *)
f''[xmax]
(* Now show everything off! Using "Show", of course! *)
Show[p1,p2,ListPlot[{{xmax,f[xmax]}}]]