BasCal Midterms

Continuity

property of functs that showcases how smoothly these functs transition between their pts w/o any abrupt changes or breaks

the basic intuition behind continuity is that if you draw the funct w/o lifting your pen off the paper, the funct is continuous

2 qualities of a continuous function

1. No breaks in the graph - a limit must exist at every x-value or the graph’ll break

2. No holes or jumps - the funct can’t have undefined pts or vertical asymptotes

polynomial function

continuous at every pt (negative infinity to positive infinity)

rational function

continuous at every real # in its domain (non-zero denominator)

continuous if there is no variable in the denominator

3 criteria of a continuous function

1. f(c) is defined

2. lim f(x) as x→c exists (both sides =)

3. f(c) = lim f(x) as x→c

2 types of discontinuity

removable

non-removable

removable

hole in the graph

point discontinuity

a discontinuity is this if a funct can be made continuous by defining (or redefining) a pt

removed by cancellation

non-removable / essential

break or vertical asymptote

infinite discontinuity and jump discontinuity

NR discontinuities are the only values of x not in the domain of the function

slope

denotes change

rise/run → m = (y₂-y₁)/(x₂-x₁)

slope of a line

always constant at any given 2 pts

slope of a curve

not constant and changes depending on the 2 pts given

tangent line

line that touches at only 1 pt that most closely approaches the curve of the given funct

since there’s only 1 pt given, it isn’t possible to find the slope of this

secant line

line that intersects the curve at 2 pts

tends to have the same orientation with the tangent line as 𝒙_𝟏 approach 𝒙_𝟎 → slope of secant line eventually = slope of tangent line

slope: m = Δy/Δx

slope of the tangent line to f at (x₀, y₀)

m_T = lim_x→x0 f(x)-f(x₀) / x-x₀

where f(x₀) = y₀

slope of the tangent line at this pt is the limit of the slope of the secant line that passes through the points (𝑥₀, 𝑦₀) and (𝑥₁, 𝑦₁) provided that the 𝒙_𝟏 approaches 𝒙_𝟎.

equation of a tangent line

y-y₀ = m(x-x₀)

make sure that x is always positive in the final answer

Limit Definition of the Derivative of a Function

uses Δx → 0

expresses the instantaneous rate of change of 1 variable w/ respect to another

the slope of a tangent line to a curve at any given point

derivative of 𝒇 at 𝒙_𝟎 is the slope of the tangent line at 𝒙_𝟎, 𝒚_𝟎 , if it exists

derivative 𝒇′ 𝒙 serves as the slope indicator of tangent lines to the function 𝒇 (𝒙)

derivative notations

the f'(x) is also called the derivative of a function w/ respect to the variable x

other notations for y=f(x): Lagrange’s Notation, Leibniz’s Notation, Newton’s Notation

Lagrange’s Notation

y', x', f'(x)

Leibniz’s Notation

dy/dx (derivative of y w/ respect to x)

d/dx (f)

Newton’s Notation

.

f

.

y

the derivative as the slope of the tangent line

f¹(x) serves as the indicator of tangent lines to f(x)

constant rule of differentiation

derivative of a constant funct is ALWAYS 0 (if a is a real no.)

power rule of differentiation

if n is rational

derivative xⁿ = nx^n-1

constant multiple rule of differentiation

derivative a*f(x) = a*f’(x)

sum and difference rule of differentiation

the sum and differences of differentiable functs must also be differentiable