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UC - MAT 1045 - Study Guide

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Applied CALCUIUS 2

test 1 Reviews Derivative written as f'(x) or dx

derivative is the slope of f(x)

• also called instantaneous rate of Change

• Rules for Derivative graphs . Direction of onging function (increasing decreasing)

tells you the sigh (position of derivative graph positive abore axis, or negative-beicw axis)

- if f(x) is decreasing, f(x) is below x axis

· concavity Of on anal funcioncconcave up/down) tells us the sign position of the second derivative

Don't forget about the age old question of How do you find the z score with the mean and standard deviation?

graph (Postir - abore x-axis or negatre-below x-axis) - it fon is concarcup, the second derivativ Don't forget about the age old question of What is an example of constant rate of change?

will be part the Cabove X X ) concavity of original functronceoncare up down) Hells us the direction cacicoring or decreasing) of the denrative graph - If f(x) is concare up the derivative We also discuss several other topics like What are the common bases?

graph will be increasing going up) If you want to learn more check out What are the types of statements and how they differ?

If you want to learn more check out What is the document where you write your code that can be saved?

• Where f(x) has a maximum or minimum

- f (x is zero a - f (x) is an inflection point where f(x) has an inflection point

-FIX has a relative maximum minimum - fll( is zero.

Study SOUTO If you want to learn more check out Why is the unemployment rate not accurate?

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Rules for Finding Derivatives: (1) Mutiply the coen crent by the power,

then lower the power

& ☆ estes 2) A constant becomes zero a

- 3x2 + 7 ex to 3) ex remains ex (If there is a coefficient

in front of it, it remains

- ex ex sex - 5ek ) in (X) becomes constans stay also)

- wine 5 by a coefficient to the X power slo efficient

comes back & is multiplied by incbare)

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) Con becomes -sin) 7) rinco becomes cosex positive chain coll! Dennotive of outside. Derivative

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of inside

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Product Rute : (17 x pervative of second

(2nd X Derivative of first) - 2 x 2 + D2 (4x3+7+) a denivation,

> (36X2+D? 18x2+1] + [6sx3+7x][60X2+1): 2x]

denvative

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Results of change

& signed area = simply take areas of parts of the graph (between the

graph a the X axis & give them a Bolitive value if they are above the x-axis and give them a negative value if they are under the x axis. Then add them Upuring those signs!!!

To find Accumulated Change : Area of A+ Area of B -

& Arca ofc t area of D Low X axis a negative

YUXUDH ☆

- Det vecane what does the area under the graph X between the axu mean?

- & accumulared after x time

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Ma pproximating Accumulated change

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using left-rectangle using right rectangle midpaint rectangle appréximation apprezinamica approximatOK *add up areas of each rectangle A if they dont girt specific height, plug

in the value for the rectangle o into the equation given to get height ☆

AX FOD AXIBOX (3)

- width of rectangler heightofthangle Sioma Noration: Riemann sum*

bare of all & f(x.). Ax rectangler

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height of different rectangle y value goes by changing x valve

from 1 to

y ccould be any # written wr ar: f(x.JAX + f(X2).DXF F(X) DX

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in Haron function

S integral sign

a,b endpoints of integral

• F(X) - integram function o dx means derivative

so f(x) dx definite integral of a function

Garea under graph

> Naans to Area Of Ror? (x2+Dax And signed area

meningited area

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xvave

from o to 2

function you will uq X valve in to

find heights

or the NO Cab

in equations

-type range, cauation, & olx

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Accumulation Functions IF(X) = distance function

f'(X)* Velocity function DCX) St VCt) dto

distance trarred during first x hours A (X) = så f(t) d +

gives the accumulation of signed area Studys under the graph

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A(X) = sa f(t) dt .: In 11 al input value

A =0 Alaf

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TA FCO, 7 A

• AN-FI

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• All=FI

If 70

A Concave up A cicavo down

f food area giren

- Xaxis

dysc

Zo

108

graph

going downo!

sketch

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☆ A alwaw starts at Zero

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Fundamental Theorem of calculus

disftcdt)f(x)

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derivative of onde = f(x) (

A f) (accumstanen function) fxQome PGT COM(3+)

d o - COS (3X)

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Annd ervative

fifunction of Fantaenvative at

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Raise the power of the function

and then divadle the coeficient by the new power #antridenvatiro may be an anracovative

to more than 1 function A SE(X) DX = F(XOFC nunt called the antiaerivative

indefinite integral

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general derivate uresc unen constant is unknown

• Specific derivative know the constant

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antidervative Of 2x is x2

un craise power by 7,

a then divide by neu power 5 2x dx = X2 +C

means general derivative SE (x dx = family of antiderivative functions -> 70 faidx= signed area, a number

1) Sf (t) at = function

2) Fdx = number ☆3) SACOH = accumulation Funchat

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