calc 1 unit 2 test

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m^3 − n^3 =

(m − n)(m2 + mn + n 2 )

arbitrary integer

+period and n, used when no interval for trig functions

critical numbers =0,18/11, and 3. the second derative is 30x^5(x-3)

- can only use the second deterative test for 18/11 to see if its a max or min - will have to use 1st derative test to find all max and min values

Find a cubic function f(x) = ax3 + bx2 + cx + d that has a local maximum value of 4 at x = −2 and a local minimum value of 0 at x = 1.

- can plug in x values - can take f'(x) at x values, euqla it to zero -system of equations

Find the points where the curve y=∣x+ 2)/(x^2+ 5)∣ has horizontal tangents.

set function to different ones, solve for each differvaite then set each one to 0 answer: (1,1/2) (-5,1/10)

The quantity (in pounds) of a gourmet ground coffee that is sold by a coffee company at a price of p dollars per pound is Q = f(p).

The rate of change of the quantity of coffee sold with respect to the price per pound when the price is $8 per pound. units: pounds/(dollar/lb)). f ′ (8) is negative since the quantity of coffee sold will decrease as the price charged for it increases. People are generally less willing to buy a product when its price increases.

where are there points of tangency to y=(1-2x)^2 that pass through (1,-3)

x=a, y=(1-2a)^2 y'(a), x=1, x=-3, plug into point slope and solve for a a=0,2 answer: (0,1) and (2,9)

Mean Value Theorem

if f(x) is continuous and differentiable, slope of tangent line equals slope of secant line at least once in the interval (a, b) f '(c) = [f(b) - f(a)]/(b - a)

orthogonal

if their tangent lines are perpendicular at each point of intersection. solve to see if their derivatives are perpendicular

when faced with a tricky product rule

remeber u can factor out terms to solve for zero

tanx

sec^2x

on mean value theorm problems

secent line and tangent line are parallel because the slope of the secent line is point slope and so it tangent

secx

secxtanx

Use L(x) to approximate the numbers 0.9^1/2 and 0.99^1/2 f(x)=(1-x)^1/2 l(x)=1-x/2

set .9^1/2 equal to f(x) to find x and then plug x into l(x). .9^1/2=.9500 .99^1/2=.9950

limh->pi/3 cos(x)-1/2/x-pi/3

f(x)=cos(x) f'(pi/3)=-3^1/2/2

how is differential same thing as linarization

f(x)=l(x) f(x)=f(a) +f'(a)(x-a) delta f=f'(a)(x-a) this is because dy=f'(a)(x-a) do dy=f'(x)dx

when it says find the local maxium and min values

find it using the 1st and 2nd derative test. make sure to label the tests and explain what ur doing and why

increasing and decreasing test min and max wise

if f' changes from + to - then there is a local max - to + then there is local min at c if it doesn't change then there is no max or min

when given a constant

plug into equation and will be treated a cinstant, either be 0 if added or constant if added

the average rate of change ofC

point slope of interval, can't check with derative

s(t) =

position (net displacement)

vertical tangents

possible when y' is undefined

increasing and decreasing test

(a) If f'(x) > 0 on an interval, then f is increasing on that interval. (b) If f'(x) < 0 on an interval, then f is decreasing on that interval.

cotx

-csc^2x

cscx

-cscxcotx

dy/dx

-inline with chain rule -don't know how to take y's derivative directly

d/dx cosx

-sinx

in order to get credit

-when taking derative, make sure you do d/dx -write out eveyrthing, espeically in related rates -

related rates

1. Draw a diagram. 2. Assign variables to all quantities that are functions of time. 3. Express the given information in terms of derivatives. 4. Write an equation that relates the various quantities. 5. Differentiate the equation with respect to time using the chain rule. 6. Substitute the given values and solve for the unknown rate.

conditions for differentiability

1. continuous 2. kinks 3. vertical tangent line

remember with related rates

1. reduce all variables if possible, u cant have two at once so you may have to sub one in for another. remember this !!! 2.

prove that x^3/3 - cos(x) +2x has exactly one root

1. use IVT to show theres atleast one root 2. cont? yes f(0) = -1 f(2) = >0 proven there is atleast one 2. show there can't be a second one 2 roots , f(a)=0, f(b)=0 so f(a)=f(b) 3. prove that f'a cant be equal to zero

Volume of a cone

1/3πr²h

Find an equation of the normal line to the parabola y = x2 − 9x + 7 that is parallel to the line x − 5y = 5.

1/5x-27/5=y -1/m = the deterative of the parabola

At noon, ship A is 90 km west of ship B. Ship A is sailing south at 25 km/h and ship B is sailing north at 5 km/h. How fast is the distance between the ships changing at 4:00 PM? (Round your answer to one decimal place.)

24 km/h

Circumference of a circle

2πr

Surface Area of a Cylinder

2πrh+2πr²

Volume of a sphere

4/3πr³

Surface area of a sphere

4πr²

as a particle moves along curve y=x^2 as the particle moves through (1,1) the x coordinate increases by 2 cm/s. how fast is the distance to the origin changing?

6 cm/ sqaure root of 2 cm/s

v(t)

= velocity (direction/sign and mag)

"Veritcal Asymptote"

A line x = a is a vertical asymptoteif lim x→a− f(x) = ±∞ or lim x→a+ f(x) = ±∞

horizontal asymptotes

A line y = L is a horizontal asymptote if limx→∞ f(x) = L or lim x→−∞ f(x) = L.

Tangent to a curve

a straight line that touches but does not cut the curve at a particular point

Area of a circle

A=πr²

If f '(3) = 0 and f ''(3) = 5, what can you say aboutf?

At x = 3, f has a local minimum.

If f '(5) = 0 and f ''(5) = 0, what can you say aboutf?

At x = 5, f has neither a maximum nor a minimum.

Fermat's Theorem

If f has a local maximum or minimum value at c, and if f'(c) exists, then f' (c) = 0.

Extreme Value Theorem

If f is continuous on [a,b] then f has an absolute maximum and an absolute minimum on [a,b]. The global extrema occur at critical points in the interval or at endpoints of the interval.

Intermediate Value Theorem (IVT)

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

Concavity Test

If f''(x) > 0 for all x in I, then the graph of f is concave upward on I. If f''(x) < 0 for all x in I, then the graph of f is concave downward on I.

2nd derivative test for local extrema

If f'(c) = 0 and f"(c)<0, there is a local max on f at x=c. If f'(c) = 0 and f"(c)>0, there is a local min on f at x=c.

where l(x) comes from

L(x)-f(a)= f'(a) (x-a) y-y1=m(x-x1) a and f(a) is tangent line

rolles theorem

Let f be continuous on [a,b] and differentiable on (a,b) and if f(a)=f(b) then there is at least one number c on (a,b) such that f'(c)=0 (If the slope of the secant is 0, the derivative must = 0 somewhere in the interval).

isosceles triangles

Most popular is the 90 degree: 45-45-90

Show that y = x 1/3 has a vertical tangent line at (0, 0).

Note that f(x) is continuous at x= 0 (since it is a root function) and from part (a) we see that limx→0 |f′(x)|= limx→0 (1/3x^2/3=∞ and so there is a vertical tangent at x= 0.

relative error and percent error

Relative: dv/v percent: dv/v times 100 write this work out, not in numbers, chekc ur answer with the numbers tho

horizontal tangents

Set numerator of derivative equal to 0. can also be solved for with implicit differentiation

Find all values c in (−27, 27) such that f '(c) = 0. (Enter your answers as a comma-separated list. c=DNE why doesnt this contradict rolles theorm

This does not contradict Rolle's Theorem, since f '(0) does not exist, and so f is not differentiable on (−27, 27).

first derivative test for local extrema

Used to determine where a function's graph has a min/max and is increasing or decreasing.

Volume of a cylinder

V=πr²h

For what values of a and b is the line 2x + y = b tangent to the parabola y = ax^2 when x = 2?

a = -1/2, b = 2

never differentiate

a constant

A baseball diamond is a square with side 90 ft. A batter hits the ball and runs toward first base with a speed of 26 ft/s. (a) At what rate is his distance from second base decreasing when he is halfway to first base? (Round your answer to one decimal place.) (b) At what rate is his distance from third base increasing at the same moment? (Round your answer to one decimal place.)

a. 11.6 b. 11.6

The circumference of a sphere was measured to be 80 cm with a possible error of 0.5 cm. (a) Use differentials to estimate the maximum error in the calculated surface area. (Round your answer to the nearest integer.) What is the relative error? (Round your answer to three decimal places.)

a. 25 b. 0.012

y=x/a(bx^2+6) a. the curve has the tangent line 2y=21+19(x-3) at x=3. find A and B b. find any HA and VA

a=12, b=4 no VA or HA

abs vs local min and max

abs: can use end points these can be more than one value if they are equal a point can be abs and local

relative change

absolute change/reference value. dA/A

a(t)

acceleration. v'(t) or s''(t). =0 means velocity is constant.

Find the linearizaton L(x) off (x) =3√1 +x at x= 0 and use it to approximate 3√0.95

answer should be written as f(.05)~~ l(.05)=59/60

differential word problems

are not taken with respect to time. differentiate everything !!!

cross section of a cylinder=

area of a circle

abs. extrema exist

at critical numbers or endpoints. make sure to check critical points are in the interval

f'(c)=

avg velcoity, mean value theorm

volume is usually equal to

base area times h, trough would be w*h*1/2*L

y=(x/x-5)^1/2, find dy/dx

cant square y, only terms of x

remember

chain rule!!!! g dawg taught u this bro

d/dx sinx

cosx

other form of derative

f(x)-f(a)/x-a

f'(x)

df/dx

when asking if an absolute value is differntiable, ex: is f(x) =x^2 |x|differentiable at x= 0?

do one sided limit notations and prove that the deratives are the same at 0, don't do a separate function and solve for the derative

the radius of a sphere was measured and found to be 21 cm with a possible error in measurement of at most 0.02 cm.

dr=.02 r=21

A spotlight on the ground shines on a wall 12 m away. If a man 2 m tall walks from the spotlight toward the building at a speed of 1.7 m/s, how fast is the length of his shadow on the building decreasing when he is 4 m from the building? (Round your answer to one decimal place.).6 m/s

draw picture. remeber to make -1.7

v(t)=

ds/dt

v(t)=

ds/dt = independent/dependent

a(t)

dv/dt and d^2s/dt^2

differential

dy/dx seperate the two. dy=f'(x)dx same thing as linarization

Find the differential dy. y = tan x

dy=sec^2(x)dx dy = y and x = dx

cos(x) even or odd?

even

inflection point

f''=0 or dne, where concavity change. is a point where f is continuous and the concavity of f changes from upward to downward or downward to upward.

(f ◦ g)'

f'(g(x)) · g'(x).

evaluate limh->0 (9+h)^1/2-3/h using a derivative

f'(x)=1/2(x)^1/2 find, f('9)=1/6

If a ball is given a push so that it has an initial velocity of 3 m/s down a certain inclined plane, then the distance it has rolled after t seconds is s = 3t + 2t2.

ignore the 3. read carefully and realize the function ignores it as well

lineratization

it uses tangent line at (a, f(a)) to estimate. l(x)= f(a) +f'(a)(x-a)

"why doesnt this contradict rolles"

just say undefined, no way of knowing what

Definition of Derivative

lim h->0 f(x+h)-f(x)/h

abs. extrema of f(x)= |2-3x| -1 on (0,1) ( meant to be brackets)

max = f(0)=1 min=f(2/3)=-1

sin (x) even or odd

odd

when making function neg for abs value function

only make top or bottom negative NOT BOTH

tan period

pi

2sec^2(x)(1+tanx)=f' critical numbers

pi/2 and 3pi/2, is undefined at pi/2 also cant apply rolles of f is undefied at pi/2

when does cos(x) =sin(x)

pi/4

before taking the derative

simplify first, then take it

abs. v(t)

speed (has no direction)

g(t) = |3t − 5| critical number

t=5/3

when see absolute value

take derative from both sides to see if it is differitable. ex: sin(1/2|x|) set up in limit defintion then do the piecewise then do two different sides of limit with two different functions see if h-> exisits, if not it is DNE, if so, then it exits

d/dx

take derivative of something with respect to x. function replaces y in dy/dx. ex: y=x^2+x, so dy/dx=2x+1 is the same thing as d/dx(x^2+x)=2x+1

How many tangent lines to the curve y = x/(x + 1) pass through the point (1, 2)?

tangent line= y=m(x-1)+2 can plug in in y' for m plug in the function for y then solve for x to find number of solutions then pthing y and x coordinates

instaneous rate of change. find the instantaneous rate of change of C with respect to x when x= 100.

the deterative, when it says " respect " u can take the derative and not do limit notation

normal line

the line that is perpendicular to the tangent one

At 2:00 p.m., a car's speedometer reads 30 mi/h. At 2:10 p.m., it reads 50 mi/h. Assuming the car has beendriven the whole time, show that at some time between 2:00 and 2:10, the acceleration is 120 mi/h2.

this is the velocity function, apply mean value theorm. time shoild be put as f(0) and f(1/6)

when doing limit defintion of derivative remember

to be careful with negatives and distribute !!

derivative

top is indepenent, bottom is depenedent top is function of the bottom. takinf derative of the top with respect to the bottom

abs. s(t)

total distance travelled, u have to find critical points and plug them in as intervals. ex: Total Distance =|s(8)−s(5)|+|s(5)−s(1)|+|s(1)−s(0)|= 120 ft

y = a^6 + cos6 x

treat a as constant when finding y'

d/dx ((x+x^1/2-1)/x)

u can plug out the x to do each one separate and not use quotient rule, usually when there is one term on the bottom. bottom can also be made neg. for product. sometimes easier. remember that

when doing differentials

u should dy= (stuff) dx

remember in linarization

use equal sign thing!!

related rates constant

usually is zero bc derative is 0

problems with implicit functions is

usually messy till end

Critical numbers are

when derative is 0 or undefinded. is a value c in the domain of f such that f'(c) = 0 or f'(c) does not exist.

which the tangent line is horizontal

when f'=0


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