Calculus pt. 1

some random precalc shit, derivatives, limits, MVT, IVT, extremas, concavity

Slope intercept form

y = mx + b

Point slope form

y-y1 = m(x-x1)

Rational Functions

a polynomial function divides another function

\-fraction with polynomials in the numerator and denominator (variables and constants)

Horizontal Asymptotes

These describe a function’s end behavior, what the function looks like / approaches as x infinitely increases or decreases

Functions sometimes can cross these, but usually won’t be able to.

BOBO BOTN EATS DC

the funny acronym:

Bigger on Bottom = 0

Bigger on Top, No Horizontal Asymptote (Oblique Asymptote, synthetic division required)

Exponents same, divide the coefficients

Continuity

A function is (this) over an interval if it is unbroken over that interval

Can I trace the graph without lifting my pencil from the paper?

Average Speed

distance travelled (rise) / elapsed time (run)

Secant line

line that intersects another line at two distinct points. what mathematicians use to find a derivative

instantaneous speed

speed at a particular time t

Sandwich Theorem

if f is between g and h for all vales x ≠ c in some interval about c, and g and h have the same limit as x → c, then f has that limit too.

Interior point continuity

Point within a function.

happens when a function y = f(x) is continuous at one of these (c) if and only if the limit as x → c is equal to f(c)

Endpoint continuity

Point at end of a function.

happens when a function f(x) is continuous at one of these (either at the left or the right) if and only if the limit as x → a+ = f(a)

OR

the limit as x → b- = f(b)

(from the right: limit as x approaches value of right = value of ___)

(from the left: limit as x approaches value of left = value of ___)

Removable Discontinuity

Nullifies continuity

Known as holes; created when the denominator of a rational function can be equal to zero

Jump discontinuity

Nullifies continuity

created through piecewise functions

infinite discontinuity

Nullifies continuity

also known as nonremoveable discontinuity, where a function goes towards infinity at a vertical asymptote.

Oscillating discontinuity

nullifies continuity

starts bouncing up and down at wild rates.

Continuous

Operations with ??? functions are also ??? if f and g are ???. Meaning this is possible:

f + g

f - g

f \* g

f / g

f(g(x))

g(f(x))

Intermediate Value Theorem (IVT)

If f(x) is continuous across \[a,b\], every value from f(a) - f(b) happens at least once.

!needs to pass vertical line test!

Guarantees that a value exists, it just doesn’t tell us where or how many times that value happens.

Limit Definition of a Derivative

lim h→0 f(x+h) - f(x) / h

OR

lim x→a f(x) - f(a) / x - a

Derivative

slope of the tangent line, instantaneous rate of change

Normal line

To a curve at a point (x,y), this line is perpendicular to the tangent line at that point (x,y).

Slope of secant line

average rate of change

slope of tangent line

instantaneous rate of change

Corner

f(x) = |x|

This function has a ___, where the derivative of the function changes abruptly

Not differentiable

Cusp

f(x) = x^(2/3)

This function has a ___, where the slopes of the secant lines approach infinity from one side and negative infinity from another side

These functions aren’t differentiable when the numerator is even and the denominator is odd.

Not differentiable

Vertical Tangent

f(x) = x^(1/3)

This function has a ___, where the slopes of the secant lines approach either infinity or negative infinity from both sides

Not differentiable

Discontinuity

f(x) = { -1, x < 0; 1, x ≥ 0

Functions like these have a ___, when one or both of the one sided derivatives do not exist.

Not differentiable

Polynomial

Rational

Trig

Exponential

Logarithmic

These functions are differentiable:

(x^2 + 1)

(1 / x^2)

(sin(x))

(x^(x^2 + 2))

(lnx)

Locally linear

Differentiable functions are ___, meaning the closer we zoom into a function, it will start to look just like its tangent line.

Polynomial

Linear

Radical

Absolute Value

Exponential

Log

Trig

Rational

These functions are differentiable:

x^2 + 2

x+2

sqrt(x)

|x| + 2

x^(x^2+2)

log base 10 of x

arctan(x)

(x^2 - 1) / (x^3)

Power rule

d/dx x^n = nx^(n-1)

x^5 → 5x^4

Provided n or x is not 0

Constant Multiple Rule

this useless derivative rule basically says you can pull out a constant (NOT FUNCTIONS) in a function and worry about them later

Sum / Difference Rule

this states that we can add / subtract derivatives provided that they are of the same function (u and v are differentiable functions of x)

Constant Rule

deriving a constant goes to 0

Product Rule

d/dx(uv) = u(dv/dx) + v(du/dx)

Quotient Rule

Lo De Hi Hi De Lo Square the bottom and away we go!

Displacement

f(t + ∆t) - f(t) given f(t) is position at time t

This can be positive or negative. Distance from the starting position to the end position.

Velocity

rate of change (derivative) of position.

A vector quantity, it commits crimes on direction and magnitude (can be negative) depending upon which direction the object moves

Speed

Absolute value of velocity, always positive or zero.

Acceleration

rate of change (derivative) of velocity

A vector quantity, it commits crimes on direction and magnitude (can be negative) depending on which direction the object moves

Speed up

Objects __ when velocity and acceleration have the same sign (both positive or negative)

Slow down

objects ___ when velocity and acceleration have the opposite signs (one positive, one negative)

cos(x)

derivative of sin(x)

\-sin(x)

derivative of cos(x)

sec^2(x)

derivative of tan(x)

Pythagorean Identities

sin^2(x) + cos^2(x) = 1

tan^2(x) + 1 = sec^2(x)

cot^2(x) + 1 = csc^2(x)

Quotient Identities

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

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

\-cot(x)csc(x)

derivative of csc(x)

tan(x)sec(x)

derivative of sec(x)

\-csc^2(x)

derivative of cot(x)

Chain Rule

Derive the outer function, then multiply it by the derivative of the inner function, loop until everything has been derived

Implicit Differentiation

allows us to find derivatives that aren’t easily solvable for y.

Will often give us a derivative that is expressed in terms of x AND y

it’s the thing where you have x and y in the equation and you have to find dy/dx

e.g y^3 + y = 3 + 2x

Steps for Implicit differentation

Derive both sides of equation with respect to x (don’t forget chain rule), then y terms (add a dy/dx through chain rule)

Isolate dy/dx to one side of equation

Factor dy/dx

solve

one-to-one function

if a functions’s inverse is also a function, it is a ___.

Horizontal line test

determines if a function is one to one. If it has two points touching the horizontal line, it is in fact not one to one.

(1,2)

If f and g are inverses, and (2,1) lies on g, that means that ___ lies on f

d/dx (1 / g(y))

provided that f and g are inverses, f’(x) is ___

hint: This just means that the derivative of g(x) at Y value is the reciprocol of the derivative of f(x) at X value

1 / (sqrt(1 - u^2)) \* du/dx

derivative of arcsin

\-1 / (sqrt(1 - u^2)) \* du/dx

derivative of arcos

1 / (1 + u^2) du/dx

derivative of arctan

\-1 / (1 + u^2) du/dx

derivative of arccot

1 / |u| *sqrt(u^2 - 1)* du/dx

derivative of arcsec

\-1 / |u| *sqrt(u^2 - 1)* du/dx

derivative of arccsc

e^u du/dx

derivative of e^u

1

lim h → 0 (e^h - 1) / h = ???

a^u \* ln(a)

derivative of a^u

1 / u du/dx

derivative of ln(u)

1 / u \* ln(a) du/dx

derivative of log base a of u

Parametric equations

these are often useful to graph curves that aren’t functions, such as circles of ellipses

define each ordered pair on a graph by relating to x and y

A parametrized curve (x(t) , y(t)) is differentiable at t if x and y are differentiable at t

what does this mean

\
e.g. dy/dx == (dy/dx) // (dx/dt)

if it asks you to derive parametrically just derive the top and the bottom separately

extrema

maximums and minimums, maxima and minima. Can be classified as absolute / global, or local / relative

Absolute minimum

lowest point on the graph

Absolute maximum

highest point of the graph

exists with bounds

Relative minimum

lowest point of a region of the graph

Relative maximum

highest point of a region of the graph

Extreme Value Theorem

If f is continuous on \[a,b\], then f has both a maximum value and a minimum value on that interval

* *if a function is continuous on a closed interval, then there is both a maximum and minimum.*
* these maximas and minimas may occur at an interior point or at the endpoints

Solving for Extrema

Identify critical points (endpoints, any value where the derivative is equal to zero or does not exist)

Plug values of candidates into the **original function,** local / absolute minimums are lower numbers, local / absolute maximums are higher.

do the stupid number line sign thing by inputting numbers before and after the critical point into the derivative to check if there is actually a minimum or maximum

!! not every critical point or endpoint signals presence of extrema.

Critical points

endpoints, any value where the derivative is equal to zero or does not exist

Mean Value Theorem

If f is continuous on \[a,b\] and differentiable on (a,b), then there is at least one value c in (a,b) where *f’(c) == f(b) - f(a) / b - a*

*if f is continuous (closed interval) and differentiable (open interval) then there’s at least one time where the derivative is equal to the average rate of change.*

increasing on \[a,b\]

if f’ > 0 in (a,b), f is ___

decreasing on \[a,b\]

if f’ < 0 in (a,b), f is ___

Concave up

if f” is positive, then f is ___

slopes of the tangent lines are increasing as we move from left to right)

Concave down

if f” is negative, then f is ___

slopes of the tangent lines are decreasing as we move from left to right

Solving for concavity (second derivative test)

provided a function is twice differentiable:

solve for inflection points (any point where concavity changes), or values of x when f” is 0 or DNE

do the stupid number line sign thing where you test the values around the inflection point candidates (plug into f”) to determine concavity

Bounce, Wiggle, Cross

a function will __ if its exponent is even

a function will __ if its exponent is odd

a function will __ if its exponent is one.

Reciprocal Identities

csc(x) = 1 / sin(x)

sec(x) = 1 / cos(x)

cot(x) = 1 / tan(x)