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memorize.aimemorize.ai (lvl 286)
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38 cards
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Section 1

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Hessian

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1 / 38

CalculusMath
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Last updated

6 years ago

Date created

Mar 1, 2020

Cards (38)

Section 1

(38 cards)

Hessian

Front

a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. https://www.khanacademy.org/math/multivariable-calculus/applications-of-multivariable-derivatives/quadratic-approximations/v/the-hessian-matrix

Back

curl

Front

http://acritch.com/media/math53/Critch_Math53_09Su_-_Nabla_Notation.pdf

Back

point of inflection

Front

the point where the graph changes concavity - find by identifying places where f''(x) = 0 or is undefined

Back

related rates

Front

an equation involving two or more variables that are differentiable functions of time can be used to find an equation that relates the corresponding rates to solve - define the equation in terms of the rates of change - find an equation relating the changing quantities - take the derivative with respect to time to get the derived equation which relates the rates of change - substitute into the derived equation the given values of the quantities & their derivatives - solve for the derivative required Examples - falling ladder https://www.khanacademy.org/math/ap-calculus-ab/ab-diff-contextual-applications-new/ab-4-5/v/falling-ladder-related-rates

Back

nabla notation for the Jacobian

Front

http://acritch.com/media/math53/Critch_Math53_09Su_-_Nabla_Notation.pdf

Back

Even and Odd Functions

Front

A function f is even if the graph of f is symmetric with respect to the y-axis. Algebraically, f is even if and only if f(-x) = f(x) for all x in the domain of f. A function f is odd if the graph of f is symmetric with respect to the origin. Algebraically, f is odd if and only if f(-x) = -f(x) for all x in the domain of f.

Back

Taylor Series expressed as function of delta x

Front

we can re-express the Taylor series from a form that emphasizes building the approximation of a function at a point p to a totally equivalent form that emphasizes using that function to evaluate other points that are a small distance delta x away

Back

Gradient

Front

a rate of inclination; a slope,

Back

empty set/null set

Front

a set with no elements

Back

acceleration due to gravity

Front

9.8 m/s^2 or 32 feet/sec^2 - the second derivative

Back

set notation

Front

A precise way of describing which items belong in a set and which do not.

Back

compound interest

Front

Back

Geometric Series

Front

In mathematics, a geometric series is a series with a constant ratio between successive terms. For example, the series 1/2 + 1/4 + 1/8 + 1/16. is geometric, because each successive term can be obtained by multiplying the previous term by 1/2. https://www.khanacademy.org/math/ap-calculus-bc/bc-series-new/bc-10-14/v/function-as-a-geometric-series

Back

exponential growth

Front

A(t) = P( 1 + r/m)^mt where m = the number of times the growth is compounded, P = initial base r = rate of growth or decay t = time

Back

Affine

Front

allowing for or preserving parallel relationships. An affine function is a function composed of a linear function + a constant and its graph is a straight line. The general equation for an affine function in 1D is: y = Ax + c. An affine function demonstrates an affine transformation which is equivalent to a linear transformation followed by a translation. difference between affine and linear? A linear function fixes the origin, whereas an affine function need not do so. An affine function is the composition of a linear function with a translation, so while the linear part fixes the origin, the translation can map it somewhere else.

Back

differentiable function

Front

a function that is differentiable at every point of its domain - it must be continuous but not all continuous functions are differentiate - for example f(x) = |x| is not differentiable but it is continuous We can test any value "c" by finding if the limit exists: limh→0 (f(c+h) − f(c) )/ h https://www.mathsisfun.com/calculus/differentiable.html

Back

Local Max/Min

Front

where df(x)=0, a critical point that is the highest/lowest over a given range but not for the entire domain (that would be the the absolute)

Back

bridging from chemistry

Front

means of keeping track of conversions - handy with related rates problems

Back

Jacobian

Front

if you have a function of many variables so f(x1, x2, x3, ...) then the Jacobian is simply a vector where each entry is the partial derivative of f with respect to each one of those variables in turn. By convention, we write this as a row vector rather than a column vector,

Back

Nabla

Front

vector of partial derivative operators - pronounced del

Back

absolute maximum

Front

a point that represents the maximum value a function assumes over its domain

Back

solving equations with unknown exponents

Front

2ish methods - #1 - get common basis "Solve 2x = 32" ... convert the 32 to 25, set the exponents equal, and solve for "x = 5" #2 - use logarithms Solve 2x = 30 logb(mn) = n · logb(m) When I take the log of both sides of an equation, I can use any log I like (base-10 log, base-2 log, natural log, etc), but some are sometimes more useful than others. Since the base in the equation "2x = 30" is "2", I might try using a base-2 log: log2(2x) = log2(30) Any log of the log's base returns a value of 1, so log2(2) = 1. Then: x · log2(2) = log2(30) x(1) = log2(30) x = log2(30) #3 - use natural log 2x = 30 ln(2x) = ln(30) x · ln(2) = ln(30) x = ln(30)​/ln(2)

Back

Omega

Front

the universal set in set notation

Back

indefinite integral

Front

An integral expressed without limits, and so containing an arbitrary constant.

Back

complement of a set

Front

an element belongs to the complement of S if x is an element of our universal set, and also x does not belong to S. Notice that if we take the complement of the complement-- we get back to the S set. So what this is saying is that the complement of the complement of a set is the set itself.

Back

Limit Formula

Front

Back

Newton-Raphson Method

Front

is a special method of solving nonlinear equations using an iterative approach based on either a constant or an updated slope in each iteration.

Back

Allocation of Labor

Front

related rates problem - use Cobb-Douglas production formula

Back

Taylor Series

Front

Taylor series is one of a family of approaches used for building approximations to functions - cooking a chicken

Back

Power Series

Front

aka Taylor series - A series which represents a function as a polynomial that goes on forever and has no highest power of x. g(x) = a + bx + cx^2 + dx^3 ..... Starting from just a single term, we call these expressions the zeroth, first, second, and third-order approximations etc. Collectively, these short sections of the series are called Truncated series.

Back

Multivariate Taylor Series - 2nd order approximation

Front

https://mathinsight.org/taylors_theorem_multivariable_introduction

Back

Extrema

Front

refers to all maximum and minimum values

Back

Elascity of demand

Front

E = (- change q/change p) * (p/q) say demand is elastic is E > 1 inelastic E < 1 and has unit elasticity if E=1

Back

objective function

Front

The function being maximized or minimized -

Back

Second Derivative Test

Front

Used to determine on what intervals a function is concave up/concave down and the points of inflections. When a function's slope is zero at x, and the second derivative at x is: less than 0, it is a local maximum greater than 0, it is a local minimum equal to 0, then the test fails

Back

Quadratic Formula

Front

Back

Concavity Test

Front

The graph of a twice-differentiable function y=f(x) is (a) concave up on any interval where y'' > 0 - that is, the slope is increasing (b) concave down on any interval where y'' < 0

Back

inflection point

Front

A point at which a function changes curvature from convex to concave or vice versa https://www.mathsisfun.com/calculus/inflection-points.html

Back