Calculus: Ch 1 & 2

Squeeze theorem

ƒ(x) ≤ g(x) ≤ h(x) for all x and if the limit of ƒ(x) as x→c = L and the limit of h(x) as x→c = L, then the limit of g(x) as x→c = L

What are the three ways to evaluate a limit?

1) Graphically 2) Algebraically 3) Numerically

Tangent line?

As one of a secant line's points is brought infinitesimally close to the other point on the line it is a tangent line. It is approximated to be touching a curve at a single point. An instantaneous value.

when to check domain

Rational Function (equation w/ x in denominator) Square root function Logarithmic function

When writing final answers for derivatives ensure

1) no negative exponents 2) no fractional exponents 3) reduce all fractions

What are three ways that a function can be discontinuous

1. f(x) approaches a different number from the right as it does from the left as x→a 2. f(x) increases or decreases without bound as x→a 3. f(x) oscillates between two fixed values as x→a

What are the three rules for a limit to be continuous?

1. lim x→a f(x) exists. 2. f(a) exists. 3. lim x→a f(x) = f(a)

Define continuity

A function f is cts at the value of a iff lim x→a [f(x)] = f(a)

What is a infinite limit?

A limit that increases or decreases without bound as the limit gets closer to the x-value. Basically, the y-values get really big in either positive/negative directions as you get close. The limit is considered to not exist as infinite is not a number.

Secant line?

A line joining two points on a curve. It is therefore is a line that touches a curve at two points. An average value.

Factoring 2 Terms

A^2 - B^2 = (A+B)(A-B)

Continuous function?

Basically if there is no discontinuity along an interval. That is, the graph can be drawn in 1 continuous motion without lifting your pencil.

What are the two main things calculus aims to answer via limits? What is their relationship?

Derivatives and integrals are inverses of each other. Derivative of an area gives a function. The integral of that function gives the area.

Factoring 3 terms

FOIL Backwards ax^2 + bx + c = 0 Add factors of c to get bx. ax^2 - bx + c = 0 → (-) (-) ax^2 + bx - c = 0 → (+) (-)

One-sided limits

For values evaluated from the left (negative superscript) and right (positive superscript) side respectively

direct Substitution property

If f is a polynomial or a rational function and a is in the domain of f , then lim x->a f(x) = f(a)

Intermediate Value Theorem

If f is continuous on [a,b] and k is a number between f(a) and f(b), then there exists at least one number c such that f(c)=k

What happens if the one-sided portions of a limit do not match? That is the curve approaches different values depending on direction.

It does not exist as it causes a discontinuity.

Infinite(asymptotic) Discontinuity?

Means the limit approaches infinite at some point. The curve has asymptotes.

How is a limit evaluated graphically?

Observe the value ON THE CURVE (ignore the point) simultaneously approached as you approach a on the x-axis from the left and right.

Jump Discontinuity?

Occurs when the left and right one-sided limits are not equal.

Removable discontinuity?

Occurs when the limit exists, but it is not equal to f(a). This results in a hole on the curve.

What is a limit EVALUATED at infinite?

Occurs when the x-value increases/decreases without bound. The y-value is ± ∞ or a finite number. It determines the END BEHAVIOR of a function.

where should you look for points of discontinuity

Piecewise: break points of piecewise https://www.kristakingmath.com/blog/how-do-you-find-discontinuities

How is a limit evaluated NUMERICALLY?

Plug into values close to a from the right and left. Observe the trend. Usually the last resort of the three ways to evaluate a limit.

When evaluating a limit analytically what should you always try first?

Simply try to solve by plugging a (what x approaches) into the function.

What is a limit? How is it denoted

Think of it as a slider along the x-axis and the corresponding curve of f(x) simultaneously.

For a rational limit EVALUATED AT infinite: what is true if the degree of numerator < degree of denominator?

The limit is equal to 0 since infinite is in the denominator.

For rational limits: when a big number such as ±∞ is approached in the denominator ______

The limit is equal to 0. Dividing by a big number produces a small number so it makes sense that it approaches 0.

For a rational limit EVALUATED at infinite: what is true if the degree of numerator = degree of denominator?

The limit is equal to the ratio of their leading coefficients.

For a rational limit EVALUATED at infinite: what is true if the degree of numerator > degree of denominator

The limit is ∞ or -∞

Definition of vertical Asymptote

The line x=a is a V.A. of f if lim(x→a⁻) f(x) = ∞ or -∞ and/or lim(x→a⁺) f(x) = ∞ or -∞

definition of horizontal Asymptote

The line y = a is called a horizontal asymptote iff lim (x→-∞) = a or lim (x→∞) = a

unit circle

a circle with a radius of 1, centered at the origin

difference of squares

a^2 -b^2 = (a+b)(a-b)

sum of cubes

a^3+b^3=(a+b)(a^2-ab+b^2)

difference of cubes

a^3-b^3=(a-b)(a^2+ab+b^2)

add/subtract rational expressions

create common denominator by finding LCD, then simplify.

avg velocity formula

displacement/time change in position/time

Find limits at infinity

divide numerator and denominator by highest power of x in the denominator note: 1/x = √1/x^2

composition of functions

f ( g(x)) means that function g is the input to function f (f⁰g)(x) = f(g(x)). plug x into g(x) then simplify, then plug g(x) in f, wherever it asks for x

differentiate f(x) = 1/x g(x) = √x h(x) = e^x y = b^x

f'(x) =-1/x^2 g'(x) = 1/2√x h'(x) = e^x y' = ln(b) * b^x

How to find tangent line of f at an x value

find f'(x) using limit definition of the derivative find f(x) at given value and f'(x) at given value write out in point slope form y-y1 = m(x-x1)

when should you look for vertical asymptotes

may occur at 1) any x value that makes the denominator = 0 2) logarithmic function

Rules of Logarithms

log a(1) = 0 log a(A) = 1 log (A X B) = log A + Log B log (A / B) = log A - Log B log (A^B) = B *log A log (1/A) = - log A ln(e) = 1

steps to find discontinuities

look at 3 requirements for continuity 1. find if f(a) exists 2. find left and right hand limits as x approaches a 3. check if lim f(x) x approaches a = f(a)

How to find domain of Polynomial function function with variable in denominator root function ln function a graph a relation

polynomial: no radicals or variables in the denominator; the domain is all real numbers. fraction with a variable in the denominator: Set the bottom equal to zero and exclude the x value you find when you solve the equation. Root function: Set the terms inside the radical sign to ≥ 0 and solve to find the values that would work for x. ln function: set the terms in the parentheses (the argument) to >0 and solve. A graph: Read graph, check where function is defined (exists). A relation: make sure no x value has two different y values

For rational limits: when a small number is being approached in the denominator______

the limit is equal to infinite. When a number is divided by a small number it produces a big number. So it makes sense that the limit approaches ±∞.

what does it mean if the lim as x approaches infinity = a constant

then there is a HA at that constant value since as x approaches infinity y is a constant

Rules of Exponents

x^A + x^B = x^(A+B) x^A * y^A = (xy)^A (x^A)^B = x^AB x^0 = 1 x^-A = 1/x^A (x^A)/(x^B) = x^(A-B) (x^A)/(y^A) = (x/y)^A